Post on 17-Feb-2018
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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:
20 Variables (10 metric; 10 matter)
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
8 Variables (4 metric; 4 matter)
4 Einstein equations2 Conservation equations
+2 Bianchi identities2 Gauge (coordinate choice 1 time, 1 space)
2 Degrees of freedom
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
8 Variables (4 metric; 4 matter)
4 Einstein equations2 Conservation equations
+2 Bianchi identities2 Gauge (coordinate choice 1 time, 1 space)
2 Degrees of freedom
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
4 Variables (2 metric; 2 matter)
2 Einstein equations0 Conservation equations+0 Bianchi identities
0 Gauge (coordinate choice 1 time, 1 space)
2 Degrees of freedom
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
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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) ,
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COBE Normalization cont.
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)
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COBE Normalization cont.
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)
=
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COBE Normalization cont.
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
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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
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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
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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
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Combining Features in LSS + CMB
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
Combining Features in LSS + CMB
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Combining Features in LSS + CMB
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
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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
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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.