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Structure Formation in the Universe
Matthias Bartelmann
Universitat Heidelberg, Zentrum fur AstronomieInstitut fur Theoretische Astrophysik
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Outline
1 Introduction
2 Evolution of Density Perturbations (1)
3 Evolution of Density Perturbations (2)
4 Evolution of Velocity Perturbations
5 Statistics
6 Large-Scale Structure
7 Nonlinear Structure Formation
8 Galaxy Distribution
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Introduction
1 IntroductionCosmic StructuresDark MatterCosmological AssumptionsExpansion, ParametersHorizon
2 Evolution of Density Perturbations (1)
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Cosmic Structures
How to get from here. . .
Planck 2015
. . . to there?
2-MASS
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Cosmic Structures
How to get from here. . .
Planck 2015
. . . to there?
VIPERS
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Dark Matter
NGC 3198 Abell 383
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Cosmological Assumptions
• general relativity plus twoassumptions:• universe is isotropic around us• our position is not preferred
• then, universe is isotropicaround any observer, and thushomogeneous
• scale factor a(t) is only degree offreedom
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Expansion, Parameters
• a(t) is described by Friedmann’s equation( aa
)2= H2
0
[Ωr0a−4 + Ωm0a−3 + ΩΛ0 + ΩKa−2
]• the Hubble constant is
H0 = 100 hkm
s Mpc= 3.2 · 10−18 h s−1
• Ωr0, Ωm0, and ΩΛ0 are density parameters
• ΩK := 1 −Ωr0 −Ωm0 −ΩΛ0 is spatial curvature parameter
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Horizon
• matter- and radiation densitiesscale differently with a; today,Ωr0 Ωm0; Ωr dominated before
aeq =Ωr0
Ωm0
= (8.3 ± 1.1) × 10−5
• the time-dependent Hubbleradius defines the horizon
rH(t) =c
H(t)
• important for structure formationis the Hubble radius at a = aeq,
rH,eq =c
H0
a3/2eq
√2Ωm0
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Evolution of Density Perturbations (1)
1 Introduction
2 Evolution of Density Perturbations (1)Basic EquationsPerturbations, CoordinatesLinearized EquationsPerturbation Equations
3 Evolution of Density Perturbations (2)
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Basic Equations
• cosmic structures (. 50 h−1Mpc) are small compared to thecurvature scale of the Universe (≈ 3000 h−1Mpc): Newtonianapproach justified
• continuity:∂ρ
∂t+ ~∇ ·
(ρ~v
)= 0
ρ,~v: density and velocity fields
• Euler:∂~v∂t
+(~v · ~∇
)~v = −
~∇pρ
+ ~∇φ
• Poisson:∇2φ = 4πGρ
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Perturbations, Coordinates
• decompose ρ,~v into background and perturbation,
ρ = ρ0 + δρ , ~v = ~v0 + δ~v
• convert to comoving coordinates: ~x = ~r/a, ~u := δ~v/a• replace gradient and time derivative:
~∇x =1a~∇r ,
∂
∂t+ H~x · ~∇x →
∂
∂t
• velocity:~v = ~r = a~x + a~x = H~r + a~x = ~v0 + δ~v
~v0 = H~r: Hubble velocity; δ~v = a~x: peculiar velocity
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Linearized Equations
• insert perturbation ansatz into continuity equation, linearize,introduce density contrast δ := δρ/ρ0
• perturbed continuity equation reads:
δ + ~v0 · ~∇δ + ~∇ · δ~v = 0
• same with Euler:
∂δ~v∂t
+ Hδ~v + (~v0 · ~∇)δ~v = −~∇δpρ0
+ ~∇δφ
• and Poisson:∇2δφ = 4πGρ0δ
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Perturbation Equations
• three perturbation equations
δ + ~∇ · ~u = 0
~u + H~u = −~∇δpa2ρ0
+~∇δφ
a2
∇2δφ = 4πGρ0a2δ
for δ, ~u and δφ
• need equation of state to linking pressure to densityfluctuations (cs: sound speed)
δp = δp(δ) = c2sδρ = c2
sρ0δ
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Evolution of Density Perturbations (2)
2 Evolution of Density Perturbations (1)
3 Evolution of Density Perturbations (2)Density PerturbationsGrowing and Decaying ModesNecessity of Dark MatterCold and Hot Dark Matter
4 Evolution of Velocity Perturbations
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Density Perturbations
• combine Euler and continuity, decompose δ into plane waves:
δ + 2Hδ = δ
(4πGρ0 −
c2s k2
a2
)• ignoring H, oscillations with frequency
ω0 :=
√c2
s k2
a2 − 4πGρ0 , kJ :=
√4πGρ0
cs
• ω0 ∈ R for k ≥ kJ: Jeans scale; H causes expansion drag
• for k kJ, for Ωm0 = 1:
δ + 2Hδ = H2δ ·
4 radiation era
3/2 matter era
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Growing and Decaying Modes
• ansatz δ(t) ∝ tn yields
n2 +n3−
23
= 0 , n2 − 1 = 0
• this translates to
δ+ =
a2 (a < aeq)a (a > aeq)
, δ− =
a−2 (a < aeq)a−3/2 (a > aeq)
• generally: linear growth factor D+(a) defined by:
δ(a) = δ0D+(a)
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Growing and Decaying Modes
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Necessity of Dark Matter
• fluctuations in radiation density scale like
δρr
ρr= 4
δTT
:
adiabatic density fluctuations δ cause temperature fluctuationsof equal order
• CMB was released at aCMB ∼ 10−3; present density fluctuationswith δ0 & 1 had δ ∼ 10−3 at aCMB: CMB temperaturefluctuations should be of order mK, not µK
• CMB fluctuations and present cosmic structures require thedominant form of matter to not interact electromagnetically!
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Cold and Hot Dark Matter
• for dissipation-free dark matter, Jeans scale is replaced by
kJ =⟨v−2
⟩1/2 √4πGρ0
• small perturbations with k > kJ cannot grow due to freestreaming
• dark matter with v→ 0, kJ → ∞ is called “cold” (CDM);
• if v is finite, e.g. for neutrinos, the matter is called “warm”(WDM) or “hot” (HDM);
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Evolution of Velocity Perturbations
3 Evolution of Density Perturbations (2)
4 Evolution of Velocity PerturbationsVelocity PerturbationsPeculiar Velocity
5 Statistics
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Velocity Perturbations
• ignoring pressure gradients:
~u + H~u =~∇δφ
a2
suggests ansatz ~u = u(t)~∇δφ• from continuity:
~∇ · ~u = −δ = −adδda
• for linear growth:
~∇ · ~u = −Hδd ln D+(a)
d ln a=: −Hδ f (Ωm) ≈ −HδΩ0.6
m
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Peculiar Velocity
• using Poisson:
u(t) =2f (Ωm)
3a2HΩm
• thus δ~v satisfying continuity must be
δ~v = a~u =2f (Ωm)3aHΩm
~∇δφ
• additional solutions possible with ~∇ · ~u = 0• since δ , 0, ~∇ · ~u = 0 only where δ = 0
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Statistics
4 Evolution of Velocity Perturbations
5 StatisticsPower Spectrum and Correlation FunctionFiltered Density ContrastGrowth SuppressionCDM Power Spectrum
6 Large-Scale Structure
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Power Spectrum and Correlation Function
• the variance of δ in Fourier space defines the power spectrumPδ(k):
〈δ(~k)δ∗(~k′)〉 =: (2π)3P(k)δD(~k − ~k′)
• correlation function of δ in configuration space is:
ξ(y) := 〈δ(~x)δ(~x + ~y)〉
• both are related by Fourier transform:
ξ(y) = 4π∫
k2dk(2π)3 P(k)
sin kyky
• the variance of δ is ξ(y = 0):
σ2 = 4π∫
k2dk(2π)3 P(k) = ξ(0)
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Filtered Density Contrast
• variance of δ in configuration space depends on filtering
δ(~x) :=∫
d3yδ(~x)WR(|~x − ~y|)
with window function WR
• for power spectrum, this implies P(k) = P(k)W2R(k)
• variance of the filtered density contrast:
σ2R = 4π
∫k2dk(2π)3 P(k)W2
R(k)
• σ8 is conventionally used for normalizing the power spectrum
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Filtered Density Contrast
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Growth Suppression
• δ grows ∝ a2 during the radiation domination, and ∝ aafterwards
• density perturbation mode “enters the horizon” when its wavelength λ = cH−1
• modes entering the horizon during radiation domination ceasegrowing until matter starts dominating
• modes small enough to enter the horizon before aeq arerelatively suppressed compared to larger modes
• dividing wave number is
k0 = 2πH0
c
√2Ωm0
aeq
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Growth Suppression
30/47
CDM Power Spectrum
• modes are suppressed by
fsup =
(aenter
aeq
)2
=
(k0
k
)2
• initial power spectrum Pi(k) is scale-free if
Pi(k) ∝ k
(Harrison-Zel’dovich-Peebles spectrum)
• suppression leads to
P(k) = Pi(k) f 2(k) ∝
k (k < k0)k−3 (k k0)
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CDM Power Spectrum
Hlozek et al. 2011
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Large-Scale Structure
5 Statistics
6 Large-Scale StructureZel’dovich ApproximationDensity EvolutionParticle Trajectories and Pancakes
7 Nonlinear Structure Formation
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Zel’dovich Approximation
• approximate kinematic treatment aiming at translinear densityfluctuations; particle trajectories:
~r(t) = a(t)(~x + b(t)~f (~x)
)from initial position ~x
• displacement field ~f assumed to be irrotational:
~f (~x) = ~∇ψ(~x)
velocity potential ψ(~x)
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Density Evolution
• density evolution given by Jacobian determinant of ~r(~x ):
ρ = ρ0 det −1(∂ri
∂xj
)= ρ0a−3 det −1
(δij + b(t) fij
)ρ0: initial mean density
• diagonalize fij := ∂fi/∂xj, eigenvalues (λ1, λ2, λ3), then
ρ =ρ0a−3∏
i(1 + bλi)
• density contrast is
δ =1∏
i(1 + bλi)− 1 ≈ −b
∑i
λi
known linear growth requires b = D+
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Particle Trajectories and Pancakes
• velocity perturbation
~u =~r − H~r
a= HD+
d ln D+
d ln a~f = HD+f (Ωm)~f
satisfies continuity equation ~∇ · ~u = −δ = −Hf (Ωm)δ• particle trajectories in Zel’dovich approximation:
~r = a(~x + D+(a)~f
)= a
(~x +
~uHf (Ωm)
)• for Gaussian random field, probability distribution of λi is
p(λ1, λ2, λ3) ∝ |(λ3 − λ2)(λ3 − λ1)(λ2 − λ1)|
• probability for two equal eigenvalues of fij is zero: isotropiccollapse is excluded!
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Nonlinear Structure Formation
6 Large-Scale Structure
7 Nonlinear Structure FormationNonlinear EvolutionNumerical TechniquesNonlinear EffectsDifferent Cosmological Models
8 Galaxy Distribution
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Nonlinear Evolution
• when δ reaches unity, linear perturbation theory breaks down;when trajectories cross, the Zel’dovich approximation breaksdown
• numerical simulations decompose the matter distribution intoparticles with initial velocities slightly perturbed according tosome assumed power spectrum
• particles experience gravity from all other particles, but directsummation of all the gravitational forces becomes prohibitivelytime-consuming; several approximation schemes are thereforebeing used
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Numerical Techniques
• particle-mesh (PM): computes gravitational potential of theparticle distribution on a grid (mesh) by solving Poisson’sequation in Fourier space
• particle-particle particle-mesh (P3M): improves PM bysumming over nearby particles
• tree codes: bundle distant particles into groups whose gravityis approximated more crudely at larger distance; the particletree is updated as the evolution proceeds
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Nonlinear Effects
• non-linear evolution couples density-perturbation modes:power moves from large to small scales as structures collapse
• beginning from a Gaussian δ, non-Gaussianities must developduring non-linear evolution
• “pancakes” and filaments form; galaxy clusters occur wherefilaments intersect; filaments fragment into individual lumpswhich stream towards higher-density regions
• giant voids form as matter accumulates in the walls of thecosmic network
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Nonlinear Effects
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Different Cosmological Models
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Galaxy Distribution
7 Nonlinear Structure Formation
8 Galaxy DistributionGalaxy Correlation FunctionPeculiar MotionInferred DensityRedshift-Space Anisotropy
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Galaxy Correlation Function
• galaxy correlation function ξ(r) quantifies excess probability forfinding a galaxy at distance r from an other:
dP = n2[1 + ξ(r)]dV1dV2
• ξ(r) is measured by counting pairs of galaxies, normalised byrandom pair counts:
ξ =〈DD〉〈RR〉
− 1
• bias factor b relates galaxy number density to density contrast:
δnn
=: δgal = bδ
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Peculiar Motion
• density perturbations δ cause displacements
δ~x =~ra− ~x
• via peculiar motion:
δ~x =~u
Hf (Ωm), δ = −~∇ · δ~x
• distance to a galaxy is inferred from its line-of-sight velocity
v = ~v · ~ez = a(H~x + ~u
)· ~ez
• interpreting ~v as Hubble flow implies apparent distance vector:
~rapp =~vH
= ~rreal +a~u · ~ez
H~ez
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Inferred Density
• apparent displacement δ~xapp is related to real displacementδ~xreal by
δ~xapp = δ~xreal + f (Ωm)(δ~xreal · ~ez
)~ez
• from Poisson with δ~x ∝ ~∇δΦ:
δ~x =i~kk2 δ
• displacements give apparent density contrast
δapp = δreal(1 + f (Ωm)µ2
)with µ := ~k · ~ez/k
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Redshift-Space Anisotropy
• apparent density contrast in galaxy counts:
δgalapp =
(b + f (Ωm)µ2
)δreal = δ
galreal
(1 +
f (Ωm)µ2
b
)• power spectra are related by
Papp
Preal=
(1 + βµ2
)2, β :=
f (Ωm)b
• redshift-space power spectrum shows characteristicquadrupolar pattern, allows to measure β
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Redshift-Space Anisotropy
• virialised motion on small scalescauses apparent line-of-sightstretching:
δ→ δ(1 + k2µ2σ2
)−1/2
• combined effect is
Papp
Preal=
(1 + βµ2
)2
1 + k2µ2σ2