structure formation in the universe - uni-heidelberg.de · structure formation in the universe...

1/47

Structure Formation in the Universe

Matthias Bartelmann

Universitat Heidelberg, Zentrum fur AstronomieInstitut fur Theoretische Astrophysik

2/47

Outline

1 Introduction

2 Evolution of Density Perturbations (1)


4 Evolution of Velocity Perturbations

5 Statistics

6 Large-Scale Structure

7 Nonlinear Structure Formation

8 Galaxy Distribution

3/47

Introduction

1 IntroductionCosmic StructuresDark MatterCosmological AssumptionsExpansion, ParametersHorizon


4/47

Cosmic Structures

How to get from here. . .

Planck 2015

. . . to there?

2-MASS

5/47

Cosmic Structures

How to get from here. . .

Planck 2015

. . . to there?

VIPERS

6/47

Dark Matter

NGC 3198 Abell 383

7/47

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

8/47

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

9/47

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

10/47

Evolution of Density Perturbations (1)

1 Introduction

2 Evolution of Density Perturbations (1)Basic EquationsPerturbations, CoordinatesLinearized EquationsPerturbation Equations


11/47

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

12/47

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

13/47

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

14/47

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

15/47

Evolution of Density Perturbations (2)


3 Evolution of Density Perturbations (2)Density PerturbationsGrowing and Decaying ModesNecessity of Dark MatterCold and Hot Dark Matter


16/47

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

17/47

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)

18/47

Growing and Decaying Modes

19/47

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!

20/47

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);

21/47

Evolution of Velocity Perturbations


4 Evolution of Velocity PerturbationsVelocity PerturbationsPeculiar Velocity

5 Statistics

22/47

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

23/47

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

24/47

Statistics


5 StatisticsPower Spectrum and Correlation FunctionFiltered Density ContrastGrowth SuppressionCDM Power Spectrum


25/47

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)

26/47

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

27/47

Filtered Density Contrast

28/47

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

29/47

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)

31/47

CDM Power Spectrum

Hlozek et al. 2011

32/47

Large-Scale Structure

5 Statistics

6 Large-Scale StructureZel’dovich ApproximationDensity EvolutionParticle Trajectories and Pancakes


33/47

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)

34/47

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+

35/47

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!

36/47

Nonlinear Structure Formation


7 Nonlinear Structure FormationNonlinear EvolutionNumerical TechniquesNonlinear EffectsDifferent Cosmological Models

8 Galaxy Distribution

37/47

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

38/47

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

39/47

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

40/47

Nonlinear Effects

41/47

Different Cosmological Models

42/47

Galaxy Distribution


8 Galaxy DistributionGalaxy Correlation FunctionPeculiar MotionInferred DensityRedshift-Space Anisotropy

43/47

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

44/47

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

45/47

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

46/47

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

47/47

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

structure formation in the universe - uni-heidelberg.de · structure formation in the universe...

Documents