04.2.26 Chris Pearson : Observational Cosmology 3 ... › ~cpp › teaching › ... · 04.2.26...

04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004

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STRUCTURE FORMATION

Observational Cosmology: 3.Structure FormationObservational Cosmology: 3.Observational Cosmology: 3.Structure FormationStructure Formation

“An ocean traveler has even more vividly the impression that the ocean is made of wavesthan that it is made of water. ”“An ocean traveler has even more vividly the impression that the ocean is made of wavesthan that it is made of water. ”

Arthur S. Eddington (1882-1944)Arthur S. Eddington (1882-1944)


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Radiation:CMB - Isotropic to 1 part in 105, 0.003%, 2µK

3.1: Isotropy & Homogeneity on the Largest Scales3.1: Isotropy & Homogeneity on the Largest ScalesIsotropy and Homogeneity on the largest scales

Cosmological Principle: The Universe is Homogeneous and IsotropicCosmological Principle: The Universe is Homogeneous and Isotropic

True on the largest Scales

Matter:Large scales > 100Mpc (Clusters / Superclusters) : Universe is smoothRadio Sources: isotropic to a few percentSmall scales : Highly anisotropic


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3.1: Isotropy & Homogeneity on the Largest Scales3.1: Isotropy & Homogeneity on the Largest ScalesIsotropy and Homogeneity on the largest scales

200M

pc

~1000M

pc

• Walls• Filaments• Clusters• Superclusters• Voids


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3.2: The Growth of Structure3.2: The Growth of StructurePrimordial Density Fluctuations Origin of LSS today - primordial density fluctuations

€

δ =ρ − ρ ρ

=Δρρ

Density perturbation

2) Cosmic Defects • Defects from phase transitions• Cosmic String• Domain Walls• Textures

1) Primordial Quantum Fluctuations • Gaussian Fluctuations from inflation

Scale Free Harrison - Zeldovich spectrum model:

• Fluctuations have the same amplitude when they enter the horizon ~ δ ~ 10-4

• Scale free Harrison-Zeldovich Spectrum of power

€

P(k) = δk2∝ kn, n =1


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3.2: The Growth of Structure3.2: The Growth of StructurePrimordial Density Fluctuations

€

δ =ρ − ρ ρ

=Δρρ


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3.2: The Growth of Structure3.2: The Growth of StructurePrimordial Density Fluctuations

€

δ =ρ − ρ ρ

=Δρρ

ρADIABATIC FLUCTUATIONSFluctuations in matter and radiation(changes in volume in the early Universe ➠ change in number densities)

ρISOTHERMAL FLUCTUATIONSFluctuations in matter ONLY➠ No perturbations in the Temperature

ISO-CURVATURE / ISENTROPIC FLUCTUATIONSNo Perturbations in the density fieldFluctuations in the matter relative to the radiation δm≡−δγ

ρ

Fluctuations in radiation field ➠ leave scar on CMB ➠ observed as deviations from 2.73K BB


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STRUCTURE FORMATION

3.2: The Growth of Structure3.2: The Growth of StructureThe Jeans Length

ρr

Mρ = ρ(1+δ)

Consider a homogeneous universe of average density

€

ρ = ρ

Embed a sphere of mass,

€

ρ = ρ (1+ δ), δ =ρ − ρ ρ

=Δρρ

<<1

€

M =4π3ρ (1+ δ)r3with over density

€

˙ ̇ r = −GΔMrt

2 = −4πGρ rt

3δtSphere collapses from rest equilibrium under self gravity 1

€

M =4π3ρ (1+ δt ) rt

3 = constant

⇒ rt =3M4πρ

1/ 3

(1+ δt )−1/ 3 = rto (1+ δt )

−1/ 3

During Collapse

€

for δ <<1, (1+ δt )−1/ 3 ≈1− δt

3, rt ≈ rto

⇒˙ ̇ r r≈ −

˙ ̇ δ t3

2

€

˙ ̇ δ t = 4πGρ δt21 = Solutions of the form

€

δt =δo2et /τ ff +

δo2e−t /τ ff

Where,

€

τ ff = (4πGρ )−1/ 2 is the dynamical free fall time

€

t→∞⇒ e−t / t ff → 0 Only exponentially increasing term survivesConclusion: Density perturbations will grow exponential under the influence of self gravity


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STRUCTURE FORMATION

3.2: The Growth of Structure3.2: The Growth of StructureThe Jeans Length

Compression

Pressure

Expansion

AcousticOscillations Pressure

GRAVITY

Equation of State for an Ideal Gas

€

P =ωρc 2 =kTµρ =

v s3ρ (v s << c)

( fundamental cosmology 5.3)

€

v s = the sound speed

In absence of pressure, an overdense region collapses on order of the free fall time

€

τ ff = (4πGρ )−1/ 2

Pressure Gradient - Resists Collapse ifa pressure gradient can be createdover a timescale given by τJ < τff

€

τ J ≈rv s

❊ Define a critical length over which density perturbation will be stable against collapse under self gravity

€

rcritical = λJ ~ v sτ ff ~v s2

Gρ

€

λJ =π v s

2

Gρ

1/ 2

= 2π v sτ ff JEANS LENGTH

€

MJ =4π3ρ λJ

3 JEANS MASS


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3.2: The Growth of Structure3.2: The Growth of StructureFormal Jeans Theory

Continuity Equation

Euler Equation

Poisson Equation

Entropy Equation


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3.2: The Growth of Structure3.2: The Growth of StructureJeans Mass, Silk Mass and the decoupling epoch

Friedmann eqn. (k=0) ➠ expansion rate of Universe given by Hubble parameter

€

H 2 =8πGρ 3

€

τ ff = (4πGρ )−1/ 2Free Fall Time

€

H−1 ≈ τ ff

€

λJ = 2π v sτ ff ≈ 2π (2 /3)1/ 2 v s

HJeans Length

€

v s =c3≈ 0.6cPhoton sound speed (ω=1/3)

This mass is larger than the largest Supercluster today !

€

λJ ,γ ≈ 3cH⇒ λJ ,γ (dec) ≈ 0.6Mpc

MJ ,baryon ≈ 36π ρ cH

3

⇒ MJ ,baryon (dec) ≈1018Mo

At decoupling (z=1089) Super-horizon scales➠ Sub horizon scales cannot grow

Before epoch of decoupling, photons and Baryons bound together as a single fluid


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€

λJ ,γ (dec) ≈ 0.6Mpc

MJ ,baryon (dec) ≈1018Mo

After epoch of decoupling, photons and Baryons behave as separate fluids

€

v s,γ =c3≈ 0.6cPhoton sound speed

€

v s,baryon =kT

mc 2

1/ 2

c ≈ 0.00001cBaryon sound speed

€

λJ =π v s

2

Gρ

1/ 2

After decouplingJean’s Length

€

λJ =v s,baryon

v s,γλJ (dec) ~ 2x10−5λJ (dec)

€

MJ ,baryon =λJ (dec)λJ

3

MJ ,baryon (dec) ~ 105MoJean’s Mass after decoupling

This mass is approximately the same mass as Globular Cluster today !

Until decoupling, structures over scales of globular clusters up to superclusters could not grow


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€

λJ ,γ (dec) ≈ 0.6Mpc

MJ ,baryon (dec) ≈1018Mo

€

λJ ,γ ≈12pc

MJ ,baryon ≈105Mo

• Close to decoupling / recombination : Baryon/photon fluid coupling becomes inefficient• Photon mean free path increases ➠ diffuse / leak out from over dense regions• Photons / baryons coupled ➠ smooth out baryon fluctuations➠ Damp fluctuations below mass scale corresponding to distance traveled in one expansion timescale

2

4

6

8

10

12

14

16

18

lg(M

J) {

Mo}

9 7 5 3 2lg(Tr) {K}

unstable

unstable

Acousticoscillation

Radiation dominated Matter dominated

Asymptotic value

Mat

ter-

radi

atio

n eq

ualit

y

reco

mbi

nati

on

€

Msilk ≈ 5x1014Mo

€

Msilk ≈ 2x1012 Ωo

Ωb

3 / 2

Ωoh2( )−5 / 4

Mo ≈ 5x1014MoTHE SILK MASS


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3.2: The Growth of Structure3.2: The Growth of StructureGrowth of Perturbations in an expanding universe : The Hubble Friction

€

τ ff = (4πGρ )−1/ 2Free Fall Time

€

H−1 ≈ τ ffGrowth of structure - competition between 2 factors:

€

H−1 =8πGρ 3

−1/ 2

Hubble Expansion

€

ρ = ρ (t)∝R(t)3Expanding Universe

€

ρ(t) = ρ (t) 1+ δ(t)[ ]Embed a sphere of density ρ, in a homogeneous universeρr

Mρ = ρ(1+δ)Total Gravitational acceleration

at surface of the sphere

€

˙ ̇ r = −GMr2 = −

Gr2

4πρr3

3

⇒

˙ ̇ r r

= −4π3

G ρ + ρ δ( ) 1

€

M = const = 4π3ρ (1+ δt ) rt

3 ⇒ rt =3M4π

1/ 3

ρ t−1/ 3(1+ δt )

−1/ 3 ⇒ rt ∝R(t)(1+ δt )−1/ 3During Collapse/expansion 2

2

€

d2

dt 2= 1

€

˙ ̇ R R−

˙ ̇ δ 3−

2 ˙ R 3R

˙ δ = −4π3

Gρ −4π3

Gρ δ

€

˙ ̇ R R

= −4π3

Gρ For homogeneous, isotropic universe, δ=0

Subtracting the homogeneous component

€

˙ ̇ δ + 2H ˙ δ = 4πGρ δ

€

˙ ̇ δ = 4πGρ δFor a static universe, H=0 ➠

Extra term in expanding universe ➠ HUBBLE FRICTION ➠ slows the growth of the density pertubations


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3.2: The Growth of Structure3.2: The Growth of StructureGrowth of Perturbations in an expanding universe

Rewrite in terms of density parameter

€

˙ ̇ δ + 2H ˙ δ = 4πGρ δ

€

˙ ̇ δ + 2H ˙ δ −32ΩmH

2δ = 0

Radiation Era Ωm<<1

€

˙ ̇ δ + 2H ˙ δ = ˙ ̇ δ +1t

˙ δ ≈ 0

Lambda Era Η=ΗΛ= Const

€

˙ ̇ δ + 2H ˙ δ = ˙ ̇ δ + HΛ˙ δ ≈ 0

Solution:

€

δ(t) ≈ A + B ln t( )Fluctuations in matter (non-baryonic) can only grow logarithmically

Solution:

€

δ(t) ≈ A + Be−2HΛ t

F;uctuations in matter tend to a constant fractional amplitude.

Matter Era (Ωm=1, H=2/3t)

€

˙ ̇ δ + 2H ˙ δ −32ΩmH

2δ = ˙ ̇ δ +43t

˙ δ −2

3t 2 δ = 0 Solution:

Growing mode and a decaying mode exist

€

δ(t) ≈ A t 2 / 3 + B t−1

€

δ ∝ A t 2 / 3 ∝R(t)∝ 1(1+ z)

, δ <<1Density fluctuations in a flat, matter dominated Universe grow as


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseGrowth of Perturbations in an expanding universe

€

δ ∝ A t 2 / 3 ∝R(t)∝ 1(1+ z)

, δ <<1Density fluctuations in a flat, matter dominated Universe grow as

• δ<<1 ➟ linear regime• δ~1 ➟ non-linear regime ➟ Require N-body simulations• Baryonic Matter fluctuations can only have grown by a factor (1+zdec) ~ 1000 by today• for δ~1 today require δ~0.001 at recombination• δ~0.001 ➟ δΤ/Τ ~0.001 at recombination• But CMB ➟ δΤ/Τ ∼10−5 !!!

DARKMATTERDARK

MATTER

Dark Matter Condenses at earlier time Matter then falls into DM gravitational wells

• MATTER PERTURBATIONS DON’T HAVE TIME TO GROW IN A BARYON DOMINATED UNIVERSE


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseDark Matter

• To be born Dark, to become dark, to be made dark, to have darkness

HOT DARK MATTER Relativistic at decoupling Light Neutrino

COSMIC RELICS Symmetry Defects

Cosmic Textures

Monopoles

Cosmic Strings

WIMPs

COLD DARK MATTER Non Relativistic at decoupling

Axions

Heavy Neutrino

SUSY Particles


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• Weakly interacting ➠ no photon damping

• Structure formation proceeds before epoch of decoupling

• Provides Gravitational ‘sinks’ or ‘potholes’

• Baryons fall into ‘potholes’ after epoch of decoupling

• Mode of formation depends on whether Dark Matter is HOT/COLD

• Hot /Cold DM Decouple at different times ➠ Different effects on Structure Formation

Chan

dra

webs

ite


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Actual picture of dark matter in the Universe !!!Actual picture of dark matter in the Universe !!!


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseDark Matter Actual picture of dark matter in the Universe !!!Actual picture of dark matter in the Universe !!!


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseHot Dark Matter

Radiation Dominates

€

ρ ∝R4 (R = R(t))

Friedmann eqn. H(t)2 =Ωr,oRo

R

4

Ho2

Radiation dominated H(t) =12t

Matter Dominates

€

ρ ∝R3 (R = R(t))

Friedmann eqn. H(t)2 =Ωm,oRo

R

3

Ho2

Radiation dominated H(t) =23t

• Any massive particle that is relativistic when it decouples will be HOT• ➠ Characteristic scale length / scale mass at decoupling given by Hubble Distance c/H(t)

Epoch of Matter/Radiation Equality

1+z ~ 3500

For radiation (photons)

Other relativistic species

Substituting for (Ro/R), The Hubble Distance at teq is

€

cH(teq )

=cHo

Ωr,o3 / 2

Ωm,o2 ≡ 2cteq ≈ 30kpc

€

Meq =4π3

cH(teq )

3

ρ(teq ) =4π3

cHo

3Ωr,o3 / 2

Ωm,o2

3

Ωm,oρc,oRo

Req

3

=π3

cHo

3Ωr,o3 / 2

Ωm,o2 ρc,o ~ 10

17MoMass inside Hubble volume

>> MSupercluster>> MSupercluster

1+z ~ >3500, MH<1017MoEpoch of equality defined when kBT~mc2

Recall Fundamental Cosmology 7.2

€

T =32πGa3c 2

−1/ 4

t−1/ 2 ≈1.5x1010 t−1/ 2At a time given by


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseHot Dark Matter

For a hot neutrino, mass mν(eV/c2) :

€

Teq ≈mν

k≈11600mν {K} ⇒ teq =1.7x1012(mν )

−2 {s}

Fluctuations suppressed on mass scales of €

λeq ≈ c teq ~ 17 mν( )−2 kpc ⇒ λo =Ro

Req

λeq ~

Teq2.73

λeq ≈

70mν

Mpc

€

M =4π3λo3Ωm,oρc,o ~

1016

mν3 Mo

• Before teq, neutrinos are relativistic and move freely in random directions• Absorbing energy in high density regions and depositing it in low density regions• Like waves smoothing footprints on a beach! • Effect ➠ smooth out any fluctuations on scales less than ~ cteq

This Effect is known as FREE STREAMING

Large Superstructures form first in a HDM Universe ➠ TOP-DOWN SCENARIO


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3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseCold Dark Matter

For a CDM WIMP, mass mCDM~1GeV :

€

Teq ≈mCDM

k≈109 {K} ⇒ teq = 5 s

€

cH

= 2ct = 3x109m⇒ λo =Ro

Req

λeq ~

Teq2.73

λeq ≈ 0.04 kpc

€

M =4π3λo3Ωm,oρc,o << Mo

Structure forms hierarchically in a CDM Universe ➠ BOTTOM-UP SCENARIO

Fluctuations λ > λο will grow throughout radiation period

Fluctuations λ < λο will remain frozen until matter domination when Hubble distancehas grown to ~0.03Mpc corresponding to 1017Mo➠ Scales > Hubble distance at matter domination retain original primordial spectrum


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STRUCTURE FORMATION

3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseStructure Formation in a Dark Matter universe

Simulation of CDM and HDM Structure formation seeded by cosmic strings (http://www.damtp.cam.ac.uk)

CDM - Bottom-Up Hierarchical Scenario

HDM - Top-Down Pancake Scenario


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3.4: The Power Spectrum3.4: The Power SpectrumQuantifying the power in fluctuations on large scales

• We would like to quantify the power in the density fluctuations on different scales

long wavelength (large scales)

Short wavelength (small scales)

High Power (large amplitude)

Low Power (small amplitude)

€

δ(r r ) =ρ − ρ ρ

=Δρρ

Density fluctuation field

€

δk = δ(r r )e− ik•r∑Fourier Transform ofDensity fluctuation field

( ) 2

kkP δ=Power of the density fluctuations


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STRUCTURE FORMATION

3.4: The Power Spectrum3.4: The Power SpectrumQuantifying the power in fluctuations on large scales

• Inflation ➠ Scale Free Harrison - Zeldovich spectrum model:

€

P(k) = δk2∝ kn, n =1

• Fluctuations have the same amplitude when they enter the horizon ~ δ ~ 10-4

lg(P

(k))

lg(k)

small klarge scales

large ksmall scales

• Inflation field is isotropic, Homogeneous, Gaussian field (Fourier modes uncorrelated)

• Value of δ(r) at any randomly selected point drawn from GPD

€

℘(δ) =1

σ 2πe−

δ

2σ 2

• ➠ All information contained within the Power Spectrum P(k)

€

σ =V(2π )3

P(k)d3k =∫ V2π 2 P(k) k 2dk∫

Average mass contained with a sphere of radius λ (=2π/k)

€

M =4π3

2πk

3

ρ

Mean squared massdensity within sphere

€

M − MM

2

∝ k 3P(k) ≡ δMM

2

Instead of simply P(k) ➠ often plot (k3P(k))1/2 the root mean square mass fluctuations


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STRUCTURE FORMATION

3.4: The Power Spectrum3.4: The Power SpectrumThe Transfer Function

• Matter-Radiation Equality: Universe matter dominated but photon pressure ➠ baryonic acoustic oscillations• Recombination ➠ Baryonic Perturbations can grow !• Dark Matter “free streaming” & Photon “Silk Damping” ➠ erase structure (power) on smaller scales (high k)• After Recombination ➠ Baryons fall into Dark Matter gravitational potential wells

• through the radiation domination epoch• through the epoch of recombination• to the post recombination power spectrum,given by TRANSFER FUNCTION T(k), contains messy physics of evolution of density perturbations

€

P(k, t) = T(k)2P(k, t primordial )

€

T(k) = f (Γ) = 1+ ((ak) + (bk)3 / 2 + (ck)2)( )ν[ ]

−1/ν

€

a = 6.4(Ωoh2)−1 b = 3.0(Ωoh

2)−1

c =1.7(Ωoh2)−1 ν =1.13

Γ = Shape ParameterCDM

€

k→ 0, T(k)2 →1 ⇒ P(k)∝ k⇒ unchanged!k→∞ T(k)∝ k−2 ⇒ P(k)∝ k−3 ⇒ Small scale power!

HDM

€

kν ≈ 0.4Ωoh2Mpc−1 (for a 30eV neutrino)

⇒ supress all fluctuation modes λ <2πkν

≈120

mν (eV )Mpc

€

T(k) =10−

kkν

1.5

The transformation from the density fluctuations from the primordial spectrum


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Primordial (P∝k)

CDM

HDM MDM

k {Mpc-1}

(k3 P

(k))1/

2

10-4

10-2

100

0.001 0.01 0.1 1 10

0.001 0.01 0.1 1 10

10-6

10-4

10-2

Primordial (P∝k)

CDM

HDMMDM

P(k)

k {Mpc-1}


10-4

10-2

100

1012 1014 1016 1018 1020

Primordial (P∝k)

CDM

HDM

MDM(k

3 P(k

))1/2

M {Mo}

BaryonsCDM

HDM MDM

0.001 0.01 0.1 1 10

T(k)

k {Mpc-1}

1016Mo

1017Mo


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Tegmark 2003


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3.4: The Power Spectrum3.4: The Power SpectrumThe Power Spectrum

Vanilla Cosmology: ΩΛ=0.72, Ωm=0.28, Ωb=0.04, H=72, τ=0.17, bSDSS=0.92 Tegmark 2003


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3.4: The Power Spectrum3.4: The Power Spectrum

Tegmark 2003

The Power Spectrum


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3.5: The Non-Linear Regime3.5: The Non-Linear RegimeThe non-Linear Regime

Primordial Fluctuations ➠ the seeds of structure formation

Fluctuations enter horizon ➠ grow linearly until epoch of recombination

Post recombination ➠ growth of structure depends on nature of Dark Matter

Fluctuations become non-linear i.e. δ > 1

How can we model the non-linear regime ?


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3.5: The Non-Linear Regime3.5: The Non-Linear Regime(1) The Zeldovich Approximation (relates Eulerian and Lagragian co-ordinate frames)

Eulerian coords (r) at time t related to Lagrangian coords (q) by initial velocity, s(q);

€

r r (q,t) = q + s(q)t

€

˙ ̇ δ + 2H ˙ δ −32ΩmH

2δ

€

r r (q,t) = R(t) q + δ(t) s(q)[ ]In an expanding Universe: where

and

€

s(q) =∇Φo(q) Provides particle displacements with respect to initial Laplacian

Mass Conservation:

€

ρ(q,t) = ρo∂r∂q

=ρ

1−δ(t) ∂ri∂qq

€

∂ri∂qq

= Deformation Tensor

In 3-D, tensor eigenvectors define 3 orthogonal axes describing contraction/expansion: α(q), β(q), γ(q),

α << β ≈ γ ⇒ Pancake / Sheet α ≈ β << γ ⇒ string / Filiament α ≈ β ≈ γ ⇒ Knot / Sphere

In the Zeldovich Approximation, the first structures to form are giant Pancakes(provides very good approximation to the non-linear regime until shell crossing)


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STRUCTURE FORMATION

3.5: The Non-Linear Regime3.5: The Non-Linear Regime(2) N-Body Simulations

PM, P3M

PP

Multipole expansion.O(N logN)ART codes

Use FFTs to invert Poisson equation.O(N logN)Particle mesh

Practical for N<104O(N2)Direct summation

❑ PP Simulations:• Direct integration of force acting on each particle

❑ PM Simulations: Particle Mesh• Solve Poisson eqn. By assigning a mass to a discrete grid

❑ P3M: Particle-particle-particle-Mesh• Long range forces calculated via a mesh, short range forces via particles

❑ ART: Adaptive Refinment Tree Codes• Refine the grid on smaller and smaller scales

• Strengths❑ Self consistent treatment of LSS and galaxy evolution

• Weaknesses❑ Limited resolution❑ Computational overheads


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3.5: The Non-Linear Regime3.5: The Non-Linear Regime(2) SAM - Semi Analytic Modelling

• Merger Trees; the skeleton of hierarchical formation• Cooling, Star Formation & Feedback• Mergers & Galaxy Morphology• Chemical Evolution, Stellar Population Synthesis & Dust

• Hierarchical formation of DM haloes (Press Schecter)• Baryons get shock heated to halo virial temperature• Hot gas cools and settles in a disk in the center of the potential well.• Cold gas in disk is transformed into stars (star formation)• Energy output from stars (feedback) reheats some of cold gas• After haloes merge, galaxies sink to center by dynamical friction• Galaxies merge, resulting in morphological transformations.

•Strengths❑ No limit to resolution❑ Matched to local galaxy properties

•Weaknesses❑ Clustering/galaxies not consistently modelled❑ Arbitrary functions and parameters tweaked to fit local properties


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3.5: The Non-Linear Regime3.5: The Non-Linear RegimeN-Body Simulations - Virgo Consortium

• two simulations of different cosmological models : tCDM & LCDM• one billion mass elements, or "particles"• over one billion Fourier grid cells• generates nearly 0.5 terabytes of raw output data (later compressed to about 200 gigabytes)• requires roughly 70 hours of CPU on 512 processors (equivalent to four years of a single processor!)

τ CDM

Ωm=1, σ8=0.6, spectral shape parameter Γ=0.21comoving size simulation 2/h Gpc (2000/h Mpc)cube diagonal looks back to epoch z = 4.6cube edge looks back to epoch z = 1.25half of cube edge looks back to epoch z = 0.44simulation begun at redshift z = 29force resolution is 0.1/h Mpc

Λ CDMΩm =0.3, ΩΛ =0.7, σ8 =1, Γ =0.21comoving size simulation 3/h Gpc(3000/h Mpc)cube diagonal looks back to epoch z = 4.8cube edge looks back to epoch z = 1.46half of cube edge looks back to epoch z = 0.58simulation begun at redshift z = 37force resolution is 0.15/h Mpc


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3.5: The Non-Linear Regime3.5: The Non-Linear RegimeN-Body Simulations - Virgo Consortium

• The "deep wedge" light cone survey from the τCDM model.• The long piece of the "tie" extends from the present to a redshift z=4.6• Comoving length of image is 12 GLy (3.5/h Gpc), when universe was 8% of its present age.• Dark matter density in a wedge of 11 deg angle and constant 40/h Mpc thickness, pixel size 0.77/h Mpc.• Color represents the dark matter density in each pixel, with a range of 0 to 5 times the cosmic mean value.• Growth of large-scale structure is seen as the character of the map turns from smooth at early epochs(the tie's end) to foamy at the present (the knot).

•The nearby portion of the wedge is widened and displayed reflected about the observer's position. Thewidened portion is truncated at a redshift z=0.2, roughly the depth of the upcoming Sloan Digital SkySurvey. The turquoise version contains adjacent tick marks denoting redshifts 0.5, 1, 2 and 3.


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3.5: The Non-Linear Regime3.5: The Non-Linear RegimeN-Body Simulations


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3.5: The Non-Linear Regime3.5: The Non-Linear RegimeN-Body Simulations - formation of dark Matter Haloes

•The hierarchical evolution of a galaxy cluster in a universe dominated by cold dark matter.･Small fluctuations in the mass distribution are barely visible at early epochs.･Growth by gravitational instability & accretion ⇒ collapse into virialized spherical dark matter halos･Gas cools and objects merge into the large galactic systems that we observe today

10Mpc

R=0.02Rot=0.002to


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3.5: The Non-Linear Regime3.5: The Non-Linear Regime

Bevis & Oliver 2002

SPH Simulations

N-Body Simulations


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3.6: Statistical Cosmology3.6: Statistical CosmologyQuantifying Clustering

• Underlying Dark Matter Density field will effect the clustering of Baryons• Baryon clustering observed as bright clusters of galaxies• Only the tip of the iceberg???

BaryonsDark Matter

Baryons

Dark Matter

We would like to quantify the clustering on all scales from galaxies, clusters, superclustersWe would like to quantify the clustering on all scales from galaxies, clusters, superclusters

Baryons may be biased


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Statistical Methods for quantifying clustering / topology

• The Spatial Correlation Function• The Angular Correlation Function• Counts in Cells• Minimum Spanning Trees• Genus• Void Probability Functions• Percolation Analysis

Generally we want to measure how a distribution deviates from the Poisson case


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3.6: Statistical Cosmology3.6: Statistical CosmologyThe Correlation Function

Angular Correlation Function w(θ) : Describes the clustering as projected on the sky(e.g. the angular distribution of galaxies, e.g. in a survey catalog)

Spatial Correlation Function ξ(r) : Describes the clustering in real space

r

V

For any random galaxy: Probability , δP, of finding another galaxy within a volume, V, at distance , r

€

δP = ndVδP = n 1+ ξ(r)[ ]dV (ξ(r) ≥ −1, r→ 0 ξ(r)→ 0)

In a homogeneous Poisson distributed field (n = the number density)

If the field is clustered

€

(ξ(r) ≥ −1r→ 0 ξ(r)→ 0)

Since Probabilities are positive

For a mean density to exist for the sample

Assume ξ(r) is isotropic (only depends on distance not direction) ➠ ξ(r) = ξ(r)

✰ In practice: the correlation function is calculated by counting the number of pairs aroundgalaxies in a sample volume and comparing with a Poisson distribution


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Similarly the angular correlation function i given by

€

δP = n 1+ w(θ)[ ]dΩ1dΩ2 n = mean surface density

For a catalog of n galaxies covering a solid angle, ΩMean number of galaxies at θ±Δθ from any randomly selected galaxy in mean solid angle <dΩ> ;

€

n −1( ) 1+ w(θ)( ) dΩ /Ω

The total number of pairs with separations θ±Δθ is therefore

€

DD(θ) =n2n −1( ) 1+ w(θ)( )

dΩΩ

DD(θ) can be measured from the catalog

Knowing DD(θ) and <dΩ>/Ω ➽ w(θ) For All Sky Survey, Ω=4π, dΩ=2πSin(θ)Δθ

For more complicated geometries, easier to calculate <dΩ>/Ω fromthe number of random-random pairs laid down over same area

€

RR(θ) =n2n −1( )

dΩΩ

€

⇒ w(θ) = (DD /RR) −1 Strictly require more random points than data pointsand need to correct for edge effects

Use DR(θ) number of pairs with separations θ±Δθ where one point is taken from random and real data set

Standard Estimator :

€

w(θ) = (2DD /DR) −1


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3.6: Statistical Cosmology3.6: Statistical CosmologyThe Correlation Function and the relation to the power spectrum

b is the bias parameter for galaxy biasing w.r.t. underlying Dark Matter Distributionb is the bias parameter for galaxy biasing w.r.t. underlying Dark Matter Distribution

The angular correlation function is found to have the relation

€

w(θ) = Aωθ1−γ

€

ξ(r) =rro

−γ

The spatial correlation function Galaxies γ=1.8, ro=5h-1 MpcClusters γ=1.8, ro=12-25h-1 Mpc

The spatial correlation function is the Fouriertransform of the Power Spectrum

€

P(k) = ξ(r)eik•r∫ d3r ≡ ξ(r) sin(kr)r∫ dr

The spatial correlation function is related to themass density variation in spheres of radius,R

€

σR2 =

ΔMM

2

R

= δ 2R

=13

ξ(Rr)r2∫ dr

σR ~ unity on scales of 8Mpc ➠ normalize power spectrum at that scale

€

b =σ 8,G

σ 8,DM

Standard Estimator :

€

w(θ) = (2DD /DR) −1

Hamilton Estimator :

€

w(θ) = 4(DDxDR) /(DR2 −1)

Landy & Szalay- SL Estimator :

€

w(θ) = (DD− 2DR + RR) /RRSmaller uncertainties on large scales

€

Δw(θ) =1+ w(θ)

DD


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€

w(θ) = Aωθ−β

Aω = 0.026 ± 0.005β = 0.94 ± 0.09

ELAIS

€

w(θ) = Aωθ−β

Aω = 0.71± 0.26β = 0.8 ± 0.09

SDF z=4 LBG

€

w(θ) = Aωθ−β

β = 0.7

APM


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3.6: Statistical Cosmology3.6: Statistical CosmologyLimber Equation

€

w(θ) = Aωθ1−γ

€

ξ(r) =rro

−γ

Limber Equation

€

roγ = Aω

−1CF(z)DA (z)0

∞

∫1−γ

N(z)2g(z)dz

N(z)dz0

∞

∫( )2

€

DA

F(z)∝ (1+ z)ε

g(z) =Ho

c(1+ z)2 (1+ zΩm + ((1+ z)−2 −1)ΩΛ[ ]

C = π1/ 2Γ (γ −1) /2[ ]Γ(γ /2)

N(z)

Angular Diameter Distance

Evolution of the spatial correlation function

Cosmology term

Numerical Contants

Redshift Distribution(measured or predicted)


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3.6: Statistical Cosmology3.6: Statistical CosmologyCounts in Cells divide the Universe into

boxes of side r and countthe number of galaxies, ni

in each cell

z

Scal

e (r

)

Boxes too sparse

Too Few boxes

OK

3~)( rzV

3min /))(()),(( rrfszL σ=Φ

22 2.0)( =zσCo

smic

Var

ianc

e to

o hi

gh

€

mean no. galaxies in volume N V = ni∑ ≡ ndV = nVV∫

Variance N 2 = ni2

i∑ + nin j

j∑

Divide into smaller and smaller boxes until ni=0 or 1

either 0 or 1

€

n2 dV1∫ dV2 1+ ξ12( )

€

N 2 = nV + (nV )2 + n2 dV1dV2∫ ξ12 Can derive

€

N − NN

=

ΔN 2

N=1N

+ 2∑ V

Σ2V = variance of the density field smoothed over the cell


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3.6: Statistical Cosmology3.6: Statistical CosmologyCounts in Cells

1

10

100

1000

0.01 0.1 1 10

SCAL

E / h

-1M

pc

Redshift

ASTRO-F

PSC-z

SWIRE

Herschel

PROBING LARGE SCALE STRUCTURE


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3.6: Statistical Cosmology3.6: Statistical CosmologyMinimum Spanning Trees


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3.6: Statistical Cosmology3.6: Statistical CosmologyGenus


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3.6: Statistical Cosmology3.6: Statistical CosmologyVoid Probability Functions


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3.6: Statistical Cosmology3.6: Statistical CosmologyPercolation Analysis


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3.7: Large Scale Surveys3.7: Large Scale SurveysLarge Scale Surveys

ASTRO-F All Sky FIR 60-170µm = 20-100mJyASTRO-F Team Pearson et al. (2004)


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3.8: Summary3.8: SummarySummary

Structure Formation in the Universe is determined by• Initial Primordial Fluctuations• Dark Matter (free streaming - Top Down / Bottom-Up Hierarchal)• Acoustic Oscillations over the Jeans Length • Photon Damping• The epoch of decoupling and recombination

Structure Formation in the Universe can be analysed by

• The Power Spectrum• N-body Simulations• Cosmological Statistics (e.g. correlation functions)• Require large scale surveys and redshifts


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3.8: Summary3.8: SummarySummary

Observational CosmologyObservational Cosmology3. Structure Formation3. Structure Formation終終終

次：次：次：Observational CosmologyObservational Cosmology

4. Cosmological Distance Scale4. Cosmological Distance Scale

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