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04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
1
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)
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.2: The Growth of Structure3.2: The Growth of StructurePrimordial Density Fluctuations
€
δ =ρ − ρ ρ
=Δρρ
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.2: The Growth of Structure3.2: The Growth of StructureFormal Jeans Theory
Continuity Equation
Euler Equation
Poisson Equation
Entropy Equation
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.2: The Growth of Structure3.2: The Growth of StructureJeans Mass, Silk Mass and the decoupling epoch
€
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.2: The Growth of Structure3.2: The Growth of StructureJeans Mass, Silk Mass and the decoupling epoch
€
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseDark Matter
• 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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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 !!!
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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 !!!
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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}
3.4: The Power Spectrum3.4: The Power SpectrumThe Transfer Function
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.4: The Power Spectrum3.4: The Power SpectrumThe Transfer Function
Tegmark 2003
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
3.4: The Power Spectrum3.4: The Power Spectrum
Tegmark 2003
The Power Spectrum
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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 ?
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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STRUCTURE FORMATION
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)
04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004
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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 RegimeN-Body Simulations
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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 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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3.6: Statistical Cosmology3.6: Statistical CosmologyQuantifying Clustering
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3.6: Statistical Cosmology3.6: Statistical CosmologyQuantifying Clustering
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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3.6: Statistical Cosmology3.6: Statistical CosmologyThe Correlation Function
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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3.6: Statistical Cosmology3.6: Statistical CosmologyThe Correlation Function
€
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