04.2.26 Chris Pearson : Observational Cosmology 3: Structure Formation - ISAS -2004 1 STRUCTURE FORMATION Observational Cosmology: 3.Structure Formation Observational Cosmology: 3. Observational Cosmology: 3. Structure Formation Structure Formation “An ocean traveler has even more vividly the impression that the ocean is made of waves than that it is made of water. ” “An ocean traveler has even more vividly the impression that the ocean is made of waves than that it is made of water. ” Arthur S. Eddington (1882-1944) Arthur S. Eddington (1882-1944)
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“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)
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
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ρ = ρ
Embed a sphere of mass,
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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
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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
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δt =δo2et /τ ff +
δo2e−t /τ ff
Where,
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τ ff = (4πGρ )−1/ 2 is the dynamical free fall time
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t→∞⇒ e−t / t ff → 0 Only exponentially increasing term survivesConclusion: Density perturbations will grow exponential under the influence of self gravity
3.2: The Growth of Structure3.2: The Growth of StructureJeans Mass, Silk Mass and the decoupling epoch
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λJ ,γ (dec) ≈ 0.6Mpc
MJ ,baryon (dec) ≈1018Mo
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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
3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseGrowth of Perturbations in an expanding universe
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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
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 !!!
3.3: Structure Formation in a Dark Matter Universe3.3: Structure Formation in a Dark Matter UniverseHot Dark Matter
Radiation Dominates
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ρ ∝R4 (R = R(t))
Friedmann eqn. H(t)2 =Ωr,oRo
R
4
Ho2
Radiation dominated H(t) =12t
Matter Dominates
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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
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cH(teq )
=cHo
Ωr,o3 / 2
Ωm,o2 ≡ 2cteq ≈ 30kpc
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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
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) :
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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
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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
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
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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
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
In the Zeldovich Approximation, the first structures to form are giant Pancakes(provides very good approximation to the non-linear regime until shell crossing)
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
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
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.
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
• 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
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
In a homogeneous Poisson distributed field (n = the number density)
If the field is clustered
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(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
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
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w(θ) = Aωθ1−γ
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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
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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
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σR2 =
ΔMM
2
R
= δ 2R
=13
ξ(Rr)r2∫ dr
σR ~ unity on scales of 8Mpc ➠ normalize power spectrum at that scale
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b =σ 8,G
σ 8,DM
Standard Estimator :
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w(θ) = (2DD /DR) −1
Hamilton Estimator :
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w(θ) = 4(DDxDR) /(DR2 −1)
Landy & Szalay- SL Estimator :
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w(θ) = (DD− 2DR + RR) /RRSmaller uncertainties on large scales
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