Cosmological Structure Formation IMichael L. Norman

Laboratory for Computational Astrophysics

mnorman@cosmos.ucsd.edu

References

• T. Padmanabhan, “Structure Formation in the Universe”, Cambridge (1994)

• S. D. M. White, “The Formation and Evolution of Galaxies”, Les HouchesLectures, August 1993, astro-ph/9410043

Outline• Examples of cosmic structure• Gravity: origin of cosmic structure• Linear theory of perturbation growth• Nonlinear simulations I. dark matter

– hierarchical clustering• Nonlinear theory of hierarchical clustering• Nonlinear simulations II. baryons

– hydrodynamic cosmology– the Enzo code

What is cosmic structure?• Inhomogeneities in the distribution of matter in

the universe at any epoch

galaxies, groups, clusters

virializedO(>100)

Lyman alpha forestnonlinearO(1-10)

galaxy LSSquasi-linearO(0.1)

CMB anisotropieslinearO(ε)

CMBhomogeneous0ExampleRegimeδ=(ρ/<ρ>)−1

Cosmic Microwave BackgroundPenzias & Wilson (1965)

back Τ=2.73 Κ

Cosmic Microwave BackgroundWMAP Year 1

∆T/T ~ δ ~ 10-4back

Galaxy Large Scale Structure:2dF Galaxy Redshift Survey

<∆ng/ng> =b<∆ρ/ρ>~0.1back

Lyman Alpha Forest:HI absorption lines in quasar spectra

Physical Origin of the Lyman Alpha Forest

• intergalactic medium exhibits cosmic web structure at high z

• models explain observed hydrogen absorption spectra

N=1283

Zhang, Anninos, Norman (1995)

5 Mpc/hδ=10

back

Galaxies, Groups & Clusters

δ(disk) ~ 106

δ(dynamical) ~ 103

The Universe Exhibits a Hierarchy of Structures

100 101 102 103 104 105 106 107 108

Light years

star clusters

dwarf galaxies

galaxies

galaxy groups

galaxy clusters

galaxy superclusters

109

Structure Formation: Goals• Understand

– origin and evolution of cosmic structure from the Big Bang onward across all physical length, mass & time scales

– Interplay between different mass constituents(dark matter, baryons, radiation), self-gravity, and cosmic expansion

– Dependence on cosmological parameters• Predict

– Earliest generation of cosmic structures which have not yet been observed

History of the Universe

linear perturbation theory nonlinear simulations

Rec

ombi

natio

n

Nuc

leos

ynth

esis

phase transitions gravitational instability

Our universe then and now

Recombination (~380,000 yr)δρ/<ρ> ~ 10-4

Present (~14x109 yr)δρ/<ρ> ~ 106

Wilkinson MAP (NASA)

Gravitational Instability: Origin of Cosmic Structure

A

B

C

A

B

C

x

x

ρ<ρ>

ρ<ρ>

very small fluctuations

gravity amplifies fluctuations

1 billionlight years

Gravitational Instability in 3-D: Origin of the “Cosmic Web”

N=5123Bryan & Norman (1998)

Evolution of Cosmic Structure: Key Issues• origin and character of primordial density

fluctuations – Inflation: scale-free, gaussian random field

• linear evolution of the power spectrumP(k) α |δ∗(k)|2 with time (redshift) for each mass component before recombination

• nonlinear evolution of density fluctuations due to gravitational and internal forces after recombination

• above in an expanding universe with known cosmological parameters (ΛCDM)

Matter Power Spectrum

• Fourier transform of linear density field

• Power spectrum is defined as:

xkk x

xx

⋅∫=

−=

iexdV

)(1

))(()(

3 δδ

ρρρδ

dkkkkP += and between allfor )( 2 kkδ

ΛCDM Matter Power Spectrum

http://www.hep.upenn.edu/~max

Mass-Energy Budget of the Universe (WMAP)

Ωcdm

Ωb

ΩΛ

Ωcdm+ Ωb+ ΩΛ=1

Linear Theory

Linear Perturbations

2/33

3/220

22

2

2

/1:mode decaying

)( :mode growing0/6/2

becomes above 1),( universe EdSfor

023

obeyson perturbati free-pressure then,8/3H ,/)( define

−− ∝∝∝

∝∝∝=

=−+

=Ω

=

Ω−+

≡Ω≡

at

atD

aa

aa

GaaH

τδ

τδτδ

τδτδδ

δδδ

ρπτ

&&&

&&&&&

&

Saturated Growth in Low Ωm Universe

0.5n factor wheexpansion of value)/1/(11

0) 0,( :universeflat )/1/(11

0) 0,( :universeopen 3///)1(

constant alcosmologic Λ10 .33

8

3331

1

2330

10

20

22

=Ω=+=Ω→∝−Ω

>Λ=+=Ω→∝−Ω

=Λ<Λ+−=−ΩΩ→

±=Λ

+−=

−

−

−

c

c

c

aaaa

aaa

aaaH

; ,κa

GH

κ

κκ

κρπ

− Ω is the driving term in growth of δ- Makes a sudden transition in Λ model- Growth like EdS for a<ac, saturates thereafter

Mass Fluctuations in Spheres• Approximate P(k) locally with power-law

• Mean square mass fluctuation (variance)

nk kD )(22 τδ ∝

[ ] 2

3

3/)3(2)3(22232

)/()cos(/)sin(3)(

,0 ,4/3

)W(

function hat window- topof ansformFourier tr is )(

)( )/(

kRkRkRkRkRW

RRR

kRW

MDRDkRWkdMM

T

T

nnkT

−=→

≥<

=

∝∝= +−+−∫

xx

xπ

δδ

CDM Power Spectrum

n=1

n=0

n=-1

n=-2

n=-3

super horizon

LSS

galaxy clusters

galaxies

firstobjects

P(k)

k

Mass Fluctuations, cont’d

• RMS mass fluctuations

• Nonlinear mass scale

6/)3(2/12)/()( nDMMMM +−∝≡ δσ

For n>-3, RMS fluctuations a decreasing function of M

EdS)for )1(( )()(get 1)(M setting

)3/(6)3/(6nl

nnnl zDM +−+ +∝∝

=

ττ

σ

For n>-3, smallest mass scales become nonlinear first

Evolution of nonlinear mass scale with redshift in ΛCDM

galaxy clusters

galaxiesfirst structures

2 4 6 8 10 12 14 16

Log(1+znl)

10

1

100

Log(Mnl)

Rare fluctuations

Typical fluctuations

Ly α forest

Structure formation is a multiscale problem

1 Gyr

0.01 Gyr

Numerical Cosmology Goals

• Simulate the processes governing the formation and evolution of the observable structures in the universe– galaxies, quasars, clusters, superclusters

• Find the “best fit” cosmological modelthrough detailed observational comparison

• Make quantitative predictions for new era of high redshift observations

Nonlinear Simulations I:Cold Dark Matter

Cold Dark Matter

• Dominant mass constituent: Ωcdm~0.23• Only interacts gravitationally with ordinary

matter (baryons)• Collisionless dynamics governed by Vlasov-

Poisson equation

• Solved numerically using N-body methods

),,( 4

0),,(32 txfdG

fftxf xt

υυπφ

φυυ υrrr

rrr

∫=∇

=∇⋅∇−∇⋅+∂

The Universe is an IVP suitable for computation

• Globally, the universe evolves according to the Friedmann equation

Hubble parametermass-energydensity

spacetimecurvaturescale factor a(t)

338)( 2

22 Λ

+−=

≡

akG

aatH ρπ&

cosmologicalconstant

Friedmann Models:The Omega Factor

Ω<1

“Flat” universeexpands forever

“Open” universeexpands forever

Ω=1

Ω>1

Ω+Λ=1“Flat” universewhich accelerates

a(t)

Ω=<ρ>/ρcrit

The Universe is an IVP...

• Locally, its contents obey:– Newton’s laws of gravitational N-body

dynamics for stars and collisionless dark matter (CDM)

– Euler or MHD equations for baryonic gas/plasma

– Atomic, molecular, and radiative processesimportant for the condensation of stars and galaxies from diffuse gas

Gridding the Universe

• Transformation to comovingcoordinates x=r/a(t)

a(t1) a(t2) a(t3)

• Triply-periodic boundary conditions

Dark Matter Dynamics in an Expanding Universe

2/12

0

22

2

1)1(11

)(4

12

+−Ω+

−Ω=

−=∇

∇−−=

=

Λ aa

Hdtda

Gaa

vaa

dtdv

vdt

dx

m

dmdm

dmdm

ρρπφ

φ&

Newton’s laws

Poisson equation

Friedmann equation

Hierarchical Clusteringof Cold Dark Matter

Hierarchical Structure Formation:Like a River

• large galaxies form from mergers of sub-galactic units

• Galaxy groups form from merger of galaxies

• Galaxy clusters form from merger of groups

• Where did it begin, and where will it end?

Lacey & Cole (1993)

Billion Particle Simulation of Large Scale Structure

P. Bode & J. Ostriker

Nonlinear Theory

Dark Matter Halo Mass Function

• Principal quantity of interest is the globally averaged number density of collapsed objects of mass M as a function of redshift=z

• Dark matter halos define the gravitational potential wells galaxies, galaxy groups, and clusters of galaxies form in

• Sensitive function of cosmological parameters

>< ),( zMn

Spherical Top-Hat Model

• Simplest analytic model of nonlinear evolution of a discrete perturbation

rateexpansion and timeinitial same but thedensity,different of universe a like evolveson perturbati

34

:isequation Friedmann thewhereas

)1(3

4:is EOM (GR), theoremsBirkhoff'by

R radius and y overdensit uniformh region wit sphericalconsider

2

2

22

2

∴

−=

+−=−=

aGdt

ad

RGR

GMdt

Rd

ρπ

δρπ

δ

R

δ

Spherical Top-Hat, cont’d

"turnaround"at timeand radius are and where

/)sin(tt ),cos1(

21

RR

obeys and bound, ison perturbati 0,E if21

:motion of integralfirst

mm

2

mm tR

ER

GMdtdR

πΘ−Θ=Θ−=

<

=−

R

+

(Ri, ti)

Rm

t

tm

E>0

E=0

E<0

Turnaround and Collapse

686.1)12(203)2(

t2 tas 0R :Collapse6.4)(

:undat turnaroy overdensit 063.1)(

)1( )/6(203

:universe EdS respect toy with overdensit mean

3/2

m

3/2

===∴

→→=

=

<<∝=

πδδ

δ

δ

δπδ

mlincollapse

mnl

mlin

EdSmlin

t

tnonlinear

t

att

linear

Virialization

mvirmvir

vir

m

RRR

GMR

GM

RGMUEK

-KKK-ER

GMU-, EK

UK-E

KU

21

53

53

21equating

53

21

21

2 :zationat viriali

sphere) us(homogeneo 530 :undat turnaro

conserved is energy totalassume

2 :destablishe mequilibriu al when virihalted collapse

22

2

2

=→=⋅

⋅==−=∴

==

−===

=

=

Evolution of Top-hat Perturbation

Padmanabhan (1994)

Statistics of Hierarchical Clustering

• Press & Schechter (1974) derived a simple, yet accurate formula for estimating the number of virialized objects of mass M as a function of τ (or equivalently, z)

• Basic idea is to smooth density field on a scale R at such that mass scale of interest satisfies

3)(3

4 RM τρπ=

Press-Schechter Theory

• Because the density field (both smoothed and unsmoothed) is a Gaussian random field, probability that mean overdensity in spheres of radius R exceeds a critical value δc is

))

where

),(2exp

),(21),(

232

2

2

(kR)P(k,τk Wd(R,τσ

RRdRp

T

c

c

∫

∫

=

−=

∞

τσδ

τσπδτ

δ

P-S Theory, cont’d

• P-S suggested this fraction be identified with the fraction of particles which are part of a nonlinear lump with mass exceeding M if we take δc=1.686

mass some of halo a tobelongsparticleevery that sofactor fudge a is 2 offactor ..

2exp22

)2) i.e.

2

2

2

bndMd

Mρ

Mp

Mρ

dMdn

MddnMdp(R,τ

MρM,τn(

cc

M

−−=

∂∂

−=∴

′′==> ∫

∞

σδσ

σδ

π

P-S theory, cont’d

cnl

nl

n

nl

n

nl

nn

MM

dMM

MM

MnM

dMdMdn

DMDRRDR

δτσ

ττρ

π

σττσ

=

−

+

=

∝∝=

+−

+−

+−

),(satisfies now where

2/)(

exp)(3

12

find we,explicitly)()(),(

33

63

2

2/1

63

23

0

Mass function is a power-law for M<Mnland an exponential for M>Mnl

Comparison with N-body Simulations

• PS under-predicts most massive objects and over-predicts “typical” objects

Jenkins et al. (1998)