Lecture 3: The Halo Model

Structure Formation and the Dark Sector

Wayne Hu

Trieste, June 2002

Outline • Spherical Collapse • Mass Function Press-Schechter Formalism Extended Press-Schechter Formalism Halo Abundance • Halo Bias • Halo Profile • Halo Model Density Field Baryonic Gas Galaxies

Closed Universe • A spherical perturbation of radius r behaves as a closed universe • Radius r ∝ a → 0, collapse in finite time • Friedman equation in a closed universe   1 da −3 −2 1/2 = H0 Ωm a + (1 − Ωm )a a dt • Parametric solution in terms of a development angle θ = H0 η(Ωm − 1)1/2 , scaled conformal time η

r(θ) = A(1 − cos θ) t(θ) = B(θ − sin θ) where A = r0 Ωm /2(Ωm − 1), B = H0−1 Ωm /2(Ωm − 1)3/2 . • Turn around at θ = π, r = 2A, t = Bπ. • Collapse at θ = 2π, r → 0, t = 2πB

Spherical Collapse • Parametric Solution:

r/A

2

t/πB

r/A

1

0.5

collapse

3

turn around

4

1

θ/π

1.5

2

Correspondence • Eliminate cosmological correspondence in A and B in terms of enclosed mass M 4π 3 4π 3 3H02 M= r 0 Ω m ρc = r 0 Ωm 3 3 8πG • Related as A3 = GM B 2 , and to initial perturbation 1 2 1 4 lim r(θ) = A θ − θ θ→0 2 4   1 5 1 3 θ − θ lim t(θ) = B θ→0 6 120 



• Leading Order: r = Aθ2 /2, t = Bθ3 /6 A 6t r= 2 B 

2/3

• Unperturbed matter dominated expansion r ∝ a ∝ t2/3

Next Order • Iterate r and t solutions 3

"

1 6t θ lim t(θ) = B 1 − θ→0 6 20 B 6t θ≈ B 

1/3 "



1 6t 1+ 60 B 

2/3 #

2/3 #

• Substitute back into r(θ) 2

θ r(θ) = A 2

6t = A B 

2

θ 1− 12 2/3 "

!

1 6t 1− 20 B 

" 2/3

= (6t)

1/3

(GM )

2/3 #

1 6t 1− 20 B 

2/3 #

Density Correspondence • Density ρm =

M 4 3 πr 3 "

1 3 6t = 1+ 2 6πt G 20 B 

2/3 #

• Density perturbation ρm − ρ¯m 3 6t δ≡ ≈ ρ¯m 20 B 

2/3

• Time → scale factor t =

2

3/2 a 1/2

3H0 Ωm 2/3 3  δ = a 4BH0 Ω1/2 m 20

Spherical Collapse Relations • A and B constants → initial cond. 1

B =

 1/2

2H0 Ωm 3 ri A = 10 δi

3 ai 5 δi

3/2

• Scale factor a ∝ t2/3  2/3 

3 a= 4

3 ai (θ − sin θ)2/3 5 δi 

• At collapse θ = 2π  2/3 

acol

3 = 4

3 ai ai 2/3 (2π) ≈ 1.686 5 δi δi 

• Perturbation collapses when linear theory predicts δc ≡ 1.686

Virialization • A real density perturbation is neither spherical nor homogeneous • Shell crossing if δi doesn’t monotonically decrease • Collapse does not proceed to a point but reaches virial equilibrium U = −2K 1 E = U + K = U (rmax ) = U (rvir ) 2 1 rvir = rmax 2 since U ∝ r−1 . Thus θvir = 32 π • Overdensity at virialization ρm (θ = 3π/2) = 18π 2 ≈ 178 ρ¯m (θ = 2π) • Threshold ∆v = 178 often used to define a collapsed object

Virialization • Schematic Picture:

3

turn around

4

r/A

2

r/A

1

0.5

t/πB virialization

1

θ/π

1.5

2

The Mass Function • Spherical collapse predicts the end state as virialized halos given an initial density perturbation • Initial density perturbation is a Gaussian random field • Compare the variance in the linear density field to threshold δc = 1.686 to determine collapse fraction • Combine to form the mass function, the number density of halos in a range dM around M . • Halo density defined entirely by linear theory • Fudge the result to get the right answer compared with simulations (a la Press-Schechter)!

Press-Schechter Formalism • Smooth linear density density field on mass scale M with tophat 3M R= 4π 

1/3

• Result is a Gaussian random field with variance σ 2 (M ) • Fluctuations above the threshold δc correspond to collapsed regions. The fraction in halos > M becomes 1 √ 2πσ(M )

Z

∞

δc

2

δ dδ exp − 2 2σ (M )

!

ν 1 = erfc √ 2 2

!

where ν ≡ δc /σ(M ) • Problem: even as σ(M ) → ∞, ν → 0, collapse fraction → 1/2 – only overdense regions participate in spherical collapse. • Multiply by an ad hoc factor of 2!

Press-Schechter Mass Function • Differentiate in M to find fraction in range dM and multiply by ρm /M the number density of halos if all of the mass were composed of such halos → differential number density of halos dn d ln M

ρm d ν = erfc √ M d ln M 2 s

=

!

2 ρm d ln σ −1 ν exp(−ν 2 /2) π M d ln M

• High mass: exponential cut off above M∗ where σ(M∗ ) = δc M∗ ∼ 1013 h−1 M

today

• Low mass divergence: (too many for the observations?) dn ∝∼ M −1 d ln M

Top Hat RMS

• RMS density reaches unity at ~8 h-1 Mpc: σ8 measure of amplitude 1010

1012

1

M (h–1 M ) 1014

1016

1018

σR

0.1

0.1

1

10

R (h–1 Mpc)

100

Non-Linear Mass Scaling

• Strong function of amplitude and growth of structure (redshift) 1014

M*(z=0,σ8)

M* (h–1 M )

1013 1012

M*(z,σ8=0.92)

1011

1010 0

1

z, σ8

2

3

Observational Mass Functions • SDSS optically identified clusters (assuming M/L; Bahcall et al 2002)

with cluster X-ray temp. function, sensitive to power amplitude σ82

Counting Halos → Dark Energy • Halo abundance exponentially sensitive to growth rate

Projected Constraints • Studies of M>2.5 x 1014 M • All other parameters known

(Haiman et al. 2000; Hu & Kravstov)

zmax=3 zmax=1.0 zmax=0.7

w

–0.4

–0.6 –0.8

–1

0.6

ΩDE

0.65

Projected Constraints • Studies of M>2.5 x 1014 M (Haiman et al. 2000; Hu & Kravstov) • Local halo abundance known + present day cosmological params zmax=3 zmax=1.0 zmax=0.7

w

–0.4

–0.6 –0.8

–1

0.6

ΩDE

0.65

Projected Constraints • Studies of M>2.5 x 1014 M (Haiman et al. 2000; Hu & Kravstov) • Present day cosmological parameters zmax=3 zmax=1.0 zmax=0.7

w

–0.4

–0.6 –0.8

–1

0.6

ΩDE

0.65

Projected Constraints • Studies of M>2.5 x 1014 M (Haiman et al. 2000; Hu & Kravstov) • Present day cosmological parameters + sample variance zmax=3 zmax=1.0 zmax=0.7

w

–0.4

–0.6 –0.8

–1

0.6

ΩDE

0.65

Extended Press-Schechter Formalism • A region that is underdense when smoothed on the scale M may be overdense on a scale of a larger M • If smoothing is a tophat in k-space, independence of k-modes implies fluctuation executes a random walk Press-Schechter prescription

δ δc

collapsed uncollapsed

M2 R(M)

Extended Press-Schechter Formalism • For each trajectory that lies above threshold at M2 , there is an equivalent trajectory that is its mirror image reflected around δc • Press-Schechter ignored this branch. It supplies the missing factor of 2 equal probability first upcrossing

δ

collapsed uncollapsed

M2

δc

M1 R(M)

Conditional Mass Function • Extended Press-Schechter also gives the conditional mass function, useful for merger histories. • Given a halo of mass M1 exists at z1 , what is the probability that it was part of a halo of mass M2 at z2

(1+z2)δc

δ

(1+z1)δc

M2

M1 R(M)

Conditional Mass Function • Same as before but with the origin translated. • Conditional mass function is mass function with δc and σ 2 (M ) shifted

(1+z2)δc

δ

(1+z1)δc

M2

M1 R(M)

Merger Simulation • Simulation by Andrey Kravstov

Magic “2” resolved? • Spherical collapse is defined for a real-space not k-space smoothing. Random walk is only a qualitative explanation. • Modern approach: think of spherical collapse as motivating a fitting form for the mass function √ 2 2 −p ν exp(−ν /2) → A[1 + (aν ) ] aν 2 exp(−aν 2 /2) Sheth-Torman 1999,

a = 0.75, p = 0.3. or a completely empirical

fitting ρm d ln σ −1 dn = 0.301 exp[−| ln σ −1 + 0.64|3.82 ] d ln M M d ln M Jenkins et al 2001. Choice is tied up with the question: what is the mass of a halo?

Numerical Mass Function • Example of difference in mass definition (from Hu & Kravstov 2002) 10–4

∆180 (wrt mean) ∆666 Jenkins et al

n(>M) (h3 Mpc–3)

10–5

10–6

10–7

10–8

Ωm =Γ =0.15; flat; h=0.65; σ8=1.07 1014

1015

M (h–1 M )

1016

Halo Bias • If halos are formed without regard to the underlying density fluctuation and move under the gravitational field then their number density is an unbiased tracer of the dark matter density fluctuation δn n

!

= halo

δρ ρ

!

• However spherical collapse says the probability of forming a halo depends on the initial density field • Large scale density field acts as “background” enhancement of probability of forming a halo or “peak” • Peak-Background Split (Mo & White 1997)

Peak-Background Split • Schematic Picture: 3 2 1

Enhanced "Peaks"

δc

Large Scale "Background"

0

x

Perturbed Mass Function • Density fluctuation split δ = δb + δp • Lowers the threshold for collapse δcp = δc − δb so that ν = δcp /σ • Taylor expand number density nM ≡ dn/d ln M "

nM

2

(ν − 1) dnM dν δ b . . . = nM 1 + + σν dν dδb

if mass function is given by Press-Schechter nM ∝ ν exp(−ν 2 /2)

#

Halo Bias • Halos are biased tracers of the “background” dark matter field with a bias b(M ) that is given by spherical collapse and the form of the mass function δnM = [1 + b(M )] δ nM • For Press-Schechter ν2 − 1 b(M ) = 1 + δc • Improved by the Sheth-Torman mass function aν 2 − 1 2p b(M ) = 1 + + δc [1 + (aν 2 )p ] δc with a = 0.75 and p = 0.3 to match simulations.

Numerical Bias • Example of halo bias from a simulation (from Hu & Kravstov 2002) 8

=

PS-based: MW96, J99 ST99 6

4

2

1014

M180 (h–1 M )

1015

What is a Halo? • Mass function and halo bias depend on the definition of mass of a halo • Agreement with simulations depend on how halos are identified • Other observables (associated galaxies, X-ray, SZ) depend on the details of the density profile • Fortunately, simulations have shown that halos take on a near universal form in their density profile at least on large scales.

NFW Halo • Density profile well-described by (Navarro, Frenk & White 1997) ρs ρ(r) = (r/rs )(1 + r/rs )2 102 101

ρ/ρs M/Ms

1 10-1 10-2 10-3 10-4 10-2

10-1

1

r/rs

101

102

Transforming the Masses • NFW profile gives a way of transforming different mass definitions 10–4

∆180 (wrt mean) ∆666 Jenkins et al. Rescaled

n(>M) (h3 Mpc–3)

10–5

10–6

10–7

10–8

Ωm =Γ =0.15; flat; h=0.65; σ8=1.07 1014

1015

M (h–1 M )

1016

Lack of Concentration? • NFW parameters may be recast into Mv , the mass of a halo out to the virial radius rv where the overdensity wrt mean reaches ∆v = 180. • Concentration parameter rv c≡ rs • CDM predicts c ∼ 10 for M∗ halos. Too centrally concentrated for galactic rotation curves? • Possible discrepancy has lead to the exploration of dark matter alternatives: warm (m ∼keV) dark matter, self-interacting dark-matter, annihillating dark matter, ultra-light “fuzzy” dark matter, . . .

Incredible, Extensible Halo Model • An industry developed to build semi-analytic models for wide variety of cosmological observables based on the halo model • Idea: associate an observable (galaxies, gas, ...) with dark matter halos • Let the halo model describe the statistics of the observable • The overextended halo model?

The Halo Model • NFW halos, of abundance nM given by mass function, clustered according to the halo bias b(M ) and the linear theory P (k)

2

∆ (k)

• Power spectrum example: 10

4

10

3

10

2

10

1

10

0

10

non-linear total linear

halo correlation

-1

10

halo profile -2

10

-1

10

0

-1

k (h Mpc )

10

1

Weak Lensing and the Halo Model Power spectrum of shear divided into the halo masses that contribute Non-linear regime dominated by halo profile / individual halos increased power spectrum variance and covariance

10–4

14

10–5

l(l+1)Clεε /2π

• •

13 12 log(M/M )