12 Dark Spherical Collapse model of Seer Rail well 45¥ · PDF file* Spherical Collapse model of halo formation ... -equation of motion: ITT =-GYES ( RH: site of the region ... But

12.

Dark matter halo

* Spherical Collapse model of halo formation

Assuming Einstein de - Seer Universe ( Rail ) ( result holds for general case as well.

)

o An isolated spherical over denseregion of she R; & density contrast Oi

Outside → spherically symmetric distribution of matter .

⇒ Birkhoff thm .

: dynamics of the regiondetermined only by inside material

.

ma' GR Version of Newton 's spherical shell theorem

.

Total mass M =45¥echo , )

A = cosmic mean density @ ti ,= 6¥42 H ;

= ¥- equation of motion : ITT = - GYES ( RH : site of the region at time t )

= - 45¥

RIICHODIE

= . Hi¥7 ( Ita )3

energy equation : IR - of = Ile - the top =E

if Kzttirifo⇒ expanding forever Cno collapseTHY? )

Ri < Hike ⇒ reallapse .

a ten HIRICHQ ) = -

1¥43C Hoi )

⇒ Rma = tthe.DK '

Set Ri=HiR;C following the Hubble How)

→ [ = - HMI o ; ; Rmax = ( t¥)R ,± Rta Cta . turnaround )

- Solution of Eom → Cycloid ! !

R - Rt÷ ( l - Gso ) i t : TTE ( O - smo )

R - Rte ( l - aso ) i t : t±I(o - sino ) → li = - 9¥

e- Itza ¥eetI÷ ){k = dodgy = .tt#Id*o = - 99ft # ,

.

" 84M¥ '

' THE

- over density OH

It OH =FCHod÷t.at#sEXf=u+o..sftEMtt3..a+oil:I*oMETosI3teioEIi+ooi

,

T

0¥Z T Font Rea = HE Ct coso , )

⇒ coso,

= II ,smo ; - T÷

to ;

i) at early times Ct=tD

to ±Itf +9T ) ~ 1+302

¥,

= effort :p )shoo '

. ¥[at⇒By= seat:*linear solution ! )

ii ) at turn around

Hota ± I . ¥ ± bets → already non -linear ! !

iii)

0=2'T → collapse to f= o ( 12=0 ) !

But in reality .the

region will be unatzed C via violent relaxation )mm

2Ekt - Ep

E

= Ep + Et = - Ek = Es ⇐ Ep ( at turn around ) = EE ( at viridization )

Epa YR ⇒ Rta '

' 2 Rvirid

Assuming vindication happens at 0=27 ( tuir = Hea )

Hair = u+o⇒fI÷Mr÷f=C#dftt÷HF÷f

= FIX 4×8 = 178 ( vkkl over density ! !)

~-2°o_! !

Corresponding" linear density contrast

' '

= Ea(#%=EoiYdIk0Pfns±EoiHoiP 's

cosmos"

a }o , HOFF Co - and"

= }(}d% ( 0*0513

at vlvklitatim ,&

A = Este) "

ftp.T (3125's

=t6864

±-

- Collapsing threshold ( critical density ) in other world models

i ) open CDM Universe ( data - - GMT : but Hubble rate different )

A = 1.686 KmC tur ) ]00185

it flat . BCDM Universe ( ¥T .

- . Gfft÷r )

fc = 1.686 [ Rnctur ) ]000$

: fat 1.686 for all realistic world models !

- Implication :

If dark matter halos form through spherical collapse , instea¥

following Its nonlinear collapse usingN -

body simulations,

we can

approximately find the collapsed spherical halos from the linear density held .

That is, if few , -4 = fc ( ~ 1.686) a spherical

regionforms a halo

.

�1�

00 At early times,

( fi > = Rs (xD% ))the amplitude of fluctuation is small

,VWHNNHMM& chance of forming halos of given mass

is tower.

0000 At later times,

R 'is larger ,MMNNNM

→ chance of forming halo is higher .

�2� At a given time, density fluctuation of spherical region

MR )

is smaller for large R ( high mass halos) .

⇒ high mass halos are ra±mm

* Recipe of findinghalos from

given linear density held.

Six ) face

5.1.2. Press-Schechter formalismLet us start the discussion from recapping the original derivation of Press-Schechter mass function (Press

& Schechter, 1974).Here, the key assumptions are A) the initial matter density �(x) contrast follows Gaussianstatistics and B) a Lagrangian (initial) region (volume V ) with linear density contrast (extrapolated to theepoch of interest) exceeding the threshold value �c are self-bounded to form a galaxy. Then, the fraction ofLagrangian volume belongs to the galaxies of mass greater than M = ⇢̄MV is given by

PG(� > �G) =1

p

2⇡�(R)2

Z 1

�c

d� exp

�1

2

�2

�(R)2

�

=1

2erfc

�cp2�(R)

�

. (5.11)

Here, the complementary error function is defined as

erfc = 1 � erf(x) =2p⇡

Z 1

xe�u2

du, (5.12)

R is the Lagrangian radius corresponding to the halo mass M

M = 3.1389 ⇥ 1011✓

⌦m

0.27

◆ ✓

R

[Mpc/h]

◆3

M�/h, (5.13)

and �(R) is a root-mean-squared value of the linear density fluctuation smoothed over radius R:

�2(R) =1

2⇡2

Z 1

0

dk k2P (k)W2R(k) . (5.14)

As the Lagrangian volume fraction is equivalent to the fraction of total mass enclosed in the halos of massgreater than M , with the mass function n(M) we write

PG(> M) = PG(� > �c) =1

⇢̄M

Z 1

MdM 0M 0n(M 0) =

1

2erfc

�cp2�(R)

�

, (5.15)

from which we find an expression for the mass function:

n(M) = � ⇢̄MM

dPG(> M)

dM. (5.16)

One problem here is, however, that integrating over masses inside all halos recovers only half of the totalmass:

Z 1

0

dMMn(M) = �⇢̄M

Z 1

0

dMdPG

dM= �⇢̄M [PG(R = 1) � PG(R = 0)] =

1

2⇢̄M . (5.17)

Here, we use the hierarchical density field without any cuto↵: �(R ! 0) ! 1, and �(R ! 1) ! 0. Thatis, the mass function we calculated above is not properly normalized. Press and Schechter [40], therefore,had to introduce an ad hoc, ‘fudge’ factor of two which reads the Press-Schechter mass function:

nPS(M) = �2⇢̄MM

dPG

dM=

⇢̄MM2

r

2

⇡⌫e�⌫2/2

�

�

�

�

dln�(R)

dlnM

�

�

�

�

, (5.18)

where ⌫ = �c/�(R) is the significance of the critical density relative to the r.m.s. variance of matterfluctuations In the literature one often finds the halo mass function in the form of

n(M) =⇢̄MM2

⌫f(⌫)

�

�

�

�

dln�(R)

dlnM

�

�

�

�

, (5.19)

Here, ⌫f(⌫) is the multiplicity function. In Press-Schechter theory, this is

⌫fPS(⌫) =

r

2

⇡⌫e�⌫2/2 . (5.20)

In principle, ⌫f(⌫) could depends on variables other than the peak significance. This is the case of densitypeaks for instance (see §6). However, when the multiplicity function depends solely on ⌫, the halo massfunction can be scaled to a universal (self-similar) form which is independent of cosmology, redshift andpower spectrum (Sheth & Tormen, 1999; Jenkins et al., 2001)

52

mm . " ;:÷ItiE.tw?uEnYEe*#Q⇒ number density of smoothedregions

= Yv=£:- 4¥ Rim

ja , ,slutty

( Wrck )=3PtgkRX ) as

- Press - Schechter mass function & Simulated mass function

⇒ Sheth - Tomen mass function C Sheth & Tomen 1999)

ofin =AF¥[ iff ] EH ( d= 0.707, Pio . } )

from f mean )dm = F → At 0.3222 ( PS ⇒ di 1

, p - o,A=k )

Note :

5TH defines

w= ( %)'

!

E) Jenkins et .

al. God )

ST

:

:

miFOF ' friends . of - friends )

solid line : mm ) = ¥slddhudm/ far )

& FCM ) = 0.301 exp [ . llnot +0.641388 ]

( -0.96 E but ' E 1.0)

* Universality of mass fn.

non ) = ni rfu ) I ghfmI|

with his GIdepending on D= £4, regardless of cosmology

,red shift

,. . .

.

( Jenkins +2

:fu) = 0.315 ( - 1 but '+0.61138 )

( -1.2 Elmo '

'

E 1.05 )

*

"

The"

factor of two.

psmoothed

dens'tfield

bn PS theory , we identify PCS >fc ) as the fraction of initial

Lagrangianvolume which belongs

¥halos of mass M > 455123.

→ result = normalization off by afactor of 2

.

fodmmf Edelmann ) = ts ! !

.

summingover all mass in halo

=

( why? )

%÷%'%OCR , ) 71.68

{o ( k ) < 1.68

⇒ excursion set theory ( Bond et al . 2001 )

halo of mass m forms at z if" the trajectoriesthat are above the threshold Scots at some mass scale

mnmtdm but are below ACA at ALL LARGER VALUES of m ?

Trajectories:p

CH) = fd3y fcg , Way ,⇒ Sick) - SCKTW

,#

you To ktsmoothingkernel

smoothed original

density fielddensity held

smoothing kernels :

:) sharp - k filter WRCA '

- Iha (3519%0) : Wrdg = 04 - KR)

ii ) spherical toftttwpcx , =*÷* of . Ie) ; WRCH =3%t"¥

ii :) Gaussian AlterWry = kw E

" ¥ "; nkck , = e

KM "

first

Crossing! !

A ± 1.68 No

decrease the

smoothing scale→

Statistics of excursion set :

( Racxtscxh . fdj÷,fifty (

snake;) eikixe:& '

i ff¥pfcffB÷ ( a'

Psck ) SKETE 's Wickwick ; eid#' *

= ffH÷g Psck) Wedd WITH

If using sharp - k Alter,

*

< s ,#sa# ) = fd¥( +32K¥) ( ray

why ? 4¥ - fkcdttkgsckyeikx

:G ,set xto ⇒ fi . fkcftftsaisten'-f¥'

gdjknsa,]⇒% '

' Atf.k" "

east ,s# ,

mm

only depend on SCID

between ktlh & k=YRtp

⇒ Markovian process .

f

£wnmMm - £ - ( f . So ) - 2£ 'S

°R

%

%⇐Press -

Schechter

they£wrmMhd- £ - ( f . So ) - 2£ 'S

° €%"÷÷±*ig¥±:* "

inthe Ps

-

calculation.

( HH =% ) : PS theory misses a factor of 2 ! !