Transcript Structure Formation - University of Minnesota

Formation and Evolution
of the
Large Scale Structure
II
Large Scale Structure of the Universe
Somewhat after recombination -density perturbations are small on nearly all spatial scales.
Dark Ages, prior to z=10 -density perturbations in dark matter and baryons grow;
on smaller scales perturbations have gone non-linear, d>>1;
small (low mass) dark matter halos form; massive stars
form in their potential wells and reionize the Universe.
z=3 -Most galaxies have formed; they are bright with stars;
this is also the epoch of highest quasar activity;
galaxy clusters are forming. Growth of structure on large
(linear) scales has nearly stopped, but smaller non-linear
scales continue to evolve.
z=0 -Small galaxies continue to get merged to form larger ones;
in an open and lambda universes large scale (>10-100Mpc)
potential wells/hill are decaying, giving rise to late ISW.
Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html
Matter Power Spectrum:
from inflation to today
A different convention:
plot P(k)k3
Evolution of matter power spectrum
d  const
log(t)
Now
z=1
P(k)
k
DM d  a  t
P(k)
2/3
CMB
k
P(k)
MRE
DM d  const
or d  ln(t )
DM d  a2  t
P(k)
P(k)
k
P(k)
k
log(rcomov)
log(k)
baryonic oscillations
appear – the P(k)
equivalent of CMB
T power spectrum
k
k
EoIn
high-k small scale
perturbations grow
fast, non-linearly
sub-horizon perturb.
do not grow during
radiation dominated
epoch
Harrison-Zeldovich
spectrum P(k)~k
from inflation
P(k) gives rms density fluctuations
on various spatial scales
V 3
k  P(k )  2 (k )
2
2
2dF
from observations :
  k 3  P (k )  0.08
from observations :
 8  k 3  P ( k )  0 .9
r  sound hor. at recomb.
r  200 h-1 Mpc
k  2 / r  0.03h Mpc-1
r  8h1Mpc
k  2 / r  0.75h / Mpc
P(k) gives rms density fluctuations
on various spatial scales
V 3
k  P(k )  2 (k )
2
2
from observations :
 8  2 (k )  0.9
from observations :
  2 (k )  0.08
r  sound hor. at recomb.
r  200 h-1 Mpc
k  2 / r  0.03h Mpc-1
Peacock; astro-ph/0309240
r  8h1Mpc
k  2 / r  0.75h / Mpc
Observed fluctuations in temperature
and matter density
Angular size of the
sound crossing horizon
at recombination
Comoving size of the
horizon at matter
radiation equality
Tegmark et al.
These two power spectra
are the main statistical
descriptors of the large
scale structure of the
Universe. Both are fully
consistent with what is
usually known as the
concordance model:
WL0.73
WDM=0.23
Wbar=0.04
flat, ni=1
Quantifying LSS on linear and non-linear scales
The power spectrum
quantifies clustering
on spatial scales larger
than the sizes of
individual collapsed
halos
The 2pt correlation fcn
is another way to quantify
clustering of a continuous
fluctuating density field, or
a distribution of discrete
objects, like collapsed DM
halos.
these are Fourier transforms
of each other
The mass function of
discrete objects is
the number density of
collapsed dark matter
halos as a function of
mass - n(M)dM.
This was evaluated
analytically by
Press & Schechter (1974)
Picture credit: A. Kravtsov, http://cosmicweb.uchicago.edu/filaments.html
Internal structure of
individual collapsed halos:
one can use an analytical
description for mildly nonlinear regimes, but numerical
N-body simulations are
needed to deal with fully
non-linear regimes.
Correlation functions
Two-point correlation function is a
measure of the degree of clustering.
It is a function of distance r only,  (r ) .
Suppose we are told that
 (r )  (r / 5Mpc)-1.8.
What does that mean?
If you are sitting on a galaxy, the probability dP
that you will find another galaxy in a volume
dV a distance r away from you is given by
dP  n dV [1   (r )]
where n = average number density
of galaxies.
best fit line
Alternative definition:
take two small volumes distance r apart;
the joint probability that you will find a galaxy
in either one of the two dV volumes a distance r
apart is given by.
dP  n2 dV1 dV2 [1  (r )]

dV2
r
r
dV
dP is the number of galaxies you expect to find
in a volume dV.
dV1
Estimating 2pt correlation function
How does one calculate the 2pt correlation function given a distribution
of galaxies is space? – Count the number of pairs of galaxies for every  ( r )
value of separation r. Then divide this histogram by the number of pairs
expected if the spatial distribution of galaxies were random, and subtract 1.
clustered
random
N real (r )
1
N random (r )
# pairs
 (r )
Correlation functions measure
the fractional excess of pairs
compared to a random distrib.

separation r
1
0
-1
separation r
Correlation fcn and correlation length
linear vertical scale
r0
N real (r )
 (r ) 
1
N random (r )
Correlation length r0 is defined as the scale where  ( r0 )  1
so expect twice the number of galaxies compared to random.
For galaxies, correlation length is ~5 Mpc,
for rich galaxy clusters it is ~25 Mpc.
2pt correlation function and power spectrum
k 3  P(k )  2 (k )
linear vertical scale
r0
Power spectrum is a Fourier transform
of the correlation function:
 ( r )   d 3k P ( k )e

ik r
 (r )  4  dk k 2 P(k ) sin(krkr)
r (Mpc) 
Mass function of collapsed halos: Press-Schechter
Smoothly fluctuating density field; randomly scattered
equal volume spheres, each has some overdensity d.
Some of these volumes will have a large enough
overdensity (dc>1.69) that they will eventually
collapse and form gravitationally bound objects.
What is the mass function of these objects at any
given cosmic epoch?
d3
d2
d1
d4
d5
rms dispersion in mass, or, equivalently,
overdensity d, in spheres of radius R
M
2
 ( A  M  ) 2
t ime dependence:  M  a  t 2 / 3
Press-Schechter (1974) main assumption:
the fraction of spheres with volume V having
overdensity d is Gaussian distributed
dx
d M  1

dM
2A
fraction of volumes
M2 
M  M 2
large R
medium R
small R
0
1.69
d
these spheres
collapse
Mass function of collapsed halos: Press-Schechter
The fraction of spheres that will eventually collapse is
The fraction of spheres that have just collapsed ( of all possible M, but same dc)
define M * : d c  2 AM *
2
2
How much mass in every unit of volume is contained in these objects?
How many of the collapsed objects are there?
power
law
exponential
M per vol 
dP
dM  
dM
Note : characteristic mass M *
is time dependent, through A :
M *  A1/( 2 )  t 2 /( 3 )  a1/ 
fraction of volumes
Press-Schechter halo mass function
large R
medium R
small R
0
1.69
d
Press-Schechter vs. numerical simulations:
solid red lines: simulations
blue dotted: Press-Schechter
green dashed: extended Press-Schechter
(takes into account non-sphericity
of proto halos.)
small R
medium R
large R
Collapse of individual DM halos
Hubble
expansion
In comoving coordinates
a sphere, centered on a local
overdensity shrinks in time;
Hubble expansion is getting
retarded by the overdensity.
At some point, the sphere’s
expansion stops (turn-around),
and the sphere starts to
collapse.
local
overdensity
rm
constant time
rm
M
Halos collapse
from inside out.
Collapse of individual DM halos
radius
at smaller radii
(larger overdensities)
halo is virialized
turn-around;
overdensity decouples
from the Hubble flow
d
 4.5

reaches asymptotic
radius
turn-around radius moves
out with time; halos collapse
and virialize from inside out.
d
 200

time
parametric
equations non-linear evolution,
shell-crossing,
apply
relaxation
Equation of motion r  GM (r ) / r 2
An overdense spherical region wil l eventually collapse,
so we can think of it as a separate closed Universel et.
Parametric solution r  A(1  cos )
t  B(  sin  )
Turn  around, rmax happens at   
KE+0.5PE=0
shell-crossing
virialized density excess
d  1  5.5  (2)3  (2) 2  176  200
at turn- internal external
around density density
increase decrease a  t 2 / 3   1/ 3
t 2   1
Collapse of individual halos:
the algebra leading to d=4.5 at turn-around
r  GM (r ) / r 2
r  A(1  cos )
t  B(  sin  )
 A3
r  2 2
Br
&
GM
r   2
r
dt
1
 B(1  cos )  
d

dr d
A sin 
r 

 A sin    
d dt
B(1  cos )
d  A sin   d  A cos
A sin 2   d



r 
d  B(1  cos )  dt  B(1  cos ) B(1  cos ) 2  dt
 A cos  A cos2   A sin 2  
A
 A3

  B 2 (1  cos ) 2  B 2 r 2
B 2 (1  cos )3

Check :
A3
M 
GB2
3M
3 A3
3
 (t ) 


4 r 3 GB 2 4 r 3 GB 2 4 (1  cos )3
3(  sin  ) 2
3(6G0 ) (  sin  ) 2 9 (  sin  ) 2



 
4G
4Gt 2 (1  cos )3
(1  cos )3 2 0 (1  cos )3
 (t ) 9 (  sin  ) 2
 
0 2 (1  cos )3
 (tmax ) 9  2 9 2
at turn  around    , and d  1 
 

 5.5
0
2 8
16
Einstein  de Sitter
background density :
2
0  3H , Ht  2
8G
3
cosmic time :
4
1
t2 

9 H 2 6G0
Why are there no galaxies with M>1013Msun ?
So far we have been mostly concerned with dark matter halos. The distribution
DM halos in mass is continuous from ~109 to ~1015 Msun. But, DM halos with
M>few x 1012Msun are not observed to host galaxies, only clusters of galaxies. Why?
~ 1011 M sun halo
~ 1014 M sun halo
about 1/10 of virial
radius, r200 for both
Cooling curve diagram
gas has cooled
gas has not cooled
Whether a galaxy forms in a given halo is
determined by the rate of gas cooling.
Etotal
3nk T
 2 B
| dE / dt |  L (T )
3 1/ 2
tdyn  (
)
16G
tcool  tdyn  T   1/ 2 L (T )
tcool 
depends on the
cooling fcn L (T )
Cosmological Parameters
From the number density
of galaxy clusters can obtain:
 8Wm0.6  0.45
Measurements of global geometry:
std candles – Supernova Type Ia
std rulers – Baryonic Acoustic
Oscillations: Wm , WL
CMB – a test of flatness
a test for Lambda – late ISW effect