Large Scale Structure
of the Universe
Evolution of the LSS – a brief history
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=2 -Most galaxies have formed; they are bright with stars;
this is also the epoch of highest quasar activity;
galaxy clusters are forming. In LCDM 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 Density Fluctuation
Power Spectrum
A different convention:
plot P(k)k3
z=1
z=1200
matter
domination
log(t)
lambda
domination
Evolution of density fluctuations: the set-up
P(k)
recombination; production of CMB
k
matter-radiation equality
inflation
radiation
domination
z=4 x 103
z>>1010
lambda-matter equality
end of inflation
P(k)
k
Planck time
Growth rate of a density perturbation depends on epoch log(rcomov)
(i.e. what component dominates global expansion dynamics at that time),
and whether a perturbation k-mode is super- or sub-horizon.
Linear growth of density perturbations:
Super-horizon, w comp. dominated, pre & post recomb.
d

d

K / a2

8G / 3
 (a  )
2
1
fluid pressure is not important
on super-horizon scales, so it
makes no difference whether
recombination has taken place
or not.
log(t)
8G
 , for K  0
3
8G
K
H2 
(   d)  2
3
a
H2 
a  t 2 /[3(1 w)]
a  3(12 w) t 2 /[3(1 w)]1
3(1 w)


a
a
2
H 
a
3(1  w)t
Ht  3(12 w)
MD
 0  a 3
growth of all
perturbations
CMB
MRE
dk  a
inflation
log(t)
Friedmann eq: different patches
of the Universe will have slightly
different average densities and
curvatures – at a fixed H:
log(rcomov)
RD
0  a 4
growth of all
perturbations
CMB
MRE
d k  a2
inflation
log(rcomov)
Linear growth of density perturbations:
Sub-horizon, radiation dominated, pre recombination
Jeans linear perturbation
analysis applies:
dk  2 Hdk  [ c a k  4G ] d k  0
2
2
s
0
log(t)
2
CMB
MRE
inflation
log(rcomov)
a  t1 / 2
a  12 t 1/ 2
1
a
H 
2t
a
Ht  12
dark matter has no pressure of its own;
it is not coupled to photons, so there is
no restoring pressure force.
2
cs k 2



d k  2 Hd k  [ a 2
 4G 0 ] d k  0
zero
radiation dominates, and
because radiation does
not cluster  all dk=0…
dk  1t dk  0
d k  A ln(t )  B
…but the rate of change
of dk’s can be non-zero
growing “decaying”
mode
mode
DM growing mode solution d k  2 ln( a)
Linear growth of density perturbations:
Sub-horizon, matter dominated, pre & post recomb.
dark matter has no pressure of its own;
a  t2/3
it is not coupled to photons, so there is
a  23 t 1 / 3
no restoring pressure force.
a
2
2 2
H 
c
s
k  2 Hdk  [ k2  4G 0 ] d k  0
a
3t
d
a
Ht  23
zero
also, can assume that total density
is the critical density at that epoch:
Jeans linear perturbation
analysis applies:
dk  2 Hdk  [ c a k  4G ] d k  0
2
2
s
0
log(t)
2
CMB
MRE
inflation
log(rcomov)
  0  38HG
4G0  32t 2
2
dk  34t dk  32t 2 d k  0
d k  At
2/3
growing
mode
 Bt
1
decaying
mode
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
DM growing mode solution d k  a
Linear growth of density perturbations:
Sub-horizon, lambda dominated, pre & post recomb.
Jeans linear perturbation
analysis applies:
dk  2 Hdk  [ c a k  4G ] d k  0
2
2
s
0
log(t)
2
CMB
MRE
2
H 2  H 0 [ ]

H  const
a  eHt
dark matter has no pressure of its own;
it is not coupled to photons, so there is
no restoring pressure force.
2
cs k 2



d k  2 Hd k  [ a 2
can assume the amplitude of
perturbations is zero, because
lambda, which dominates,
does not cluster:
 4G 0 ] d k  0
zero
dk  0
inflation
log(rcomov)
dk  2 Hdk  0
d k  A  Be2 Ht
“growing” decaying
mode
mode
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
DM "growing" mode soln d k  const
log(t)
Linear growth of density perturbations:
dark matter, baryons, and photons
d  const
dark matter d  a  t 2 / 3
baryons fall in to potential wells
photons
freestream
CMB
dark matter d  a  t 2 / 3
baryon - photon fluid oscillates
dm d  const or ln(t )
bar - phot fluid
oscillates
inflation
dark matter d  a  t 2 / 3
baryons
same
photons
same
MRE
dark matter d  a 2  t
baryons
same
photons
same
log(rcomov)
Evolution of matter power spectrum
d  const
log(t)
Now
z=1
On sub-horizon scales
DM d  a  t 2 / 3
CMB
MRE
DM d  const
or d  ln(t )
DM d  a 2  t
EoIn
log(rcomov)
log(k)
growth of structure
begins and ends with
matter domination
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  a 2  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
Transfer
Functions
Transfer function is defined
by this relation:
PMRE (k )  Tk  Pinf (k )
2
Peacock; astro-ph/0309240
Growth of large scale structure
Dark Matter density maps from N-body simulations
Lambda (DE)
spatially flat
matter=0.3
fractional
overdensity
~const
Standard
spatially flat
matter=1.0
fractional
overdensity
~1/(1+z)
350 Mpc
the Virgo Collaboration (1996)
Growth of large scale structure
In linear theory gravitational potential decays if DE or negative curvature
dominate late time expansion
Lambda (DE)
spatially flat
matter=0.3
gravitational
potential
~(1+z)
Standard
spatially flat
matter=1.0
gravitational
potential
~const
350 Mpc
the Virgo Collaboration (1996)
Late Integrated Sachs-Wolfe (ISW) Effect
If a potential well evolves as a photon transverses it, the photon’s energy will change
 T 
 d
 2 


 T  ISW
Sachs & Wolfe (1967) ApJ 147, 73
Crittenden & Turok (1996) PRL 76, 575
photon gains
energy after
crossing a
potential well
Energy
potential well
Look for correlation between CMB temperature fluctuations and nearby structure.
Detection of late ISW effect in a flat universe is direct evidence of Dark Energy
Detecting late ISW
Late ISW is detected as a cross-correlation, CCF
on the sky between nearby large scale structure
and temperature fluctuations in the CMB.
HEAO1 hard X-rays full sky
median z~0.9
NVSS 1.4 GHz nearly full sky
radio galaxies; median z~0.8
Lines are LCDM predictions, not fits to data
Note: points are highly correlated
Boughn & Crittenden (2005) NewAR 49, 75, astro-ph/0404470
Baryonic Acoustic Oscillations
One wave around one center:
Wave starts propagating at Big Bang;
end at recombination. The final length
is the sound crossing horizon at recomb.
(Change of color means recombination.)
Many waves superimposed
Matter power spectrum - observations
Baryonic Acoustic Oscillations (BAO)
from
k-space to
real space
Eisenstein et al. astro-ph/0501171
Percival et al. astro-ph/0705.3323
gal. corr. fcn.
SDSS and 2dF galaxy surveys
BAO bump
Narrow feature:
standard ruler
comoving r (Mpc/h)
(sound crossing horizon at recombination)
Recombination affects
the matter power spectrum too
galaxy correlation function
Luminous SDSS red galaxies, z ~ 0.35
mh2=0.12,
Eisenstein et al. astro-ph/0501171
0.13,
0.14
mh2=0.130+/-0.011
sound horizon size at
recombination
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
time 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)
2
define M * : d c  2 AM *
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 : characteri stic 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 cos 2   A sin 2  
A
 A3

  B 2 (1  cos  ) 2  B 2 r 2
B 2 (1  cos  )3

Check :
A3
M 
GB 2
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 |   (T )
3 1/ 2
tdyn  (
)
16G
tcool  tdyn  T   1/ 2  (T )
tcool 
depends on the
cooling fcn  (T )
Cosmological Parameters
From the number density
of galaxy clusters can obtain:
 8m0.6  0.45
Measurements of global geometry:
std candles – Supernova Type Ia
std rulers – Baryonic Acoustic
Oscillations:  m ,  
CMB – a test of flatness
a test for Lambda – late ISW effect