1. Large Scale Structure of the Universe

2. 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,

3. Matter Density Fluctuation Power Spectrum
P(k)~kn Harrison-Zel’dovich n=1
A different convention:
plot P(k)k3

4. Evolution of density fluctuations: the set-up
z=1200
z=4 x 103
z=1
z>>1010
log(t)
log(rcomov)
lambda-matter equality
recombination; production of CMB
matter-radiation equality
end of inflation
Planck time
domination
matter
radiation
lambda
infla-
tion
sub-horizon
super-horizon
P(k) k
P(k) k
Growth rate of a density perturbation depends on epoch
(i.e. what component dominates global expansion dynamics at that time),
and whether a perturbation k-mode is super- or sub-horizon.

5. Linear growth of density perturbations: Super-horizon, w comp
Linear growth of density perturbations: Super-horizon, w comp. dominated, pre & post recomb.
fluid pressure is not important
on super-horizon scales, so it
makes no difference whether
recombination has taken place
or not.
Friedmann eq: different patches
of the Universe will have slightly
different average densities and
curvatures – at a fixed H:
CMB
MRE
inflation
log(t)
log(rcomov) MD
CMB
MRE
inflation
log(t)
log(rcomov) RD

6. Linear growth of density perturbations: Sub-horizon, radiation dominated, pre recombination
dark matter has no pressure of its own;
it is not coupled to photons, so there is
no restoring pressure force.
Jeans linear perturbation
analysis applies:
log(t)
zero
CMB
radiation dominates, and
because radiation does
not cluster  all dk=0…
MRE
inflation
log(rcomov)
…but the rate of change
of dk’s can be non-zero
growing “decaying”
mode mode

7. Linear growth of density perturbations: Sub-horizon, matter dominated, pre & post recomb.
dark matter has no pressure of its own;
it is not coupled to photons, so there is
no restoring pressure force.
Jeans linear perturbation
analysis applies:
log(t)
zero
also, can assume that total density
is the critical density at that epoch:
CMB
MRE
inflation
log(rcomov)
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
growing decaying
mode mode

8. Linear growth of density perturbations: Sub-horizon, lambda dominated, pre & post recomb.
dark matter has no pressure of its own;
it is not coupled to photons, so there is
no restoring pressure force.
Jeans linear perturbation
analysis applies:
log(t)
can assume the amplitude of
perturbations is zero, because
lambda, which dominates,
does not cluster:
zero
CMB
MRE
inflation
log(rcomov)
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
“growing” decaying
mode mode

9. Linear growth of density perturbations: Sub-horizon, curvature dominated, pre & post recomb.
dark matter has no pressure of its own;
it is not coupled to photons, so there
no restoring pressure force.
Jeans linear perturbation
analysis applies:
log(t)
can assume the amplitude of
perturbations is zero, because
curvature, which dominates,
does not cluster:
zero
CMB
MRE
inflation
log(rcomov)
Two linearly indep.
solutions: growing
mode always comes
to dominate; ignore
decaying mode soln.
“growing” decaying
mode mode

10. Linear growth of density perturbations: dark matter, baryons, and photons
log(t)
CMB
MRE
inflation
log(rcomov)

11. Evolution of matter power spectrum
log(t)
Now
z=1
On sub-horizon scales
growth of structure
begins and ends with
matter domination
CMB
Evolution of matter power spectrum
MRE
EoIn
log(rcomov)
log(k)

12. Evolution of matter power spectrum
log(t)
P(k)
high-k small scale
perturbations grow
fast, non-linearly
Now
z=1 k
P(k) k
baryonic oscillations
appear – the P(k)
equivalent of CMB
T power spectrum
CMB
P(k)
Evolution of matter power spectrum
MRE k
sub-horizon perturb.
do not grow during
radiation dominated
epoch
P(k) k
P(k) k
Harrison-Zeldovich
spectrum P(k)~k
from inflation
P(k)
EoIn k
log(rcomov)
log(k)

13. Transfer Functions Peacock; astro-ph/0309240
Transfer function is defined
by this relation:
Peacock; astro-ph/

14. Growth of large scale structure
Dark Matter density maps from N-body simulations
Lambda (DE)
spatially flat
Wmatter=0.3
fractional
overdensity
~const
Standard
spatially flat
Wmatter=1.0
fractional
overdensity
~1/(1+z)
350 Mpc
the Virgo Collaboration (1996)

15. Growth of large scale structure
In linear theory gravitational potential decays if DE or negative curvature
dominate late time expansion
Lambda (DE)
spatially flat
Wmatter=0.3
gravitational
potential
~(1+z)
Standard
spatially flat
Wmatter=1.0
gravitational
potential
~const
350 Mpc
the Virgo Collaboration (1996)

16. Late Integrated Sachs-Wolfe (ISW) Effect
If a potential well evolves as a photon transverses it, the photon’s energy will change
Sachs & Wolfe (1967) ApJ 147, 73
Crittenden & Turok (1996) PRL 76, 575
Energy
Energy
Energy
Energy
photon gains
energy after
crossing a
potential well
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

17. 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/

18. 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

19. Matter power spectrum - observations
Baryonic Acoustic Oscillations (BAO)
SDSS and 2dF galaxy surveys
from
k-space to
real space
BAO bump
gal. corr. fcn.
Narrow feature:
standard ruler
(sound crossing horizon at recombination)
comoving r (Mpc/h)
Eisenstein et al. astro-ph/
Percival et al. astro-ph/

20. Recombination affects the matter power spectrum too
sound horizon size at
recombination
Luminous SDSS red galaxies, z ~ 0.35
Wmh2=0.12,
0.13,
0.14
galaxy correlation function
Wmh2=0.130+/-0.011
Eisenstein et al. astro-ph/

21. 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)
Internal structure of
individual collapsed halos:
one can use an analytical
description for mildly non-
linear regimes, but numerical
N-body simulations are
needed to deal with fully
non-linear regimes.
Picture credit: A. Kravtsov,

22. Correlation functions
Two-point correlation function is a
measure of the degree of clustering.
It is a function of distance r only,
Suppose we are told that .
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
where n = average number density
of galaxies.
dP is the number of galaxies you expect to find
in a volume dV.
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. r dV r
dV1
dV2

23. 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
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
# pairs
separation r
clustered
random
random 1 -1
Correlation functions measure
the fractional excess of pairs
compared to a random distrib.
separation r

24. Correlation fcn and correlation length
linear vertical scale
Correlation length is defined as the scale where
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.

25. 2pt correlation function and power spectrum
linear vertical scale
Power spectrum is a Fourier transform
of the correlation function:

26. 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?
rms dispersion in mass, or, equivalently,
overdensity d, in spheres of radius R
large R
medium R
small R d
fraction of volumes
Press-Schechter (1974) main assumption:
the fraction of spheres with volume V having
overdensity d is Gaussian distributed
these spheres
collapse

27. 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)
How much mass in every unit of volume is contained in these objects?
How many of the collapsed objects are there?
power exponential
law

28. Press-Schechter halo mass function
large R
medium R
small R d
fraction of volumes
power law
exponential cut-off
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

29. 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
time rm
constant time rm
Halos collapse
from inside out.

30. Collapse of individual DM halos
at smaller radii
(larger overdensities)
halo is virialized
turn-around;
overdensity decouples
from the Hubble flow
radius
shell-crossing
turn-around radius moves
out with time; halos collapse
and virialize from inside out.
reaches asymptotic
radius
time
parametric
equations
apply
non-linear evolution,
shell-crossing,
relaxation
KE+0.5PE=0
at turn-
around
internal
density
increase
external
density
decrease

31. Collapse of individual halos: the algebra leading to d=4
Collapse of individual halos: the algebra leading to d=4.5 at turn-around

32. 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?
about 1/10 of virial
radius, r200 for both
Cooling curve diagram
galaxies
tcool < tdyn
gas has cooled gas has not cooled
galaxy clusters
tcool > tdyn
Whether a galaxy forms in a given halo is
determined by the rate of gas cooling.

33. Cosmological Parameters
From the number density
of galaxy clusters can obtain:
Measurements of global geometry:
std candles – Supernova Type Ia
std rulers – Baryonic Acoustic
Oscillations:
CMB – a test of flatness
a test for Lambda – late ISW effect