Transcript Document 7312476

The Theory/Observation connection
lecture 3
the (non-linear) growth of structure
Will Percival
The University of Portsmouth
Lecture outline
 Spherical collapse
– standard model
– dark energy
 Virialisation
 Press-Schechter theory
– the mass function
– halo creation rate
 Extended Press-Schechter theory
 Peaks and the halo model
Phases of perturbation evolution
Inflation
Transfer function
Matter/Dark energy
domination
linear
Non-linear
Linear vs Non-linear behaviour
z=0
non-linear
evolution
linear
growth
z=1
z=2
z=3
z=4
z=5
z=0
large scale power
is lost as fluctuations
move to smaller scales
P(k) calculated from Smith et al. 2003, MNRAS, 341,1311 fitting formulae
z=1
z=2
z=3
z=4
z=5
Spherical collapse
 homogeneous, spherical region in isotropic
background behaves as a mini-Universe (Birkhoff’s
theorem)
 If density high enough it behaves as a closed
Universe and collapses (r0)
 Friedmann equation in a closed universe (no DE)
 Symmetric in time
 Starts at singularity (big bang), so ends in singularity
 Two parameters:
– density (m), constrains collapse time
– scale (e.g. r0), constrains perturbation size
The evolution of densities in the Universe
Critical densities are
parameteric equations
for evolution of universe
as a function of the scale
factor a
All cosmological models
will evolve along one of
the lines on this plot
(away from the EdS
solution)
Spherical collapse
Set up two spheres, one containing background,
and one with an enhanced density
Contain equal mass
collapsing perturbation
Radius ap
Background
Radius a
Spherical collapse
For collapsing Lambda Universe, we have Friedmann equation
p is the curvature of the perturbation
And the collapse
requirement
Can integrate numerically to find collapse
time, but if no Lambda can do this analytically
ap
tcoll
Spherical collapse
Problem: need to relate the collapse time tcoll to the overdensity of the
perturbation in the linear field (that we now think is collapsing).
Spherical collapse
Problem: need to relate the collapse time tcoll to the overdensity of the
perturbation in the linear field (that we now think is collapsing).
At early times (ignore DE), can write Friedmann equation as
For the background,
Different for perturbation p
Obtain series solution for a
So that
Spherical collapse
Problem: need to relate the collapse time tcoll to the overdensity of the
perturbation in the linear field (that we now think is collapsing).
Can now linearly extrapolate the limiting behaviour of the perturbation at
early times to present day
Can use numerical solution for tcoll, or
can use analytic solution (if no Lambda)
If k=0, m=1, then we get the solution,
for perturbations that collapse at present
day
Evolution of perturbations
Wm=0.3, Wv=0.7, h=0.7, w=-1
evolution of
scale factor
top-hat collapse
virialisation
Spherical collapse
Cosmological dependence
of c is small, so often
ignored, and c=1.686 is
assumed
Spherical collapse: how to include DE?
If DE is not a
cosmological constant,
its sound speed controls
how it behaves
DE
DM
high sound speed
means that DE
perturbations are
rapidly smoothed
on large scales dark
energy must follow
Friedmann equation
– this is what dark
energy was
postulated to fix!
quintessence has
ultra light scalar
field so high
sound speed
DE
DM
low sound speed
means that large
scale DE
perturbations are
important
The effect of the sound speed
provides a potential test of gravity
modifications vs stress-energy.
Spherical collapse: general DE
For general DE, cannot write down a Friedmann equation for perturbations,
because energy is not conserved. However, can work from cosmology equation
cosmology equation
depends on the
equation of state of dark energy p = w(a) 
homogeneous dark energy
means that this term depends on
scale factor of background
“perfectly” clustering dark
energy – replace a with ap
can solve differential equation and follow growth of
perturbation directly from coupled cosmology equations
Evolution of perturbations
Wm=0.3, Wv=0.7, h=0.7, w=-1
evolution of
scale factor
top-hat collapse
virialisation
Evolution of perturbations
Wm=0.3, Wv=0.7, h=0.7, w=-2/3
evolution of
scale factor
top-hat collapse
virialisation
Evolution of perturbations
Wm=0.3, Wv=0.7, h=0.7, w=-4/3
evolution of
scale factor
top-hat collapse
virialisation
virialisation
 Real perturbations aren’t spherical or homogeneous
 Collapse to a singularity must be replaced by virialisation
 Virial theorem:
 For matter and dark energy
 If there’s only matter, then
comparing total energy at
maximum perturbation size
and virialisation gives
virialisation
The density contrast for a virialised perturbation at the
time where collapse can be predicted for an Einstein-de
Sitter cosmology
This is often taken as the definition for how to find a
collapsed object
Aside: energy evolution in a perturbation
in a standard cosmological constant
cosmology, we can write down a Friedmann
equation for a perturbation
for dark energy “fluid” with a high sound
speed, this is not true – energy can be lost or
gained by a perturbation
the potential energy due to the matter UG and
due to the dark energy UX
Press-Schechter theory
 Builds on idea of spherical collapse and the overdensity field
to create statistical theory for structure formation
– take critical density for collapse. Assume any pertubations
with greater density (at an earlier time) have collapsed
– Filter the density field to find Lagrangian size of
perturbations. If collapse on more than one scale, take largest
size
 Can be used to give
– mass function of collapsed objects (halos)
– creation time distribution of halos
– information about the build-up of structure (extended PS
theory)
The mass function in PS theory
Smooth density field on a mass scale M, with
a filter
Result is a set of Gaussian random fields
with variance 2(M).
For each location in space we have an
overdensity for each smoothing scale: this
forms a “trajectory”: a line of  as a
function of 2(M).
The mass function in PS theory
For sharp k-space filtering, the overdensity of the field at any
location as a function of filter radius (through 2(M) ), forms
a Brownian random walk
We wish to know the probability that we
should associate a point with a collapsed
region of mass >M
At any mass it is equally likely that a
trajectory is now below or below a
barrier given that it previously crossed it,
so
Where
The mass function in PS theory
Differentiate in M to find fraction in range dM and multiply
by /M to find the number density of all halos. PS theory
assumes (predicts) that all mass is in halos of some (possibly
small) mass
High Mass: exponential cut-off for M>M*, where
Low Mass: divergence
The mass function
The PS mass function is not a great match to
simulation results (too high at low masses and low
at high masses), but can be used as a basis for
fitting functions
Sheth & Tormen (1999)
Jenkins et al. (2001)
PS theory - dotted
Sheth & Toren - dashed
Halo creation rate in PS theory
Can also use trajectories in PS theory to calculate when
halos of a particular mass collapse
This is the distribution of first upcrossings,
for trajectories that have an upcrossing for
mass M
For an Eistein-de Sitter cosmology,
Creation vs existence
Formation rate of galaxies per
comoving volume
Redshift distribution of halo
number per comoving volume
Extended Press-Schechter theory
Extended PS theory gives the
conditional mass function, useful for
merger histories
Given a halo of mass M1 at z1, what
is the distribution of masses at z2?
Can simply translate origin - same formulae as
before but with c and m shifted
Problems with PS theory
 Mass function doesn’t match N-body simulations
 Conditional probability is lop-sided
f(M1,M2|M) ≠ f(M2,M1|M)
M
M1
M2
 Is it just too simplistic?
Halo bias
 If halos form 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
 This is not expected to be the case in practice:
spherical collapse shows that time
depends on overdensity field
 A high background enhances the
formation of structure
 Hence peak-background split
Peak-background split
Split density field into peak and background components
Collapse overdensity altered
Alters mass function through
Peak-background split
Get biased formation of objects
Need to distinguish Lagrangian and Eulerian
bias: densities related by a factor (1+b), and
can take limit of small b
For PS theory
For Sheth & Tormen (1999) fitting function
Halo clustering strength on large scales
Bias on small scales comes from halo profile
N-body gives halo profile:

= [ y(1+y)2 ]-1 ; y = r/rc (NFW)
 = [ y3/2(1+y3/2) ]-1; y = r/rc
(Moore)
(cf. Isothermal sphere 
= 1/y2)
The halo model
Simple model that splits matter
clustering into 2 components
• small scale clustering of
galaxies within a single halo
• large scale clustering of
galaxies in different halos
galaxies
non-linear
linear
M=1015
small scale
clustering
M=1010
large scale
clustering
bound
objects
Predicts power
spectrum of the form
Further reading
 Peacock, “Cosmological Physics”, Cambridge University Press
 Coles & Lucchin, “Cosmology: the origin and evolution of cosmic structure”,
Wiley
 Spherical collapse in dark energy background
– Percival 2005, A&A 443, 819
 Press-Schechter theory
– Press & Schechter 1974, ApJ 187, 425
– Lacey & Cole 1993, MNRAS 262, 627
– Percival & Miller 1999, MNRAS 309, 823
 Peaks
– Bardeen et al (BBKS) 1986, ApJ 304, 15
 halo model papers
– Seljak 2000, MNRAS 318, 203
– Peacock & Smith 2000, MNRAS 318, 1144
– Cooray & Sheth 2002, Physics reports, 372, 1