The TheoryObservation connection lecture 3 the nonlinear growth of structure

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The Theory/Observation connectionlecture 3the
(non-linear) growth of structure

• Will Percival
• The University of Portsmouth

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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

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Phases of perturbation evolution
Inflation
Matter/Dark energy domination
Transfer function
linear
Non-linear
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Linear vs Non-linear behaviour
z0
non-linear evolution
z1
z2
z3
linear growth
z4
z0
z5
z1
large scale power is lost as fluctuations move to
smaller scales
z2
z3
z4
z5
P(k) calculated from Smith et al. 2003, MNRAS,
341,1311 fitting formulae
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Spherical collapse

• homogeneous, spherical region in isotropic
  background behaves as a mini-Universe (Birkhoffs
  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

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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)
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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
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Spherical collapse
For collapsing Lambda Universe, we have Friedmann
equation
?p is the curvature of the perturbation
And the collapse requirement
ap
Can integrate numerically to find collapse time,
but if no Lambda can do this analytically
tcoll
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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).
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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
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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 ?k0, ?m1, then we get the solution, for
perturbations that collapse at present day
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Evolution of perturbations
Wm0.3, Wv0.7, h0.7, w-1
evolution of scale factor
limit for collapse
top-hat collapse
virialisation
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Spherical collapse
Cosmological dependence of ?c is small, so often
ignored, and ?c1.686 is assumed
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Spherical collapse how to include DE?
If DE is not a cosmological constant, its sound
speed controls how it behaves
on large scales dark energy must follow Friedmann
equation this is what dark energy was
postulated to fix!
DE
DM
low sound speed means that large scale DE
perturbations are important
quintessence has ultra light scalar field so high
sound speed
DE
DM
high sound speed means that DE perturbations are
rapidly smoothed
The effect of the sound speed provides a
potential test of gravity modifications vs
stress-energy.
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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
homogeneous dark energy means that this term
depends on scale factor of background perfectly
clustering dark energy replace a with ap
depends on the equation of state of dark energy p
w(a) ?
can solve differential equation and follow growth
of perturbation directly from coupled cosmology
equations
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Evolution of perturbations
Wm0.3, Wv0.7, h0.7, w-1
evolution of scale factor
limit for collapse
top-hat collapse
virialisation
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Evolution of perturbations
Wm0.3, Wv0.7, h0.7, w-2/3
evolution of scale factor
limit for collapse
top-hat collapse
virialisation
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Evolution of perturbations
Wm0.3, Wv0.7, h0.7, w-4/3
evolution of scale factor
limit for collapse
top-hat collapse
virialisation
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virialisation

• Real perturbations arent spherical or
  homogeneous
• Collapse to a singularity must be replaced by
  virialisation
• Virial theorem
• For matter and dark energy
• If theres only matter, then
• comparing total energy at
• maximum perturbation size
• and virialisation gives

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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
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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
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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)

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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).
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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
gtM
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
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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 MgtM, where
Low Mass divergence
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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
PS theory - dotted Sheth Toren - dashed
Sheth Tormen (1999)
Jenkins et al. (2001)
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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,
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Creation vs existence
Redshift distribution of halo number per comoving
volume
Formation rate of galaxies per comoving volume
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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
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Problems with PS theory

• Mass function doesnt match N-body simulations
• Conditional probability is lop-sided
• f(M1,M2M) ? f(M2,M1M)
• Is it just too simplistic?

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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

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Peak-background split
Split density field into peak and background
components
Collapse overdensity altered
Alters mass function through
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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
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Halo clustering strength on large scales
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Bias on small scales comes from halo profile
N-body gives halo profile r y(1y)2 -1 y
r/rc (NFW) r y3/2(1y3/2) -1 y
r/rc (Moore) (cf. Isothermal sphere r 1/y2)
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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
small scale clustering
large scale clustering
bound objects
Predicts power spectrum of the form
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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, AA 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