the theory/observation connection lecture 3 the (non-linear) growth of structure
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The Theory/Observation connectionlecture 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
linear Non-linear
Transfer function Matter/Dark energydomination
Linear vs Non-linear behaviour
z=0
z=1
z=2z=3z=4z=5
lineargrowth
non-linearevolution
z=0
z=1
z=2z=3z=4z=5
large scale poweris lost as fluctuationsmove to smaller scales
P(k) calculated from Smith et al. 2003, MNRAS, 341,1311 fitting formulae
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 (r0) 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
Contain equal mass
collapsing perturbationRadius ap
BackgroundRadius a
Set up two spheres, one containing background, and one with an enhanced density
Spherical collapse
For collapsing Lambda Universe, we have Friedmann equation
And the collapse requirement
Can integrate numerically to find collapse time, but if no Lambda can do this analytically
ttcollcoll
aapp
p is the curvature of the perturbation
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
Obtain series solution for a
For the background,
Different for perturbation p
So that
Spherical collapse
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
Problem: need to relate the collapse time tcoll to the overdensity of the perturbation in the linear field (that we now think is collapsing).
Evolution of perturbations
top-hat collapse
limit for collapse
evolution ofscale factor
m=0.3, v=0.7, h=0.7, w=-1
virialisation
Spherical collapse
Cosmological dependence of c is small, so often ignored, and c=1.686 is assumed
Spherical collapse: how to include DE?
high sound speedmeans that DE perturbations are rapidly smoothed
DE
DM
on large scales dark energy must follow Friedmann equation – this is what dark energy was postulated to fix! low sound speed
means that large scale DE perturbations are important
DE
DM
quintessence has ultra light scalar field so high sound speed
The effect of the sound speed provides a potential test of gravity modifications vs stress-energy.
If DE is not a cosmological constant, its sound speed controls how it behaves
Spherical collapse: general DE
cosmology equation
depends on theequation 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
For general DE, cannot write down a Friedmann equation for perturbations, because energy is not conserved. However, can work from cosmology equation
Evolution of perturbations
top-hat collapse
limit for collapse
evolution ofscale factor
m=0.3, v=0.7, h=0.7, w=-1
virialisation
Evolution of perturbations
top-hat collapse
limit for collapse
evolution ofscale factor
m=0.3, v=0.7, h=0.7, w=-2/3
virialisation
Evolution of perturbations
top-hat collapse
limit for collapse
evolution ofscale factor
m=0.3, v=0.7, h=0.7, w=-4/3
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 - dottedPS theory - dottedSheth & Toren - dashedSheth & 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 Formation rate of galaxies per comoving volumeper comoving volume
Redshift distribution of Redshift distribution of halo number per comoving halo number per comoving volumevolume
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)
Is it just too simplistic?
MM1MM2
MM
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
M=1015
M=1010
linear
non-linear
bound objects
galaxies
large scale clustering
small scale clustering
Predicts power spectrum of the form
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
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
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