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

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