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CELL COUNT MOMENTS IN THE HALO MODEL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
S2007 Anand Balaraman
To my parents and to all my teachers
I owe my heart full gratitude to my advisor Dr. James N. Fry for all his assistance
and patient guidance. I thank the members of my thesis committee Dr. Fredrick Hamann,
Dr. Paul Avery, Dr. Pierre Sikivie and Dr. Steve Detweiler for their interest, precious
time and cooperation. I thank K~en Booth and Stacy Wallace for guiding me patiently
in formatting this document. I thank my brother and sister for being very supportive at
times of difficulties.
TABLE OF CONTENTS
ACK(NOWLEDGMENTS ......... . .. .. 4
LIST OF TABLES ......... .... .. 8
LIST OF FIGURES ......... .. . 9
ABSTRACT ......... ..... 10
1 HOMOGENEOUS COSMOLOGICAL MODELS .... ... .. 11
1.1 Standard Models of Cosmology . ...... .. 11
1.1.1 Cosmologfical Principle ........ .. .. 11
1.1.2 Weyl's Postulate ......... .. 12
1.1.3 Robertson-Walker Aletric . ...... .. 12
1.1.4 Fr-iedman Models ......... ... 1:3
1.2 Cosmological Parameters ......... .. 16
1.2.1 Hubble Constant ......... ... 16
1.2.2 Deceleration Parameter ....... .. .. 17
1.2.3 Density Parameter ......... ... 17
2 LARGE SCALE STRITCTITRE OF THE UNIVERSE .. .. .. 19
2.1 Large Scale Distribution Of Galaxies ...... .. 19
2.2 Structure Formation ......... ... 19
2.2.1 Linear And Non Linear Regimes ..... .... 20
2.2.2 Super-horizon And Sub-horizon Modes ... ... .. 21
2.2.3 Composition Of The Cosmic Fluid And Growth Dynamics .. .. 21
2.2.4 Statistical Properties Of Inhomogeneities ... ... .. 2:3
22.214.171.124 Ergodicity . ..... .... 2:3
126.96.36.199 Auto correlation function and Power spectrum .. .. .. 24
2.2.5 Primordial Perturbations . ..... .. 25
188.8.131.52 Primordial spectrum .... .... . 25
184.108.40.206 Adiahatic and Isocurvature perturbations .. .. .. .. 26
2.3 Growth Of Perturbations In The Linear Regime ... ... .. 27
2.3.1 Gravitational Instability And Jean's Analysis .. .. .. 27
2.3.2 Jean's Mass ......... .. 28
2.3.3 Baryonic Theories Of Structure Formation .. .. .. 29
220.127.116.11 Baryonic perturbations before recombination .. .. .. 29
18.104.22.168 Baryonic perturbations after recombination .. .. .. :30
2.:3.:3.3 Silk damping . ...... ... .. :31
2.3.4 Dark Matter Theories Of Structure Formation .. .. .. .. :32
22.214.171.124 Dark matter perturbations in radiation-dominated era :32
5.3.1 Corrections To The Point Cluster Result
126.96.36.199 The bias corrected point cluster expressions
188.8.131.52 Backgfround particles and statistics of mass .
5.4 Full Halo Model.
184.108.40.206 Dark matter perturbations in matter-dominated era.
220.127.116.11 Fr-ee Streamingf
18.104.22.168 HDM and CDM scenarios
2.3.5 Processed Power Spectrum And Transfer Function.
2.3.6 Problems In SCDM And The Emergence Of ACDM
22.214.171.124 OCDM models
126.96.36.199 Mixed Dark Matter (ill)M) models.
188.8.131.52 TrCDM model.
184.108.40.206 ACDM model.
2.4 Growth Of Perturbations In Non Linear Regfime
2.4.1 Spherical Collapse Model.
2.4.2 Press Schecter Formalism.
2.4.3 Halo Density Profile.
2.4.4 Halo Correlations
2.4.5 Halo Occupation Moments.
3 GALAXY CLUSTERING IN THE NON LINEAR REGIME ......
3.1 Galactic Distribution And Redshift Surveys
3.2 Non Linear Gravitational Clustering (NLGC)
3.2.1 Cosmologfical BBGK(Y Equations
3.2.2 Scale Invariant Model.
3.2.3 Hierarchical Model
3.3 Cell Count Moments
3.3.1 Fundamental Equations.
4 NON LINEAR CLUSTERING AND THE HALO MODEL ....._
4.1 Power Law Clustering Model .......... ...
4.2 Halo Model
4.3 Halo Model Formalism.
4.4 Model Ingredients .... .... ......
4.4.1 Halo Mass Function .. ...... ....
4.4.2 Halo Density Profile .. ...... ....
4.4.3 Halo Correlations And Halo Bias ..........
4.4.4 Halo Occupation Distribution
4.5 Halo Model And Non Linear Clustering ..........
5 CELL COUNT MOMENTS IN THE HALO MODEL ......__
Point Cluster Model
5.4.1 Point Cluster Limit Of The Full Halo Model ... .. .. 79
5.4.2 Resolved Halo Expressions And Form Factor .. .. .. 80
5.5 Numerical Simulations ......... .. .. 81
5.6 Galaxy Surveys ......... .. .. 82
5.6.1 Angular Surveys ......... .. .. 83
5.6.2 Redshift Surveys ........ ... .. 84
6 RESULTS AND DISCUSSION ....... ... .. 89
6.1 Hierarchical Amplitudes For Scale Free Power Spetra. ... .. .. .. 90
6.2 Numerical Simulation Results . .... .. 91
6.3 Hierarchical Amplitudes For A CDM Power Spectrum .. .. .. .. .. 93
6.3.1 Statistics Of The Mass Distribution ... .. .. 94
6.3.2 Statistics Of The Galaxy Number Counts .. .. . .. 95
7 SUMMARY AND CONCLUSION ...... ... .. 111
7.1 Future Directions ........ .. .. 113
REFERENCES ......... . .... .. 114
BIOGRAPHICAL SK(ETCH ......... . .. 117
LIST OF TABLES
2-1 HOD parameters from SPH and SA simulations ... .. .. 48
6-1 Distribution of halos and substructures in the simulation .. .. .. 97
6-2 Bias parameters for mass in LCDM . ..... .. 97
6-3 Bias parameters for galaxies in LCDM ...... .. . 98
LIST OF FIGURES
Halo density profiles .....
Halo hias .....
Halo occupation number ......
Hierarchical amplitudes for the galaxies in the APhi catalog ..
Variance of the galaxy distribution in the APhi angular survey .
Hierarchical amplitudes for the galaxies in the 2dFGR S catalog
Hierarchical amplitudes for scale free power spectra ......
Mass spectrum of the objects in the simulation .....
Variance of the mass distribution observed in the simulation ..
Hierarchical amplitudes from the simulation .....
Variance of the mass distribution in the halo model ......
Variance of the mass distribution (Model vs. Simulation). ...
Hierarchical amplitudes, S, in the Halo model .....
Variance of mass distribution (APhi vs. Model) .....
Hierarchical amplitudes for the galaxy number counts (Model vs
Hierarchical amplitudes for the galaxy number counts (APhi vs i
Hierarchical amplitudes for the galaxy number counts (APhi vs t
Hierarchical amplitudes for the galaxy number counts (APhi vs i
. . 101
Simulation ) .107
Model) .. .. 108
simulation) .. 109
dFGR S). .. 110
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
CELL COUNT MOMENTS IN THE HALO MODEL
Cl.! I!1-: Prof. James N Fry
One of the main goals of the study of Large Scale Structures is to understand the
clustering properties of the galaxy distribution front fundamental principles. Purely
haryonic theories of Structure Formation failed to produce the desired results, -II__- -r;i.-
strongly that the Universe is filled with some kind of dark matter which is non haryonic
in nature and interacts weakly with ordinary haryonic matter. At this point of time it
is not possible to determine the distribution of dark matter observationally. However we
can observe the distribution of galaxies. Observations show that galaxy clustering on
small scales is non-linear and an analytical understanding of Non Linear Gravitational
Clustering (NLGC) is difficult to achieve. The higher order correlation functions cannot
he determined analytically and the information available in full correlation functions
derived front galaxy catalogs, if not beyond the reach of our computational abilities, is
beyond our ability to handle. Count in cells offer a manageable alternative statistical
measure and the Halo model a useful phenomenological model for studying NLGC. We
study the moments of Count in cells in the Halo model. Halo model is used to predict,
calibrate and interpret redshift distortions and finite volume effects in the measurements of
galaxy correlation functions, weak gravitational lensing, the Lynian-alpha forest and C \lli
Sunyaev-Zel'dovich foregrounds. In order to be confident of these results it is important to
test the model in as many r-:--4~ as is possible.
HOMOGENEOUS COSMOLOGICAL MODELS
The Cosmos was an enigma until Albert Einstein formulated the General theory of
Relativity which enabled the construction of self consistent models of the Universe and
opened the possibility of studying the Universe for the first time. This resulted in the
emergence of a standard model of cosmology, popularly known as the Big Bang theory.
There are at least three independent pieces of evidence in support of the standard model,
Isotropic Cosmic Microwave Background Radiation and its black body spectrum,
Observed abundance of light elements like 2H, 3He & 7Li.
In this C'!s Ilter we will discuss the essential elements of the Standard cosmologfical models.
This would serve as the background material for the later C'!s Ilters.
1.1 Standard Models of Cosmology
The standard models of cosmology rest on the assumption that our understanding of
the correspondence between gravitation and the geometry of space-time as described by
general relativity is applicable on cosmic scales. Though not a well tested assumption, it
provides a framework within which one can study the dynamics of the Universe and make
predictions that can be verified by direct observations. General relativity is a geometric
theory of gravitation which relates the geometry of space-time to the energy content
of space-time. To write down the space-time metric of the Universe we need two more
assumptions. One is called as the Cosmological principle, which determines the form of
the space-time metric and the other one is called Weyl's postulate which resolves the
ambiguity in the interpretation of space and time coordinates in a dynamic Universe by
definingf a set of fundamental observers and a universal time called the cosmic time.
1.1.1 Cosmological Principle
Cosmological principle is the assumption that the Universe is spatially homogeneous
and isotropic. Proving the homogeneity assumption is not easy. The assumption of
isotropy, which has an observational' '.11,~! when combined with Copernican principle
imply homogeneity. The best evidence for isotropy comes from the Cosmic IVicrowave
Background Radiation (C \!1IR). Copernican principle is the philosophical prejudice that
our own location in the Universe is not a special point. So if the Universe is isotropic
about our location it can be so from any other location. Thus the cosmological principle
is a prejudice and its only scientific value is that it is in reasonable agreement with
observation and makes further progress possible.
1.1.2 Weyl's Postulate
Weyl's postulate is the assumption that the worldlines of the cosmic fluid particles
diverge from a singular point. This implies that the geodesic passing through each point in
space-time is unique except at the singular point from where they diverge. This makes it
possible to define a set of fundamental observers, for whom the Universe ahr-l-w look
isotropic, and a universal time called cosmic time upon which all the fundamental
observers agree. Cosmic time is the proper time as measured by the fundamental
observers. Since their geodesics meet only at the point of singularity in the past, the
cosmic time can be measured with reference to that singular point.
1.1.3 Robertson-Walker Metric
In an expanding Universe the physical coordinates of fundamental observers is not
fixed and changes with time. But the uniformity of expansion enables one to resolve
the physical coordinate into a time dependent scale factor, R(t) and a time independent
comoving coordinate. The comoving coordinate is a grid system that expands along with
the Universe keeping the coordinate positions of the fundamental observers to a fixed
The most general form of the metric that embodies the cosmologfical principle and the
ideas of fundamental observers and cosmic time was derived, independently, by Robertson
and Walker and is called the Robertson-Walker metric. The Robertson-Walker metric is
dS2 =C22 _it p2 1 kT~d2 2 mo lo, 2 Sn2
Here (r, 8, 4) are the comoving coordinates, t is the cosmic time, R(t) is the scale factor
and k is the curvature term which defines the geometry of the spatial subsection.
There are three possible geometries for the spatial subsection of the Universe :
+1 for a positively curved space :Spherical geometry;
k = for a flat space :Flat geometry;
-1 for a negatively curved space :Hyperbolic geometry.
Alternatively one can make the scale factor dimensionless by appropriate rescaling.
dS2 = C2 2 t2 2 2 2 Sn22)(1-2)
where r = xx, a = 1/R(to) and a(t) = xR(t); a(to) = 1.
1.1.4 Friedman Models
To write down the field equations of general relativity we need the metric of
space-time and the energy-momentum tensor of the cosmic fluid. While the Cosmological
principle determines the form of the metric, Weyl's postulate simplifies the form of the
energy-momentum tensor of the cosmic fluid. Since the geodesics of the cosmic fluid
particles do not intersect, they move along streamlines and so behave like perfect fluids for
which the energy-momentum tensor can be written as,
T"" = (p + p)u~u" py"'' (1-3)
Here p is the energy density of the cosmic fluid and p is its pressure. In this context,
Weyl's postulate is referred to as the perfect fluid approximation.
The perfect fluid approximation and the assumptions of homogeneity and isotropy
simplify the Einstein's field equations drastically and lead to the following pair of
independent equations called the Friedman equations from which the standard cosmologfical
models are constructed.
(a 2 8xTG x2
a/ 3 a
ii 4rG 3
(p + 2 ). (1-5)
a 3 c
Fr-iedman equations are a pair of independent equations on three variables, a(t), p(t)
and p(t). Thus it is not possible to solve these equations without a third independent
equation. For perfect fluids, pressure of the fluid is related to its energy density through
the equation of state which is written as p = wpC2. Using the equation of state as the third
independent equations we get the following solutions for the Friedman's equations.
p(t) oc a-3(1+w)(); _6)
a(t) oc t2/3(1+w). _n7
The cosmic fluid, is a multi component system with each component having their
own equation of state. However most of the time only one of these components will be
dominant and will determine the dynamics of expansion. The scale factor dependence of
energy density is different for each component and these differences lead the Universe to
undergo phase transitions during which one component replaces another component as
the dominant component driving the expansion. The cosmic fluid is made up of radiation,
I l..10.10, dark matter and vacuum energy. Dark matter can either be relativistic or non
relativistic and are pressure free. The vacuum energy serves as an effective Cosmological
constant, Ai whilchl defines lthe vac~uuml energy density as pAh -- A/8x.ii
O for dust (pressure less non-relativistic particles);
S= 1/3 for radiation and relativistic particles;
-1 for vacuum energy.
Setting c = 1 for convenience and by defining the Hubble parameter as H(t) a(t)/a(t),
the first of Friedman's equations can he rewritten as
H2(1 __ O 2
The Hubble parameter determines the rate of expansion of the Universe at a given point of
There are three possible dynamics for the expansion of the Universe:
Closed : Expansion stops, turns around and finally collapses.
Critical :Expansion stops and the Universe remains static forever.
Open :Universe expands forever.
The Friedman equation in this form, illuminates the relation between the dynamics
of the Universe and its geometry. Defining the critical density as pe. 3H2/8xrG and
rearranging the terms,
1 50- 1. (1-9)
where R p/p,. is called the density parameter.
Looking at Eq. 1-9, the correspondence between the dynamics of the Universe and its
geometry is clear.
Positive curvature R > 1 (CLOSED).
Flat n 0= 1 (CRITICAL).
Negative curvature W R < 1 (OPEN).
However this correspondence is valid only if the vacuum energy density is zero. For a
non-zero vacuum energy, the dynamics depends on the vacuum pressure and the value of
vacuum energy density. If the vacuum has positive pressure then the Universe is definitely
closed. For the case of negative vacuum pressure, the Universe is definitely open if its
geometry is flat or hyperbolically curved. On the other hand if the Universe is spherically
curved and the vacuum pressure is negative then the dynamics is closed if the turnaround
happens before the vacuum energy becomes dominant and open otherwise.
1.2 Cosmological Parameters
The Friedman models of cosmology describe a set of possible Universes whose
dynamics is determined by a number of cosmological parameters. To single out the model
that better describes our Universe, one has to determine these parameters from direct
observations. The following is an incomplete set of fundamental observable parameters.
Hubble Constant, Ho.
Deceleration parameter, go.
Density Parameter, R.
1.2.1 Hubble Constant
Hubble constant is the present d :-- value of Hubble parameter and describes the
present expansion rate of the Universe.
Ho = ~a/ (1-10)
To determine the Hubble constant it is important to determine the recession velocities
of fundamental observers precisely in all directions and over a wide range of depths. This
has ahr-l- .- been a difficult task because galaxies have peculiar velocities and are not
fundamental observers. Also there was a difficulty in measuring the distance to galaxies
accurately. For these reasons the value of Hubble's constant was ah-- .1-< a source of
controversy. During the Princeton meeting on Critical 17.:rl~~l;,. in C --I,;;..1.-,;; (1996), two
schools of observational cosmologists came up with two different values for the Hubble's
constant. One group claimed a value of Ho = 70 + 10 km s- Mpc while the other group
claimed a value of Ho = 55 + 10 km s- Mpc. Because of these discrepancies it is a common
practice among cosmologists to write the Hubble's constant as Ho = 100h kms- Mpc
and express all the estimates of cosmological distances and dimensions by including a
h-l factor and the value of h is left to the choice of the readers. For example distances
are indicated in units of h-1 Mpc. One of the main goals of the Hubble Space Telescope
was to determine the value of Ho to lo- precision. Tod w- we know the value of Hubble's
constant much more precisely but it is advisable to continue with the h-l factor. The
most recent measurements of Hubble's constant by WMAP gives us a value of h = 0.71.
1.2.2 Deceleration Parameter
The deceleration parameter, go, is defined as the present dimensionless deceleration of
the Universe given by the expression,
to = .C (1-11)
In the Friedman models the deceleration parameter is related to the density parameter in
mass and vacuum through the relation go = Rm/2 RA. Since go and Rm are separately
measurable quantities it is possible to determine RA, if Only We Succeed in determining
go. Measuring qo requires observation of objects at high redshifts. However the evolution
of the internal properties of galaxies like its intrinsic luminosity etc. evolve with cosmic
epoch, making it difficult to determine go precisely.
1.2.3 Density Parameter
Density parameter is defined as the ratio of the energy density of the cosmic fluid
at a given epoch to the critical density at that epoch. Since the cosmic fluid is a multi
component system, it is important to determine the density parameter in individual
components separately. The cosmic fluid is made up of radiation, matter and vacuum
energy. The matter component can further be resolved into baryonic and non baryonic
(dark matter) components. Baryons are collisional fluids while the non baryonic dark
matter are collisionless and so act as dust.
Otos =Prot Pr + Pm + PA
= R + Rm + RA.
Here OR, Rm and RA are the density parameters in radiation, matter and vacuum. Rm
includes both baryons and non baryonic dark matter. If Rb is the density parameter in
the haryonic component separately, then R,, Rb is the density parameter in the non
haryonic dark matter. The primary constraint for the energy density in haryons come
from Big Bang Nucleosynthesis (BBN). The observed abundance of light elements like
2H (deuterium), 3He and 7Li sets a firm upper limit for b S2 ab S62 I 2.0 x 10-2. This
following are the estimates of WMAP (Wilkinson Microwave Anisotropy Probe) for the
present dwi composition of the Universe.
Or 2 = 2.56 x 10-5 Ob 2 = 0.0224 R,7h2 = 0.135 RA = 0.73
LARGE SCALE STRUCTURE OF THE UNIVERSE
2.1 Large Scale Distribution Of Galaxies
The Cosmological principle, at the best, can only be a zeroth order approximation
of the real Universe. On small scales we know that the Universe is inhomogeneous. Stars
congregate into galaxies. Galaxies, which are the building blocks of the structures in
the distribution of visible matter, are themselves clustered. The associations range from
tiny groups to giant clusters. A mapping of the galaxy distribution in our immediate
neighborhood shows even greater structures like super clusters, great walls and voids.
Structures whose size range from 10 kpc to 100 Mpc are called Large Scale Structures
(LSS). However deeper surveys like the Sloan Digital Sky Survey (SDSS) and the 2 degree
Field Galaxy Redshift Survey (2dFGRS) show that the fluctuations decline if we smooth
out the matter distributions over large scales (~ 200 Mpc) and the Universe appears to be
homogeneous. Thus the Cosmological principle is applicable only on large scales and that
too in a statistical sense.
The Cosmic Microwave Background Radiation (Cill 11R) is a fossil record of the
Universe at a very early epoch. Cil IlR is remarkably isotropic, to 1 part in 10s on angular
scales ranging from 1' to 180. The isotropy of C'j !liR implies that the Universe was very
smooth at the time of decoupling and it is very lumpy today-. So the question that rises
naturally is, how did these structures come into being from such smooth initial conditions?
The branch of Cosmology that addresses this question is called Structure Formation.
2.2 Structure Formation
The current paradigm on structure formation is that, there were primordial
inhomogeneities at the time of decoupling and these inhomogeneities grew by gravitational
instability, giving rise to the structures that we see badl-ly. Structure formation can begin
as soon as the Universe become matter-dominated. So if there were any primordial
inhomogeneities when the Universe became matter-dominated, then they will manifest as
anisotropy in C'j !liR. If we assume that the perturbations are purely adiahatic and the
decoupling is instantaneous, then the density contrast at the time of decoupling is related
to the temperature anisotropy in C1111lR hy the simple relation,
(6 Tj 1t 6 p\
With instruments of improved sensitivity, the space satellite COsmic Background Explorer
(COBE) detected anisotropy in C1111 IR of the order of 10-5 which is consistent with
primordial adiahatic perturbations as large as 10-4
2.2.1 Linear And Non Linear Regimes
Given the primordial perturbations one can follow their evolution due to gravitational,
hydrodynamical and Hubble expansion processes and see how we can explain the
structures that we see tod w-. This is usually done by expressing the perturbations from
the homogeneous background as a density contrast field defined at every point as,
p(x, t) p(t)
6(x, t) -(2-1)
and then study the evolution of this field and its associated quantities like its variance
etc. using various analytical and numerical techniques. Clearly one can distinguish two
regimes based upon the amplitude of the density contrast field and the methods that
are emploi-, .1 to study the evolution of the field are different for the two regimes. In the
linear regime (6
possible to find analytical solutions. Once the perturbations grow in amplitude and enter
the non linear regime (6 > 1), their growth becomes non linear and they rapidly evolve
towards forming bound structures. It is hard to find an exact analytical solution in this
regime except in highly idealized models. Numerical simulations are emploi-. I1 to evolve
the Boltzmann equations beyond the linear regime and then analyze the results of the
simulation assisted by some modeling.
2.2.2 Super-horizon And Sub-horizon Modes
Due to the non local nature of gravitational dynamics it is not easy to follow the
evolution of perturbations in position space. The problem becomes tractable if the density
contrast field is resolved into its Fourier modes and then follow the evolution of each mode
separately. This works as long as the modes do not couple. This condition is met in the
At any epoch the modes 6k, can be categorized into super-horizon modes or sub-horizon
modes depending on whether they are bigger than or smaller than the horizon scale at
that epoch. To study the evolution of super-horizon modes we need a general relativistic
treatment of the problem. For sub-horizon modes, a Newtonian treatment is sufficient. All
the modes that correspond to interesting structures tod w- are sub-horizon modes.
2.2.3 Composition Of The Cosmic Fluid And Growth Dynamics
The dynamics of growth of sub-horizon perturbations depend on the composition of
cosmic fluid. Gravity tends to amplify the perturbations, pressure opposes gravity and the
Hubble expansion tends to suppress the growth. In multi component systems we run into
situations where one component may be gravitationally dominant while a sub dominant
component provides the pressure. Under such circumstances the relative importance of
the growth and suppression mechanisms depend on the composition of the fluid. Also the
expansion rate changes as one component replaces another as a dominant component.
The cosmic fluid is made up of photons/electro-magnetic radiation, generally called as
radiation, collisional matter, generally called baryons, collisionless relativistic particles
called as Hot Dark Matter (HDM), collisionless non relativistic particles called as Cold
Dark Matter (CDM). To this one should add vacuum energy, also called as dark energy
which acts like a perfect fluid with a negative pressure. The pressure and energy densities
of these components are connected through the equation of state, p = wp. Relativistic
particles have the same equation of state as radiation. The nature of dark energy has
not been understood properly, yet, and so its equation of state is model dependent. If we
assume that the vacuum behaves like an effective cosmological constant (A) with energy
density A/8xrG then it has w = -1.
S= 1/3 : HDM, radiation
-1 : Dark energy(Cosmological constant)
In the Friedmann models energy density in a particular component changes with
the scale factor as p(a) oc a-3(1+w). The differences in the scale factor dependence of
energy density dilution between various components lead the Universe to undergo phase
transforms during which time one component replaces another as a dominant component
driving expansion. Since the energy density in radiation dilutes as a-4 and that in matter
dilutes as a-3, it is clear that the Universe should have undergone a transition from a
radiation dominated phase to a matter dominated phase. So the history of our Universe
can be divided into a Radiation Dominated era (RD) and a Matter dominated era (ill)).
It is possible to determine the epoch of this transformation if we know the composition
of cosmic fluid tod w-. During the RD era the Universe expands at a vary high rate. A
high expansion rate implies a stronger Hubble drag on the perturbations that results in
reducing their amplification substantially. So during the RD era the perturbations in the
dark matter component do not grow much. The baryons which are tightly coupled to
the photons via Thompson scattering get pressure support from radiation and so do not
grow at all. The coupling between radiation and baryons continue in the MD era until
the time when the Universe cools down to a temperature of around 4000.K( allowing the
electrons to become bound to nucleus and form neutral atoms. This decouples the photons
from baryons and results in the Cosmic Microwave Background Radiation. This is called
recombination. After the epoch of recombination photons withdraw their pressure support
allowing the haryonic perturbations to grow. This means that perturbations in the
haryonic component cannot grow until the time of recombination. This is an important
conclusion that has great implications for Structure Formation.
2.2.4 Statistical Properties Of Inhomogeneities
Structure formation theories can make definitive predictions about the large scale
structures, once the initial conditions and the evolution dynamics are given. To compare
these predictions with actual observations, neither can we follow the evolution of a single
system nor can we predict the density contrast at a given point in space-tinte. The
problem with the former approach is that the time scale of evolution of the system is much
larger than the time scale of observation. The problem with the latter approach is that
it requires the precise statement of initial conditions in the form of spatial distribution of
primordial seed perturbations. This requirement cannot he met even in principle, because
the primordial seed perturbations are generated by mechanisms that are inherently
random resulting in seed perturbations that are randomly distributed in space. So instead
of predicting the exact mass distribution we shall predict the average statistical properties
of the mass distribution.
Employment of statistical methods necessitates the construction of an ensemble of
identically prepared system. One can think of the observable Universe as a particular
realization of a statistical ensemble of possibilities and all interesting statistical properties
of the density contrast field can he considered to be average across ensembles. But the
Universe by definition is one single system. Even if there were an ensemble of "IUni"verses
we can observe only one realization of that ensemble. To resolve this issue, it is reasonable
to assume that the Universe is Ergodic. A field cp(x) is ergodic if its ensemble average
(cp(x)) equals the spatial average cp(x).
p (x) lim pxd.
Since the Universe is homogeneous at large scales we expect the ensemble average of the
density contrast field to vanish.
(6())= p(x) -p (p(x)) p
The assumption, that the Universe is ergfodic, is essential for comparing theories with
observation. Observations measure statistical properties by spatial averaging. Popular
theories of Structure Formation assume Gaussian initial conditions. Adler(1981) proved
that Gaussian random fields are ergodic iff their power spectrum is continuous . We
assume that this criteria is met for our Universe.
220.127.116.11 Auto correlation function and Power spectrum
The two point correlation function for the density field 6(x) is defined as
I(r) = (6(x)6(x + r)). (2-2)
The Power spectrum of the density field is the fourier transform of its two point
i(r) = |6k~ 2 erxp(-ik x)d~k.
P(k) (|Sk 2) is the power spectrum.
The 2-point correlation function and power spectrum are Fourier pairs and they carry the
same information. ((r)/P(k) exhaust all the statistical information about the field 6(x),
if it is Gaussian. Primordial perturbations are believed to be Gaussian. Even when the
perturbations grow, Gaussianity is preserved as long as the growth is linear. Tod w- the
scale of non-linearity is around 10h- Mpc. The density field when smoothed out on scales
much larger than 10h-1 hpe is Gaussian and it would have been Gaussian on all scales in
the early stages. A Gaussian random field with a vanishing mean and a variance of a2 is
characterized by its PDF,
p  =(2;2 -1/2exp? [S'2/2e2
a2(R) = F(R) where R is the smoothing scale.
So a Gaussian field with vanishing mean is completely characterized by its variance or the
two point correlation function.
2.2.5 Primordial Perturbations
To understand the formation of structures in all its details one requires the specification
of initial conditions in terms of the statistical properties of primordial inhomogeneities.
The physics that explains the origin of primordial perturbations should also give its
statistical properties. The Inflationary models offer a mechanisms through which
primordial inhomogeneities can he generated. In the inflationary mechanism, quantum
fluctuations get amplified and evolve into seed perturbations with random phases.
Perturbations generated through the inflationary mechanism are Gaussian distributed
and have a Harrison-Zel'dovich spectrum (|6k 2 ock).
18.104.22.168 Primordial spectrum
Primordial power spectrum is the quantification of primordial inhomogeneities.
Various surveys of galactic distributions have shown that the observed two-point
correlation function for galactic distributions is a smooth function of position and can
he well represented by a power law over a wide range of scales. This clearly -II- -- -;
that fluctuations on a very wide range of scales must have been present in the initial
perturbation spectrum. If the primordial power spectrum were very broad with no
preferred scales, it is natural (in the absence of a better assumption) to begin with a
power law spectra of the form
P(k) = Ak". (2-3)
Based on certain physical requirements we can constrain the range of values that a can
Asymptotic homogeneity requires n > -3 as the lower bound.
An upper bound of a < 4 comes from an argument due to Zel'dovich assuming small
scale graininess of the system and requiring mass and momentum conservation. This
is called the minimal spectrum.
The value of a must lie within these bounds. The case of a = 1 is of special interest
and is called the Harrison-Zel'dovich spectrum This spectrum has a number of appealing
features including self similarity. Under this spectrum the Universe looks the same when
viewed on the scale of horizon. For this reason it is also called scale invariant spectrum.
The simplest of inflationary theories come out with the Harrison-Zel'dovich spectrum
Ch.~ ~ -!ni the index n only fixes the shape of the primordial spectrum and not its
amplitude. The power spectral amplitude, A is fixed by normalizing the power spectrum
either at a high redshift of z ~ 1100 (COBE normalization) or by observing the clustering
of structures at z = 0 (o-s normalization).
22.214.171.124 Adiabatic and Isocurvature perturbations
There are two modes of perturbations for a multi component fluid system. The one
that preserves the specific entropy of the perturbation is called adiabatic perturbation and
the one that keeps the metric curvature invariant is called isocurvature perturbations. The
isocurvature perturbations perturb the metric so that the Riemannan tensor changes in
such a way as to keep the Ricci scalar invariant. Effectively isocurvature perturbations are
perturbations in the equation of state. For a binary system of radiation and matter with
densities pr, pm and density contrasts 6,, 6m, respectively, the adiabatic and isocurvature
fluctuations are characterized by the following relations.
6, ~6m = 0 (adiabatic mode). (2-4)
6rpr + 6mpm = 0 (isocurvature mode) (2-5)
A general perturbation need not be purely adiabatic or isocurvature, but a superposition
of both modes. C'!LI:R polarizations show that the isocurvature component of primordial
perturbations is negfligfible.
2.3 Growth Of Perturbations In The Linear Regime
2.3.1 Gravitational Instability And Jean's Analysis
The problem of gravitational instability in a static Universe was first studied by
James Jeans (1902) who derived the criterion for collapse. Lemaitre and Tolman studied
the gravitational collapse of a spherically symmetric perturbation in an expanding
background. But the Jean's analysis in a dynamic background for the most general
case was solved by Evgenii Lifschitz (1946). The main difference between the results
of Jean's analysis for static and dynamic backgrounds is that, in a static Universe the
unstable modes grow exponentially while in the dynamic Universe the unstable modes
Jean's analysis starts by writing the hydrodynamical equations for a fluid in a
gravitational field. For an expanding Universe these equations are transformed to
comoving coordinates. Then consider perturbations from background and expand the
hydrodynamical equations up to first order and subtract the zeroth order solution to get
the linearized hydrodynamical equations in comoving frame. For adiabatic perturbations
with c~ = 8p/8p, these equations simplify to the following evolution equation for the
6, + 2 bk ( ~2 -4a k=0 26
a a ~~), ,(6
2.3.2 Jean's Mass
In Eq. 2-6 the second term is the Hubble drag term that indicates the expansion
of the Universe. For a static Universe this term vanishes and yields the classical Jean's
analysis for a static background. The last term called the dispersion term describes
oscillation or instability depending upon whether the pressure term, csk2 is dominant or
the gravitational term, 4xrGp. If the pressure term csk2 is dominant then the dispersion
term is positive and the equation has an oscillatory solution. In that case the perturbed
dense region has enough pressure support to overcome gravitational collapse and remains
stable. On the other hand if the gravitational term 4xrGp is dominant, then the dispersion
term is negative and the perturbation is unstable under gravitational collapse and the
perturbation has a growing solution. For a static Universe for which the hubble drag
term vanishes, the growth is exponential. However for a dynamic Universe the growth is
The criteria for instability is that the size of the perturbation should be such that the
gravity part in the dispersion term should be dominant. This introduces a characteristic
size of the problem called Jean's wavelength.
As = r=cs G ) (2-7)
One can define a Jean's mass which is the total mass enclosed inside the Jean's wavelength.
My = p (2-8)
All the perturbations that are less massive than My5 are stable and execute acoustic
oscillation, while those that are more massive than My5 are unstable and collapse
gravitational. The Jean's mass at a given epoch is dependent on the equation of state
of the cosmic fluid at that epoch. So during the events of phase transition the Jeans mass
changes abruptly thereby deciding the size of structures.
2.3.3 Baryonic Theories Of Structure Formation
The natural starting point for the theories of Structure Formation is to assume that
all the matter, visible and invisible, are in some haryonic form. But all such attempts
failed and the failure of these models -II__- -r that the dominant component of matter in
the cosmic fluid must he in some non haryonic form.
126.96.36.199 Baryonic perturbations before recombination
B l1i0.10- are collisional fluids and are tightly coupled to radiation through Thompson
scattering. So radiation provides the pressure support for haryons. To follow the evolution
of sub-horizon sized haryonic perturbations we need to compare their .Jeans wavelength to
the Horizon scale. Horizon scale. Horizon scale at a given epoch is the nmaxiniun distance
over which causal coninunication could have taken place by that epoch.
The Hubble radius dH = H-l is a measure of horizon size.
dH = H-1 (in RD era).
8xG'(p,n + pr) 8xGpr
The haryonic .Jeans length is
Af=c\ ( for a relativistic gas ).
The ratio of haryonic .Jeans mass to the mass inside the horizon is,
AfB 2H-1 3p, i
Thus the haryonic .Jeans scale is outside the horizon in the RD era, meaning haryonic
perturbations that enter the horizon in the RD era have pressure support against gravity
and they stop growing and becomes stable modes on entering the horizon. Since the stable
modes have oscillatory solutions, the haryonic perturbations that enter the horizon start
oscillating. These oscillations are called Sakharov oscillations.
188.8.131.52 Baryonic perturbations after recombination
Since' l'i-10.10 remain coupled to radiation until decoupling, they continue to get the
pressure support from radiation even after the epoch of matter-radiation equality. As the
temperature cools electrons get bound to the nucleii to form neutral atoms. This is when
the photons decouple from baryons. After decoupling, radiation withdraws the pressure
support. Now the pressure support should come only through inter baryonic collisions. So
there is an abrupt drop in the baryonic Jeans length.
To calculate the baryonic Jeans mass after decoupling, we can assume the Universe
to be dominated by hydrogen which is a monoatomic gas with adiabatic sound speed
c~ = 5ksT/(3mH).
M r5/2 CY
6 G3/2 1/2
1.3 x105 2 -/2 .(2-10)
Thus there is a substantial drop in Jeans mass following decoupling and the Jeans
mass after decoupling is ~ 10s which is close to the mass of globular clusters. Now
baryonic perturbations with mass M~ > M y" can grow and collapse to form structures.
The evolution equation for such modes simplify to,
6kI + 2a 4xrGpmbk, = 0. (2-11)
For an Einstein-deSitter Universe (Rm = 1, a = 0), this equation has a growing solution
with 6 oc a(t) oc t2/3. After decoupling the unstable baryonic perturbations grow linearly
with the scale factor. Modes that entered the horizon before the epoch of recombination
are executing Sakharov oscillations. Their amplitudes at the epoch of recombination
depends on the phase of their oscillations. Some of them will have zero amplitude at the
epoch of recombination and those modes do not survive. Those modes that complete an
integral number of oscillation at the time of recombination will appear with maximum
amplitude. All other modes will have amplitudes between these values. These acoustic
oscillations will leave an imprint in the C1111:R. Modes with mass M~ > My5 = 3.75 x
1015 oRb 2 2A that enter the horizon after the epoch of recombination never go through
oscillations and continue to grow.
184.108.40.206 Silk damping
Among the modes that enter the horizon before recombination, not all of them
survive. Joseph Silk (1968) described a mechanism by which baryonic perturbations can
be damped through the diffusion of photons. As the epoch of recombination approaches,
the mean free path of the photon-baryon collisions become large enough for the photons to
diffuse out of the over dense regions into under dense regions.The diffusing photons drag
matter with itself, diluting the inhomogeneities. This damping mechanism is called Silk
The photon mean free path for Thompson scattering is given as
o-r = 6.665 x 10-29 2 1S the Thompson cross-section. Total number of collisions in time at
is, NeouI = at/\,. The mean square distance of random walk for NeouI is
(Ar) 2 = Vcoll t =axt
Integrating this over time we get the Silk damping scale as,
A ik dt~Tj (2-12)
3 teech (taec)
5 a2 tdec
Taking zaec ~ 1100 and Xe = 0.1,
A2 il =3. 1 2 m2-3/4 MPc
Msilk = 62 102 2 5/
M~silk ~ 1013 t xdec
Among the modes that entered the horizon before decoupling, all the modes with
M~ < Msilk are dissolved completely and the rest of them survive. The Silk damping scale
is close to the mass of galaxy clusters, implying all the modes that are smaller than galaxy
clusters are wiped out.
2.3.4 Dark Matter Theories Of Structure Formation
Silk damping mechanism wipes out all inhomogeneities with masses below 1013Me0
which is close to the mass of galaxy clusters. After decoupling the perturbations grow
linearly with the scale factor. That means the present day- density contrast of the
surviving modes must be zaec times their amplitude at the time of decoupling. Tod~i-
the overdensity in galaxy clusters is O(1) which should generate an anisotropy of O(10-2)
in C\! IlR which is ruled out. Thus the upper limits of C'j llR anisotropy were in conflict
with the purely baryonic theories of structure formation. This conflict with observation
is a strong reason for taking seriously the proposition that the Universe is dominated
by some unknown form of non-baryonic matter. Since dark matter do not interact with
radiation they are free to grow as soon as the Universe become matter dominated.
By the time baryons decouple from photons and are ready to form structures, dark
matter perturbations would have already grown ~ 10 times larger than their baryonic
counterparts. The dark matter perturbations create potential wells into which the baryons
fall and quickly catchup with the dark matter perturbations.
220.127.116.11 Dark matter perturbations in radiation-dominated era
Dark matter perturbations in RD era are not easy to handle due to the fact that they
interpenetrate each other making it impossible to treat them as a fluid. But if we assume
a smooth background of radiation that affects only the expansion rate, then the evolution
equation for the sub-horizon modes is
C'I lIngus;! the variables to y -- pm/pr and by using the Friedmann equation H2
8xrG(pr + pm)/3, the growth equation can be recast in the form
2 + 3y 3
6" + 6' 6 = 0. (2-14)
2y(1 + y) 2y(1 + y)
which has a growing solution 6 oc 1 + 3y/2. This means that throughout the RD era
the perturbation amplitude has grown only by a factor of 3/2. This result is known as
Meszaros effect. In the absence of the Hubble drag term the growth is exponential. This
shows that during the RD era radiation drives the expansion so fast that matter has no
time to respond. A relativistic treatment of the problem arrives at almost same conclusion
except that it predicts a logfarithmic growth.
18.104.22.168 Dark matter perturbations in matter-dominated era
Since DM are collisionless they are not bound to radiation and can start to grow
as soon as the Universe becomes matter dominated unlike the baryons which have to
wait until decoupling. The evolution equation is the same as that for dust, which has a
growing solution 6 oc a(t) oc t2/3 for the special case of an Einstein-deSitter Universe
(Rm = 1; is = 0). The Einstein-deSitter model is a good approximation when the Universe
is matter dominated. For a Universe with non vanishing vacuum energy, the growth
factor is calculated by numerically integrating the equation. Carroll et al. (1992) provide
a fitting formula that connects the growth factor for a Universe with non zero vacuum
energy to that of Einstein-deSitter Universe.
6(z = 0; R = Om + RA) 9 m(2, A)6(z = 0; Om, = 1);
g(Rm, RA) m 42 76~ A + (1 + O/2)(1 +C/0].(-5
The growing dark matter perturbations create a potential well for the baryons to fall after
22.214.171.124 Free Streaming
Fr-ee Streaming is the damping niechanisni in dark matter perturbations. Collisionless
particles do not interact with anything and so can free stream out of the overdense
regions, thus smoothing out the perturbations. Since the montentunt of particles get
redshifted with expansion, the relativistic dark matter particles become non relativistic
beyond a point. The free streaming niechanisni is effective in wiping out the perturbations
until the dark matter particles become non-relativistic. The epoch at which the dark
matter particles become non-relativistic determines the free streaming length scale. The
conioving distance that the free streaming particles can travel by the epoch t is
r7 = dt' (2-16)
Now partition this interval into two parts. The first part is in the epoch when the particle
is relativistic and the second part is in the epoch when it has become non-relativistic.
= 2 +,t d t .
an zr /n a 2 /i t ,)
The nmaxiniun free streaming length is
17. =cor, .7. ~0.5Ale(DIh2 /: -/:
The free streaming mass is,
Mf, 4 x101"( 30eV -2 AL., _UeutrinOSi
6 x10"Ms.. (1 K~eV- WIMPs)
126.96.36.199 HDM and CDM scenarios
If we take the idea of non-baryonic dark matter seriously then there are two categories
of them. HDM and CDM. Lighter particles are relativistic when they decouple front
the thermal background. Zel'dovich coined the term Hot Dark Matter (HDM) for such
particles. Massive particles are non-relativistic when they decouple from the thermal
background. Peebles coined the term Cold Dark Matter (CDM) for such particles.
HDM particles have large free streaming length scales. In a Universe dominated by
HDM particles, all the modes smaller than the size of super clusters will be wiped out by
free streaming mechanism. So the first structures to form will be of super cluster scales
and smaller perturbations like galaxies should form by the fragmentation of such large
structures. So the HDM approach of Structure Formation is called top-down scenario.
The absence of power in small scales introduce a small wavelength cutoff in the HDM
power spectrum. Large scale perturbations have smaller amplitude and so should form
structures very late. In this scenario galaxies should have formed very recently (z < 1).
The weakness of this scenario is the observed fact that galaxies are present at earlier times
(z > 1). So HDM models of structure formation are out of fashion.
CDM particles are non relativistic when they decouple from the thermal background
and so have negligible free streaming lengths. So power survives on all scales and all the
modes start growing simultaneously. In this scenario small scale structures form first and
they cluster to form larger structures. This is referred to as hierarchical clustering or
bottom-up scenario. The greatest advantage of this model is its ability to produce small
scale structures earlier. But it does not explain the presence of very large scale structures
that are observed today.
2.3.5 Processed Power Spectrum And Transfer Function
The goal of Linear Perturbation theory is to predict the amplitude of all the linear
modes at some particular epoch given their primordial amplitudes. The amplification of
primordial perturbations is expressed in terms of the linear growth factor D(z). However
various dissipation processes like Silk damping and free streaming distort the primordial
perturbation spectrum substantially and these changes are recorded in the form of a
Transfer Function, Tk~. So the amplitude of a linear mode at the present epoch can be
k61 =0=D2X) Tk2 k rmz (2-17)
Here x is the primordial redshift and D(x) is the linear growth factor given by Eq. 2-15.
The shape of transfer function depends on the characteristic lengths of the problem which
in turn depends on the composition. One characteristic length that is independent of
composition is the horizon size at the epoch of matter-radiation equality. All the modes
that entered the horizon in the matter-dominated epoch grow at the same rate whereas
those that entered the horizon when the Universe was radiation dominated froze until
the Universe became matter-dominated when it started growing again at a different rate.
So corresponding to the scale of horizon at the epoch of matter-radiation equality, the
transfer function of all the components should show a bend.
Baryonic perturbation modes that entered the horizon in the RD epoch start
oscillating and enter the AID epoch at various amplitudes. So the transfer function will
have an imprint of these oscillations. Further the transfer function for harvonic fluids will
have a lower wavelength cutoff characterized by the Silk damping scale. Since dark matter
perturbations do not oscillate in the RD epoch their transfer function will be smooth. Free
streaming is important for HDM perturbations. So the HDM transfer functions will have
a small wavelength cutoff characterized by the free streaming scale. CDM perturbations
have negligible free streaming lengths and so have no cutoff wavelengths. Power survives
on all scales.
Calculating the transfer function for multi component system involving radiation,
'I l.i iwis and dark matter is a non trivial business and involves solving the Boltzmann
equations numerically and matching boundary conditions. This was first done rigorously
by Bond & CI. II li- (1983). Seli I1: & Zaldarriaga (1996) developed a code CMBFAST to
do this calculation for any cosmological model and the code is now available as a
public-domain software. Bardeen, Bond, K~aiser & CI. I1 li- (1986) found an analytical
fitting formulae for the standard cases like adiahatic CDM, HDM scenarios which are
accurate to about 101' The analytical fitting formula for the CDM transfer function as
given by BBK(S is,
Tk = (1":'" [1 3.89q (16.1q)2 (5.46iq)3 (6.71q)4]L/ (2-18)
k Rmb2 MpC-1 f Rbh ~ 0
Ombhexp[-~ob mJR) f Rbh / 0.
2.3.6 Problems In SCDM And The Emergence Of ACDM Paradigm
Among the many models of structure formation, CDM emerged as the only model
that successfully predicts the observed small scale structures. However it failed to explain
the observed large scale structures. One of the characteristics of the CDM model is the
turn over of the power spectrum at a characteristic scale corresponding to the horizon
size at the time of matter-radiation equality. Large scale and small scale normalization
of CDM power spectrum produces inconsistent results. The CDM power spectrum when
normalized at small scales, based on the observed galaxy clustering, underpredicts the
power on large scales. When normalized at the largest scale, based on the Cill IR data
(COBE normalization), it over predicts small scale structures. The Standard Cold Dark
Matter (SCDM) model requires modification that changes the shape of transfer function to
explain all the observed phenomena. There seems to be more than one way to accomplish
this and so we have different flavors of CDM.
188.8.131.52 OCDM models
Sacrificing the inflationary prediction of flat spatial geometry if one accepts Rm = 0.3,
then it increases the horizon size at matter-radiation equality. This enhances the power on
184.108.40.206 Mixed Dark Matter (MDM) models
The idea behind this model is to retain power on large scales by introducing
calculated amounts of HDM so that power survives on small scales too. This model
requires fine tuning of the HDM and CDM compositions. This places stringent constraints
on the mass range of various neutrino species.
220.127.116.11 TCDM model
The goal of TrCDM model is to prolong the RD epoch by introducing an unstable
relativistic particle (-r neutrino) which mostly decays into radiation. This model requires
fine tuning of the neutrino mass so that its decay is de 1li- II to leave the predictions of Big
Bang Nucleosynthesis (BBN) unmodified. However TrCDM is not favored anymore.
18.104.22.168 ACDM model
If one does not want to sacrifice the inflationary prediction of flat spatial geometry,
this can be achieved by a combination of Rm and RA Such that am + RA = 1. Introduction
of cosmologfical constant pushes back the age of the Universe. So there is more time to
build the largest structures through hierarchical clustering. ACDM solves many other
problems in various branches of cosmology. Observation of distant supernovae finally
brought a paradigm shift in structure formation and made ACDM emerge as the most
2.4 Growth Of Perturbations In Non Linear Regime
2.4.1 Spherical Collapse Model
The spherical collapse model is an idealized situation of the collapse of a spherically
symmetric perturbation in an expanding background. Consider a spherical region with
average over density A(r) in an otherwise homogeneous background with density p = Ope.
Using the Friedmann equation the background density can be written as
p = Ope
II ( 3H2
Total mass inside the overdense region is
0 [1 + a(r)]
O6[1 + a(r)]
The equation of motion is
d2r GM1 r2G
dt2 72 dt r
Initially the overdense region expands along with the background and the peculiar velocity
component is negligible. So the kinetic and potential energy components are
1= dr 2 H2r2 GMn
Tu |U| TR[1 + a(r)].
2dt 2 r
The criterion for collapse, E < 0 gives
T[1 R(1 + a(r))] < 0.
Since T is positive semi definite, the collapse criterion becomes
This criterion is satisfied by all 6(r) > 0 in a closed and critical Universe. For the case of
E < 0, there is a cycloidal parametric solution for the equation of motion.
r(0) (1 cos 8) t(0) =(8 sin 8).
From this solution it is clear that the overdense region expands along with the background
initially, reaching a maximum size of rmax and turns around at time tturn corresponding to
the parametric value of 8 = xr. For an Einstein-deSitter (a = 0, Rm = 1) background,
rm= E hon=(2E)3/2
a = $ pur 9iT2Ib ~ 5.55.
If the over density is perfectly spherical then it would collapse to a singular point at
ther = 2ttrn corresponding to 8 = 2xr. However the overdense regions are most likely
triaxial systems and they virialize into a blobs of bound objects called dark matter halos.
Collisionless particles virialize by a process called violent relaxation. Violent relaxation
allows a system of collisionless particles to reach a equilibrium configuration by means of
strong fluctuations in the mean gravitational potential. Such virialized systems are called
dark matter halos. Dark matter halos are bound system with a well defined density profile
that can be used to model the structures that we see in the Universe. Relaxation of the
baryonic component is non trivial and a thorough understanding of that mechanism will
greatly improve our understanding of galaxy formation.
For a virialized system one can apply the virial theorem to obtain the virial radius,
rver. The virial theorem ;7io that for a virialized system (U) + 2 (T) = 0 -> | (U) | = 2 (T).
Energy conservation condition gives
GMn GM G
-- = -- +(applying virial theorem)
Tmaz rvir 2re
rvir = ; iir = 2tturn.
Since r,i, = rmam/2, density of the system after virialization is 8 times larger than
its density at turn-around. In an Einstein-deSitter Universe, a oc t2/3 implying that the
background density would have diluted hv a factor of 4. So the density contrast of the
system after virialization is
Avir = =18xr2 ~ 180.
2.4.2 Press Schecter Formalism
The Spherical Collapse 1\odel describes the formation of dark matter halos. Press &
Schecter (1974) described the formation of large scale structures based on the hierarchical
clustering model. In particular, they arrived at an analytical expression for the mass
spectrum of virialized objects. Spherical collapse model predicts that overdense regions
collapse and form hound objects when their amplitude, as predicted by the linear theory,
approach the value 6,. = (3/20) [12xr]2/:3 1.686. PreSs & Schecter assumed that objects
on a particular scale start collapsing when the density field, smoothed out on that scale,
exceed the critical value be.. Press-Schecter analysis starts with the assumption that the
primordial density fluctuations are Gaussian. Then the probability that a fluctuation has
an amplitude 6 is
P(6; f) = x-p [-62/2es(2Q)
If we consider fluctuations of mass Af, then the fraction of those fluctuations which
have their amplitudes exceeding the critical amplitude 6,. and hence have become bound is
F(Af) = P(6; Af)d6,
1 rfv ;a- (2-20)
So the fraction of perturbations with masses in the range At and At + dM~, that have
formed bound objects is
In the hierarchical model o-(R) becomes arbitrarily large as R 0 So we would expect
F(0) =1 But,
F(0) =[1 erf (0)]
So Eq. 2-20 under predicts the number density of virialized objects by a factor of 1/2.
This is because a perturbation may have an amplitude less than the critical amplitude
6c when smoothed out on a scale R but may have an amplitude greater than be on a
larger smoothing scale. The Press-Schecter analysis ignores this fact. This is called
"cloud-in-cloud" problem. But Press & Schecter recognized this problem in their
approach and assumed that this discrepancy can be explained in terms of halo mergers
and multiplied their result by a factor of 2 and proceeded without any further analysis.
The number density of bound objects with masses in the range M~ and M~ + dM~ is
du p 8F
(M~)dM~ = 1~Id M
dur 2, dllnv p
(M) =x vexp/2 (-v2/2
Low resolution N-body simulations of hierarchical clustering can provide us the mass
functions for CDM halo distributions. The mass spectrum of bound objects as predicted
by Press-Schecter analysis matched reasonably well with the simulation results in the
high mass end of the spectrum but predicted more low mass objects than are seen in the
simulations. The source of this discrepancy can be traced to the fact that the spherical
collapse model assume the halos to be perfect spheres. But the halos are more likely to be
triaxial system. Sheth & Tormen (1997) extended the analysis to account for ellipsoidal
collapse and derived an expression for the mass function which included additional
parameters which requires to be fixed through the results of N-body simulation.
dn p dln u
(M~) = VfsT (u) (2-22)
v fs() = 2A[1+ (qu2)' -p 2 Oq p -qu2/2) (2 23)
Press-Shecter form : The Sheth-Tormen form of mass function includes the
Press-Schecter form as a special case. The Press-Schecter form is recovered through
the following choice of parameters.
q = 1, p = 0, A = 1/2.
Sheth-Tormen form : The following choice of parameters provide a good fit for
the simulation results
A(p) =I 1 +' 2- 0.3222,
pm 0.3, qm 0.707.
2.4.3 Halo Density Profile
Halos are virialized system whose average density is a,,, times the background
density. When dark matter perturbations grow and become non linear they virialize
to form bound objects with a density profile. The virial radius, r,, which marks the
boundary of a halo is defined as the distance from the halo center within which the
average density is 6,i, times the background density. The virial overdensity Asi, is
dependent on cosmology. For an Einstein-deSitter Universe (a = 0, am = 1),
a,,r ~ 180. In principle it is possible to find analytical expressions for the density profiles
of objects that result as an end product of violent relaxation. However deriving analytical
expressions for the halo density profile proved difficult. All lli-; under ideal conditions
such as self similar collapse of spherical perturbations in Einstein-deSitter Universe
-II- -_ -rh I1 power-law profiles. The simplest model for the density profile of virialized
system is that of an Isothermal sphere (p OC T-2). However Efstathion et al. (1988)
and Frenk et al. (1988) observed significant departures from the power-law in the halos of
CDM simulations. High resolution N-body simulations make it possible to determine the
halo density profiles empirically. Studies by T N.1- Iro, Frenk & White (NFW) and Moore et
al. (j1\')9) indicate that the density profile is universal, meaning the shape of the density
profile is independent of the halo mass. But the density profiles showed a characteristic
scale. So the density function can be written as a mass dependent amplitude times a
normalized profile, which is universal in units of scale radius r,.
NFW and M99 -II_t---- -1.* the following forms for the density profiles respectively,
p(r) (NFW) (2-24)
(r rs [ +(r /rs)]2
p(r) (:\!')9) (2-25)
(r/rs)3/2 [I (Ts 3/2i
where the scale radius, r,, and amplitude, p,, are functions of the halo mass. The
scale radius, r, is related to the the virial radius r,, through the concentration parameter
c = re/r,. For normalization we make use of the fact that the total mass inside the halo is
.r 4xiper~ [1n(1 + c) ir](NW)
o4xipsr~ [~ In(1 + c3/2) .~l)9).
This fixes the amplitude p, in terms of the concentration parameter c as,
In (l+c) -c/(l+c)i-p (NW)
Here, the density contrast at virialization is approximated as 6, ~ 200
N i.1- Iro et al. (2004) -11---- -r another form for the density profile.
=exp (-2/a~ [(r/r-2a) 1]); a~ ~ 1.7 (2-26)
r-2 is defined as the distance from the halo center at which the density profile locally
resembles an isothermal profile, i.e. p(r-2) c -2 lOCally.
Refer to Figure 2-1 for a comparison of the three profiles.
2.4.4 Halo Correlations
The halo correlations are not as well studied as the other two quantities. As long as
our interest is in a regime dominated by the 1-halo term, the halo correlations are not a
source of concern. However once we are beyond that regime the only prescription that
is available is that of Mo & White. They developed a simple analytical model to study
the gravitational clustering of DM halos. They found the statistical distribution of DM
halos using the extended Press-Schecter formalism and followed the modification of this
distribution by the gravity induced motions, using the spherical collapse model.
Their studies indicated that the DM halos do not cluster the same way as the mass
density field. They studied how the spatial distribution of DM halos are biased relative to
that of mass and arrived at a relation for the linear hias. According to their formalism the
correlation function of a set of clusters with masses mi and 177 separated by a comoving
distance r is given by,
fix (m77, m722 r) a b(mi)b(1772 in (r *
where (u,z is the mass correlation function predicted by the linear theory and
b(m) = (1 +).
Jing (1998) found some disagreement between the Mo-White results and the results of
numerical simulations at the low mass end and so modified the original formula of Mo and
White to fit the simulation results at lower mass end.
b(m) = (1 + ) (1 + )0.06-0.02/1
6,. 2 174
Refer to Figure 2-2 for a comparison of the values of Mo-White and Jing hias
parameters for the case of Press-Schecter and Sheth-Tormen mass functions. There is
no Jingf correction available for Sheth-Tormen mass function. Observe that below a mass
of 5 x 1013Me, the bias parameter is less than unity indicating that structures below this
scale are anti-biased.
2.4.5 Halo Occupation Moments
Studies by Berlind et al.(2003)  and K~ravatsov et al.(2004)  showed that the
description of P(N|MI) is considerably simplified if one distinguishes the contribution of
central galaxies from satellite galaxies. To model the observed (2 for SDSS galaxies, Zehavi
et al.  parameterized (NV)M foT galaXieS brighter than a threshold luminosity as,
(N)Mn/ = (nc>)M s1V)M (2-27)
This model has three free parameters, Mmm,, M~ and a~. Halos of mass less than Mmen
cannot host a galaxy. My is the normalization mass of the power law for satellite galaxies
and a~ is the power law index.
However Berlind et al.  noticed that (Ne,)M iS HOt StriCtly a Step function but gets
smoothed out because of the scatter in the relation between the baryon mass of the central
galaxy and the mass of the halo. So the (Ne,)M iS expressed as,
(nc~/r -1 log M logf M~,1 2
2 Elog M
Also at low masses (Ns,)M drops below a power-law extrapolation of (Ns,)M for higher
masses, leading to the following expression for (Ns,)M/
(Nis) M M (2-29)
Thus the halo occupation numbers are described fairly accurately by a five parameter
(l,4,, i1,, M(;, alogMr, a) model. The five parameter model gives a near perfect fit for the
results of SPH and SA simulations.
Zheng zheng et al.(2005)  extract these parameters from the SPH and SA
simulations for various values of galactic number densities(n,). Refer to Table 2-1 for
The other halo occupation moments written down in terms of (Nc,) and (Ns,),
(NV) __ ) 2(Nc,1s) + (N B2])
(Ni) __ [4l]) [3i~]iVs c N[3i])) + 6(N,2]N[~2] [4S~])
(Ni[s], = Nvs]) + 5((N[4l]Ni~s + 1~cIV[4)) 10 ((N, [~2]) 1) [t(i5])
If we assume that the satellite galaxies are Poisson distributed, then these expressions
(NE*) = (Ns)" + p(Ne,) (Ns,)"-l (2-30)
This is a very usefkil result which we will make use of later. Refer to Figure 2-3 for a plot
of (NV)M/ RS a function of M~
Comparison of the Halo density profiles of NFW, Moore et al(j1'9) &
Navarro et al.. NFW profile is shown in red color, M99. profile in green
and T i.1- lI ro et al. in blue. Observe that the Navarro et al. profile is in good
agreement with M99. profile in the core but deviates from both NFW and
M99 at the halo boundaries.
Table 2-1. HOD parameters from SPH and SA simulations
log ii t,
1 101 10" 103 104 105 106 107 108 109 1010 101 1012 10'3 1014 1016
Figure 2-2. Comparison of the Halo hias parameters. The blue curves are for
Press-Schecter mass function and the red curves are for the Sheth-Tornien
mass function. The solid lines are for Mo-White model of hias and the dashed
curves are for Jingf corrections. There is no Jingf correction available for the
Sheth-Tornien mass function.
1011 10"' 1013 1()14
Figure 2-3. The Halo occupation number is plotted as a function of halo mass. The red
dotted curve represents the contribution, 1V due to the central galaxy and the
cyan curve represents the contribution, 1V due to the satellite galaxies. The
blue curve is the sum of 1V and 1V.
GALAXY CLUSTERING IN THE NON LINEAR REGIME
3.1 Galactic Distribution And Redshift Surveys
Galaxy clustering studies go back to the d .1-< Of Hubble (19:34), who studied the
frequency distribution of the count NV of galaxies in a telescope field and found it to be
log-normal. Bok (19:34) and Mowbray (19:38) compared the variance of NV with what would
be expected for a statistically uniform Poisson distribution and found it to be considerably
Cosmographical studies of Shapley and his associates showed a complex distribution
of galaxies. Cosmography of the early d .1-4 suffered the 1 Iin .]ur handicap of lacking a three
dimensional picture. In an attempt to circumvent this problem, Limber (195:3, 1954)
arrived at a relation connecting the angular correlation function w(0) with the spatial two
point correlation function ((r). Using that relation he estimated the two-point correlation
function for the Lick survey.
Large amounts of data generated hv the Lick survey motivated Neyman and Scott to
devise statistical methods to quantify the galactic distributions. Among the many choices
that were available to quantify clustering statistics, the N-point correlation functions
method emerged as everyone's favorite. The cell count method developed by Abell (1958)
to study the distribution of galaxy clusters is also very useful.
Completion of Lick and Zwicky surveys and the improvement in computational power
took the cosmologists to a position where they could extract the first few correlation
functions from these catalogs with some precision and reproducibility. Peebles &
Groth(1975) extracted the two and three point correlations and found that the three
point correlation function can he expressed as a simple function of two point correlation
(123 = (12 13 + 21 23 31 32). (3-2)
y ~ 1.77 + 0.04; ro ~ 5 h-lrMpc;
d2~ 0.85 ( Zwicky catalog );
1.24 (Lick catalog ).
Fry & Peebles (1978) calculated the four point correlation functions which was the next in
the hierarchy. They too found that the four point correlation function can be expressed as
a function of two point correlation function.
rl1234 a 12 2a~3 34 + + 11 terms]
+Rb ~12 13 14 + -+ 3 terms]. (3-3)
R, = 2.5 +0.6,
Rb = 4.3 + 1.2 (Fry & Peebles 1978).
All these improved our understanding of galaxy clustering substantially and
constrained the models of galaxy clustering. However our understanding of galaxy
clustering is not yet complete.
3.2 Non Linear Gravitational Clustering (NLGC)
Galactic clustering is strongly non-linear. An analytical understanding of Non-Linear
Gravitational Clustering(NLGC) is difficult to achieve. High resolution N-body simulations
help us to track down the evolution at each stage but they do not offer any physical
insight. So an analytical understanding of the problem is alr-ws- desirable.
3.2.1 Cosmological BBGKY Equations
The physics of NLGC is described by the BBGK(Y hierarchy of equations for the
phase space distribution function. Davis and Peebles (1978) attempted to get an exact
solution for NLGC by solving the BBGK(Y equations. Since the galaxies are strongly
clustered, they treated the Universe as a strongly non-ideal gas of galactic particles which
interact only through gravity. The BBGK(Y equations are infinite set of coupled equations
involving all order of correlation functions. To close these equations one has to terminate
the hierarchy by dropping the correlation functions beyond some chosen order.
For this part they applied the empirical result that the three point function ( can he
expressed as a simple function of the two point function I.
(123 = (12 13 + 21 23 + 31 32) -3
This is a K~irkwood-like approximation except for the term (12 23 31 which is missing. If
it were present it would have dominated at small scales and observations do not indicate
that. This closes the BBGK(Y hierarchy leaving a set of non-linear integro-differential
equations which can he solved in principle. But in practice they are intractable. However
the concepts that they introduce along the way like the Stable Chal-I i !t!g hypothesis and
the corollary results are invaluable and are useful till date.
The stable clustering hypothesis ;7in that at small scales gravity induced peculiar
velocities exactly cancel the Hubble flow and as a result high density regions decouple
from the Hubble flow and maintain a constant physical size thereafter. An important
result that came out of their analysis is that for scale free initial conditions (P(k) ~ k")
in an Einstein-deSitter Universe (Rm = 1, a = 0) under the Stable clustering limit, the
logarithmic slope of the non-linear two point correlation function is related to the spectral
index n as
The implication of this result is that if stable clustering hypothesis holds true, then the
non-linear density field retains the memory of initial conditions and one can extract the
primordial spectrum which is one of the main goals of Structure Formation theorists.
3.2.2 Scale Invariant Model
Though it is not possible to achieve an analytical understanding of NLGC at this
moment, we can infer something from the scaling properties di;11l-p II by it. Since gravity
is scale free, gravity driven clustering should display simple scaling properties. This would
manifest in the behavior of N-point correlation functions. This leads to the formulation
of the the scale invariant models which are characterized by the scaling property, which is
written as the hierarchical ansatz.
(a (Axx, -, xN) = A-(N-1)YN (x1, -, xN) (3-5)
This agrees with the existing data for NV = 3, 4, 5 and with numerical simulations
for NV = 2. It is expected to hold over the entire range of scales where the two-point
correlation function follows a power law I oc r-Y which is 0.1 h-1Mpc < r < 100 h-1Mpc.
Though this relation is entirely empirical it has the appealing feature that it admits a
self similar solution to the BBGK(Y equations under the stable clustering assumption in an
Einstein-deSitter Universe which has no characteristic length or time scales
3.2.3 Hierarchical Model
Observations of galaxy clustering at small scales and the self similar solution of
BBGK(Y hierarchy equations motivate the Hierarchical model which was written in its
most general form by Fry(1984b).
E(Xl.. (xX, ,x) = Q,,,p C AB. (3-6)
a=1 labels edges
So the p-point correlation function is constructed from the linear superposition of the
product of (p 1) pair correlation functions summed over all possible topologies. This
expression has a tree graph representation called p-trees which are graphs linking p
vertices(galaxies) with (p 1) edges.
Each vertex is a galaxy.
To each edge a pair correlation function is associated.
A p-tree has tp = pp-2 distinct topologfies.
An amplitude Q,~, is associated to each topology.
The functional form of two point correlation function is known and so what needs to be
determined are the amplitudes Q,~,. This can be done either analytically or empirically.
Attempts to derive Q,~, from basic principles was strongly resisted by the complexity of
the problem. Fry(1982, 1984) took the first step. To reduce the complexity of the problem
he made an assumption that the amplitudes Q,~, are independent of topology, scale and
configuration. Solving the BBGK(Y equations assuming a hierarchical form for the phase
space p-point distribution function along with other symmetry assumptions he arrived at
the following relation for Q,~, = Q,
Q, = (3-7)
2 1p-1 i(~p-
Relaxing some of the symmetry assumptions that Fry made, Hamilton (1988) arrived at
the following relations.
But the applicability of these results is severely restricted by the assumptions that are
embodied in them. To fit the numerical results Balian and Schaeffer (1989) wrote down.
Qp = ,, V ~ +1. (3-9)
The computational labor required to extract the correlation functions from the galaxy
catalogs become increasingly difficult as we go up in the order. So, that is not an efficient
way of handling the problem. Determining Qp,a's empirically is ruled out.
To completely characterize the statistical properties of galactic distribution we need
all the finite order correlation functions. Since calculating them from the galaxy catalogs
is not within the reaches of our computational abilities one has to look for alternative
statistical measures. One of the most promising alternative statistical measures are the cell
3.3 Cell Count Moments
Cell count moments were introduced by Abell (1958) as a statistical measure of the
distribution of discrete objects. In the Count-in-cells analysis the statistical properties
which is the probability distribution function of finding NV objects within a volume V
placed at a randomly chosen position. The general properties of the CPDF have been
studied by White (1979) and Peebles (1980).
3.3.1 Fundamental Equations
One can define a smooth number density field n(r) and the galactic distributions
are defined through a random sampling of this continuous field. The factorial moments
(NE~) of the number of galaxies in a randomly placed cell of volume V are related to the
moments of the underlying continuous density field. To find this relation the procedure is
as follows. Divide the cell into infinitesimal elements such that the elements have either
one object or no object at all. Now the total number of objects in the cell is given by
The summation is over all the elements in the cell and as is the number of objects in
cell i. Since the probability of finding more than one object in a cell is infinitesimal, ni has
the following properties
= n36VibV26V3( 3 i 1 a ,,
(ni_ sin)= n46VI 6V4 4
nl)(n2 82 n>
na2 3 3 >
ns3 4 4 >
(r1234 12 34 ~
n2 VibV2 12
n3 VibV2 V3 123
+ 13 24 + 14 23 -
((1 n1 82
If 5%/' is the volume of element i, then (n ) = ab%~.
adV = nV = N.
(N) = (~ us)
The first five moments are
Here p, and (p are the
2 312, 3 13~,
2 712, + 63 3, p4 14~,
1V 151V2 2 + 251V3 3 + 101V4 4 p5 155
raw and irreducible moments, respectively, and are defined as
1 d37 0ppE, p)
and the are related to each other through the following relations.
p4= 1 + 662 + ~3 +43 4
I = 1 +102 + 1"52 + 103 + 10(2 3 + "54 5.
The factorial moments of the count are defined as
NW1l N!/(NV p)! = NV(N 1) (NV p + 1). (3-17)
It can be verified from Eqs. 3-13 3-16, that the factorial moments are related to the
volume averaged moments, py, of the continuous number density field as
(NW1 = 1~pp.(3-18)
NON LINEAR CLUSTERING AND THE HALO MODEL
4.1 Power Law Clustering Model
The power law cluster model is one of the earliest models used to study the
galaxy correlation functions. This was pioneered by Neyman & Scott (1957) and was
generalized by Peebles (1974) and McClelland & Silk (1977) . In this model galaxies
are distributed in clusters that are randomly located. The clusters are assumed to have a
power law density profile given by
n(r) = nAr-t. (4-1)
where A is a normalization constant that depends on the cluster mass.
The conditional probability of finding an object within a spherical shell of radius r
and thickness 6r, from a randomly chosen object is
P(2|1) = n[1 + ((r)]6V, 6V = 4xrr26r. _2)
Since the clusters are randomly distributed the excess neighbors to a given object arise
from objects in the same cluster. Now the total number of pairs with separations rl2 and
T12 + 12l from ally given object from a randomly chosen cluster is
26NV,(rl2 7 78718Y)DE1 2-7
Averaging this over a spherical shell of radiuS T12 and thickness 6r12,
where r, = r~ + rf2 2rlrlalp. On substituting Eq. 4-1 for the density profile in Eq. 4-3
26NV,(rl2) 8 r2A2n2r~' r~ 5-e71
If NV is the total number of objects in the cluster,
N= nrR d 4xurAr3-e
Now the correlation function is given by
(3 )Ar -2e e-3 6)
In the limit r
((r) OC T3-2t _n7
Now one can choose e to be 0.6 to match the behavior of observed galaxy correlation
function on small scales (~(r) oc r-ls), but the higher order correlation functions for that
value of a show large deviations from the observed value.
Peebles showed that the power law cluster model does not work. In the power law
cluster model galaxies are assumed to be distributed in clusters with a power law density
profile from the cluster centers. Such a model predicts power law correlation functions
that are inconsistent with observations. In hind-sight we know the reason why the power
law cluster model did not work. Numerical simulations show that the density profiles
of halos are not smooth power laws but have a characteristic scale. That makes all the
difference. Numerical simulations have shown that if we start with a smooth distribution
of matter, under gravitation, they evolve into a complex network of sheets, filaments
and dense knots. These knots are identified as virialized Dark Matter halos. Further the
clusters themselves cluster contrary to the assumption of the power law cluster model,
that the cluster positions are uncorrelated.
4.2 Halo Model
Halo model is a semi analytical model used to study Non linear Gravitational
Clustering (NLGC). In the halo model the density field is decomposed into a distribution
of virialized dark matter halos with some density profile. This is more or less what
Neyman & Scott (1952)  so-----~ -rh I1 earlier. They found that galaxy distributions are
made up of distinct clusters with a range of sizes. The power law cluster model studied by
Peebles (1980) and Mc Clelland & Silk (1977) too, is a primitive halo model.
4.3 Halo Model Formalism
The formalism of Halo model in its present form can be traced to the works of
Scherrer & Bertschinger (1991) who derived the fundamental expressions for the p point
correlation functions and in terms of halo correlation functions, halo mass function and
halo density profile.
Consider a distribution of dark matter halos with the halo centers positioned at xi.
Let mi be the mass of the halo whose center is located at xi and p(mi, x xi) be its
density profile. The density fluctuation field at an arbitrary point x is then
b(x) = p(m4 x x) (4-8)
where the summation is over all the halo centers. The summation over the halo centers
can be replaced by integrals using the Dirac delta notation as
dm d3 j X D XiuUD 0 Ili
Identify that the ensemble average of the sum over Dirac deltas is simply the mass
function of the halo distribution.
(r "3 (x- x )s~ bD0-mi)) =(mn) (4-9)
a(x) = d3Zim" 6 X-Xi)6D 0 i) p 0, X Xi (410)
The 2-point correlation function of mass
I(2 1, a2 (6(1 72~)
Consider the ensemble average over the product of the Dirac delta terms,
(C 63~ X1~ i6 1-M) :b6x X2 X)bnD 2 -m,)
It can be resolved into a 1-halo term (i = j terms) and a 2-halo term (i / terms).
dm 1n dm 2 ) t h ) (11 1 41 2; X 1 X 2)
dm\I1/VD X1 X2)bD 01 m2)
(60D "")) (2h)
The first term corresponds to the case when the density perturbations at xl and x2
are inside two different halos and the second term corresponds to the case when they are
inside a single hlalo. J"( (my,n; 2X1,xn X2 1 the h~alo correlation function. So the twvo point
correlation function can be written as a 1-halo term plus a 2-halo term.
W1 3 1 W1) ; l 1 X1) Wl, 72 X1) (4-12)
dm d dm d
x d31 32 0) E 1 0) E 2 d) 0.1T"1, MX1, X2) 13
l 2h) (r1, r2
Similarly the expression for the p-point correlation function involves terms that
can be resolved clearly into 1-halo term, 2-halo term, ... etc. upto the p-halo term. The
1-halo term corresponds to the case when all the p points are inside a single halo, the
2-halo terms correspond to the case when all the p points are distributed between 2 halos
and so on up to the p-halo term which corresponds to the case when all the p points are
distributed in p separate halos.
On the smallest scale the 1-halo term is dominant and on the largest scale the p-halo
term is dominant. It is obvious why it should be so. On the smallest scale all the p-points
are more likely to be inside a single halo while the likelihood of them being in distributed
in multiple halos increases with scale until we reach a state where they are most likely
to be distributed in p separate halos. Though the expressions are lengthy they display
beautiful symmetry properties and one can write down the expression for any order guided
purely by the regularity of pattern and intuition.
The expression for the p-point correlation function in the halo model is,
(p( x --,x ) I1.. (4-14)
q=1 ji=1 jp=1
l= 1 i= 1
4.4 Model Ingredients
Fr-om these expressions it is clear that the halo model requires the following quantities
Halo mass function, du dm,
Halo density profile, p(m, x),
Halo correlation functions, p(ml, m,; xl, x,),
Halo occupation moments, (1 #1).
Low resolution N-body simulations provide us the halo mass function while the high
resolution N-body simulations provide us the halo density profile. If our interest is only on
small scales we do not need the halo correlation functions. This is because on small scales
all the points (galaxies) are most likely to be inside a single halo and the 1-halo term is
4.4.1 Halo Mass Function
Low resolution hierarchical simulations give us the mass spectrum of collapsed
objects. An analytical expression for the mass spectrum as derived through the Press-Schecter
formalism predicts the mass spectrum reasonably well in the large mass end but deviates
significantly front the simulation results for the small mass end. Sheth & Tornien corrected
this generalizing the Press-Schecter fornialism to triaxial system and their formula
provides a better fit for the simulation results.
The Press-Schecter and Sheth-Tornien mass functions are written in the general form,
dn p d 1n u
( Af) = v f (u)
Sheth-Tormen form : The following choice of parameters provide a good fit for
the simulation results
v fS() = 2A [1 +(qu2D -p 2 Op ( -qua2/2 (4-15)
A(p) = 1 + 2 m 0.3222,
p a 0.3, qm 0.707.
Press-Shecter form : The Sheth-Tornien form of mass function includes the
Press-Schecter form as a special case. The Press-Schecter form is recovered through
the following choice of parameters.
q = 1, p = 0, A = 1/2.
fPS = exp (-v2/2)
4.4.2 Halo Density Profile
High resolution simulations carried out hv ii.- I1ro, Fr-enk & White (NFW) and
1\oore et al. (j1!19) indicate that the density profile is universal, meaning the shape
of the density profile is independent of the halo mass. But the density profiles
showed a characteristic scale. So the density function can he written as a mass
dependent amplitude times a normalized profile, which is universal in units of scale
NFW and M99 -II---- -rh I1 the following forms for the density profiles respectively,
u~) (1+2)2 (4-16)
232 3/ MOOre et al.
4.4.3 Halo Correlations And Halo Bias
Mo & White showed that the DM halos do not cluster the same way as the mass
density field. They studied how the spatial distribution of DM halos are biased
relative to that of mass and arrived at a relation for the linear bias. According to
their formalism the correlation function of a set of clusters with masses mi and m2
separated by a comoving distance r is given by,
(a (m m2, r) a b(mi)b(m2 ldm 7 -
where (am is the mass correlation function predicted by the linear theory and
b(m) = (1 + ).
Jing (1998) modified the original formula of Mo and White to fit the simulation
results at lower mass end.
b(m) = (1 + ) (1 + )0.06-0.02n
4.4.4 Halo Occupation Distribution
For the Halo occupation distribution I use the five parameter model developed by
Berlind et al.  and is discussed in section 2.4.5 of ('! .pter 2. According to this
model the galaxies are divided into a central galaxy and satellite galaxies and the
average halo occupation number is written as,
where the occupation number of the central galaxy, No, is,
(ni~1/ I e log M logf Mmin)
2 Elog M
Also at low masses (NVs)M/ drops below a power-law extrapolation of (Ns,)M foT
higher masses, leading to the following expression for (Nis)M
Thus the halo occupation numbers are described fairly accurately by a five
parameter (Mmin,,, Ml ~, alogM/, a~) model. The five parameter model gives a
near perfect fit for the results of SPH and SA simulations.
If we assume that the satellite galaxies are Poisson distributed, then the halo
occupation moments can be cast into the following form,
(NW1)> = (Ns)" + p(Nc,) (Ns,)"
4.5 Halo Model And Non Linear Clustering
Halo model cannot be literally true. It is an idealization of the real world which
is complex. However the halo model has successfully explained the evolution of
power spectrum in the non-linear regime. Prior to the advent of Halo model the
evolution of power spectrum was explained by the Linear Non linear mapping
technique formulated first by Hamilton, K~umar, Mathew and Lu (HK(LM) 
and later generalized by Peacock & Dodds (PD) . But this technique had the
stable clustering hypothesis in-built. The validity of Stable clustering hypothesis
has been questioned in the recent years. Ma & Fry (2000)  showed that the
stable clustering is applicable only under limited conditions and is not true in
general. Further this technique can be applied only for the power spectra and not for
Halo model has been applied successfully by Sheth & Jain  to study the evolution
of two point correlation function in the non linear regime. Similar studies have
been done on power spectra (,  & ). Applicability of Halo model is not
restricted to power spectra. It has been used to study poly spectra of all order and
has successfully explained the evolution of poly spectra. Ma & Fry 2000b  have
demonstrated that the Halo model can retrace the power spectrum and bispectrum
generated by cosmological simulations for various models.
CELL COUNT MOMENTS IN THE HALO MODEL
I study the statistics of the distribution of mass (dark matter) and the galaxy
number counts in the halo model and compare the results with the results of
numerical simulations and the corresponding quantities extracted from galaxy
catalogs. On scales larger than the largest halo, the counts are well explained by
the point cluster model which ignores the internal structure of the halos. In the
intermediate scale the one halo term of the point cluster model is dominant and
one can still ignore the internal structure of halos so that the one halo term of the
full halo model reduces to the one halo term of the point cluster model. On small
scales the internal structure of halos become important and a full halo model with
its internal structure resolved is required to explain the counts. The halo model
calculations of hierarchical amplitudes for scale free power spectra produced results
that are in reasonable agreement with the results of numerical simulations  and
with the results of calculations based on the Hyper Extended Perturbation Theory
(HEPT) . As a next step I apply the halo model expressions to the CDM power
spectrum to study the volume averaged moments of galaxy counts and that of mass
distribution. On large scales the naive point cluster calculations agrees qualitatively
with the results of numerical simulations but fails in the details. This source of this
discrepancy can he traced partly to the assumption that the halo number and halo
occupancy are statistically independent, which is found to be incorrect. I relax this
assumption and quantify the correlations in terms of hias parameters and correct
the point cluster expressions that includes the bias parameters. On the smallest
scales the internal structure of halos become important which would result in the
introduction of a form factor to the point cluster results.
5.2 Statistical Definitions
Galaxies being point particles, their distribution is a point process. However it is
convenient to describe their distribution by means of a continuous random field
n,(x). L?- cr (1956)  proposed an algorithm called Poisson process to construct
a discrete distribution from a continuous field. Poisson process assumes that the
probability of finding a galaxy in an elemental volume 6V surrounding a point x is
proportional to the the value of the continuous galaxy number density field n,(x)
at that point and is independent of the occupational status of the neighboring
volume elements. There is no unique way of constructing discrete distributions
from continuous fields. But Poisson process produces discrete distribution with
desired properties and has the advantage of giving a discrete distribution whose
ensemble averaged spatial correlation functions coincide with those of the underlying
continuous density field.
One can define the density contrast in the continuous number density field as
by (x) = (-1
where n, is the mean number density of galaxies. Poisson process acts locally
implying that the number count of objects within a finite volume V is a random
variable whose statistical properties are determined by the locally averaged value,
by of 6,(x). In a Poisson process NV is Poisson distributed with a mean of X
n,(1 + 6v)V.
Fry (1985)  showed that the moment generating functions (mgf) of the
continuous field and that of discrete counts are related by the following replacement.
Maisrete1) AT .,et -1),(5-2)
This gives the moments of discrete counts as derived in Peebles (1980) 
(NV) = NV, (5-3)
((NV 1V)2) __ V2 2, (5-4)
((NV NV) ) = NV + 3NV2 2 13 3, (5-5)
((NV 1V)4) _(1 V2) __ 2 1 2~ + 6V3~ 3 V4 4. (5-6)
Observe that in the discrete limit (small NV ) the first term on the right hand side
which is called the shot noise, is the dominant term and the distribution of counts
reduce to Poisson distribution with mean NV. In the continuum limit (NV c o) it
approaches the continuous distribution. The factorial moments of the discrete counts
are defined as
NWB1 I = N(N-1) ---(NV- p +1).
It can be verified from Eqs. 5-3 5-6, that the factorial moments of the discrete
counts are related to the volume averaged raw moments, py, of the continuous
number density field as
(NE'1) = NW,(5-7)
p, d371 d 30prl p 1 p)
The raw moments p, are related to irreducible moments (p as
S4= 1 + 6 (2 3 4 ,
I = 1 +102 + 1"52 + 103 + 10(2 3 + "54 55-
These results can be summarized by writing the exponential generating functions for
the raw and irreducible moments
M~t) = Np~t1, (5-8)
K((t) =i NW,.~ (5-9)
where K(t) = log(M~(t)).
For discrete counts with probabilities PN Szapudi & CI I1 li- (1993)  use the
G~z) = CPz". (5-10)
G(z) is related to the exponential generating function through the relation
M~(t) = G(z)zt+i. (5-11)
In the hierarchical models the p-point correlation functions are related to the 2-point
correlation function. The hierarchical amplitudes, S, are then defined by the relation
(p = A'- .(5-12)
But (p is not strictly proportional to ("-1 on all scales. So the hierarchical
amplitudes, S, are scale dependent and I study the scale dependence of the
hierarchical amplitudes in the context of halo model.
5.3 Point Cluster Model
On large scales, when the cell volume includes many halos, the contribution to the
galaxy counts from those halos that are on the cell boundaries is negligible. So the
internal structure of the halos can be ignored. Counts are then explained by the
point cluster model. In the point cluster model, total count NV in the cell volume is
N = Ns.(5-13)
Nsh is the total number of halos in the cell volume V and Nsi is the occupancy
number of a given halo. Nsh and Nsi are random variables. In the point cluster model,
counts are explained by convolving the statistical properties of the halo number, Nsh
and the halo occupancy, Nsi. Given PN one can use the generating function G(z)
to characterize the distribution of total count. If p, is the probability of finding a
halos in a cell and if qm is the probability of finding m galaxies in a halo, then the
probability for the total count, NV for the first few NV assuming that no halo is empty,
Po = po (5-14)
Pi = plqg (5-15)
P2 =2 p9192~ (5-16)
P3 P3ql + 2q192 + 193 (5-17)
Pq4 p449 3p34 92 + 2(2qlq3 + 42" 19pl4 (5-18)
Ps = psts + 4p49 92 93 p3(3 3 341422 + 2(2qlq4 + 2q243) p195. (5-19)
Szapudi & CI. I1 li- (1993)  show that the generating function for PN is G(z)=
gh[gi(z)]. To find the exponential generating functions M~(t) and K(t), use the
relations M~(t) = G(z)|z=l+t and K(t) = log(M~(t)). The correlation functions could
then be extracted using the following relations,
NV"Ip, = |=0 (5-20)
d K (t)
NVp( = |=0 (5-21)
We can write the moments of the total count in terms of the distribution of the halo
number and occupancy distribution. The irreducible factorial moments of the total
count are dependent on the connected moments (p~a of the halo distribution and on
the raw moments py,s of the occupation distribution. The terms with Nsh reflect the
discreteness of the halo number distribution.
N = N N(5-22)
2a 2 ,h 2~,i (5-23)
3p2,i 2,h (5-24)
6p2,i 3 ,h (13,i ,
~4 = 4,h +++(5-25)
10pi ,h (10p~3,i + 15p ~,)(3,h (1p2i3, 2h 5,i
s ~ V =V sV" +V + (5-26)
The expression for (p has contributions due to all moments of occupation number
I-l,i convolved with the moments of the halo number (q,a. For instance, in the
expression for 1s, the first term correspond to the case of 5 separate halos each
with an occupancy 1 and the last term correspond to the case of a single halo with
occupancy 5. The other terms correspond to 4 halos with occupancy (2, 1, 1, 1) and
3 halos with occupancies (3, 1, 2) and (2, 2, 1) and 2 halos with occupancies (3, 2)
and (4, 1). One can follow this pattern and write the expression for the most general
term. The numerical factors are known as Stirling numbers of the second kind
S(n, m), the number of owsi~ of arranging a distinguishable objects into m cell with
no cells left empty.
Hierarchical ansatz imply that (p ~ r-CP-1). Since TVh ~ T3, the dominant
contribution to (p on large scale comes from the halo correlations (p~a. If halos are
poisson distributed, (p~a = 0 for p > 2, for which
(1 = pp.(5-27)
5.3.1 Corrections To The Point Cluster Result
The naive point cluster model assumes that the total number of halos inside the
cell, Nsh and the occupancy of those halos, Nsi are independent random variables.
Statistical correlations between them can be quantified in terms of bias parameters.
So the point cluster expressions need to be corrected for bias. The bias parameters
that appear in the bias corrected point cluster expressions are calculated analytically.
22.214.171.124 The bias corrected point cluster expressions
The results of the naive point cluster calculations do not fit the simulation results
even in the point cluster regime. The results agree with the simulations only
qualitatively and not quantitatively. This failure is partly attributed to the fact that
the point cluster model assumes the halo number, Nsh and halo occupation number,
Nsi to be statistically uncorrelated which is know to be incorrect. The statistical
dependence of these quantities reflect in the halo correlations and the occupation
moments. Massive halos are strongly correlated. The clustering properties of halos
are related to that of background density through the halo-bias parameters. The
clustering of p-point correlation functions of halos up to the linear term is
6,a(xi, mi) = b(mi) b(m,)&(xl, x) (5-28)
where b(m) are called the bias factors. The linear bias factors for halos of mass m
are given by Mo & White  as
Jing  corrected the Mo-White formula to find a better fit for the simulation
results and the corrected bias factor is
b (u) v (s 0.06-0.02n
These bias formulae are applicable for halos of a given mass. When halos with a
wide range of masses are considered, b(m) should be replaced by the value obtained
by averaging it over all the mass range.
E S-1-m4 L w(m)2)b (m2) dm~
where w(m) is the weight given for halos in the mass range m and m + dm and ifl
is the cutoff mass that is used to censor halos that are less massive than Mc. For
galaxy number counts the mass cutoff changes with occupation moments. To find
the bias parameter corresponding to the pes occupation moment we only consider
halos that can host p galaxies and all the halos that are not massive enough to hold
p galaxies are censored. Thus the weight factor depends on the mass function and
the occupation moments. The bias parameters corresponding to the pes occupation
moment for mass and galaxy number counts are
w(m) (dn/dm)m : for mass, (5-30)
(dn/dm) (NE 1) : or number counts.
So we have a hierarchy of mean bias parameters which, for mass and galaxy counts
are as follows,
J: dmn(d4ldn/d) m2'bb~ m)
f, dm (dn/dm) (NE 1)b(m)
S Jp dm~(du/dlm)( (NW1)
The cutoff mass for the bias parameters (b), of number counts The bias corrected
point cluster expressions can be written down by inspection,
:2 = (b)l~ (22, (5-33)
g=(b)" 3(b)L(b)2 2,~iG 2 "; 3,i
E3 (b):t (3jh 2 (5-34)t_3b)l~i 5
(s=(b); I(s + Ns
(10(b)2( b)3 2,,i 3, i + 5(b)4 (b) l1p4,i ~2 I-5,i
Observe that the one-halo terms are independent of bias parameters.
126.96.36.199 Background particles and statistics of mass
The assumption that objects outside the halo are statistically insignificant applies
only for galaxies and not for mass. In the continuum limit the background objects
significantly affect the statistics and a complete description should include the
background objects. So Eq. 5-13 is applicable only for the number count of
galaxies/substructures and not for mass. With this modification the cell count
N~b =V Ns +Nbb. (5-37)
Here NV' is the total number of objects inside halos and NVb is the total number of
background objects. It is reasonable to assume that NV' and NVb are StatiStiCally
uncorrelated. The generating function of the sum of two independent random
variables is simply the product of the individual generating functions.
G(z) = gh[gi(z)] x gb(z) (5-38)
K((t) = log(G(z)|z=1+t),
=log(gh(gi(1 + t))) + log(gbl + :)
= K'(t) + Kb t).
Using 5-9 we find the following result for the irreducible factorial moments.
d K (t) d K' (t) d Kb~t
|<=0 = |=0 + |<=0, (5-39)
dtp dtp dtp
niP(, = I'"( + R~~bpb (5-40)
If we further assume that NVb is POiSSon distributed, then (p~b = 0, p > 2.
z)' = ', (5-41)
Thus the point cluster results for the continuum limit differs from the point cluster
results of the discrete limits by a factor of (NV'/N) This changes the hierarchical
amplitudes for mass in the intermediate scales to
( 4 p-2lrv S'. (5-42)
where S, and S, are the hierarchical amplitudes in the intermediate scales with and
without the background objects respectively.
For mass the suppression factor is given as,
N dm(dn/dm)m. (5-43)
Similarly one can define the suppression factor for galaxy number counts but it
requires hypothesizing the existence of galaxies outside halo. This idea is against
the modern paradigm of galaxy formation, introduced by White & Rees (1978) ,
that galaxies can only form inside dark matter halos which provide the gravitational
potential required to cool the baryonic material.
NV1 n md/m)Nm.(5-44)
5.4 Full Halo Model
The point cluster model ignores the internal structure of halos and so are bound to
fail on small scales when the internal structure of the halo becomes important. We
need a full halo model which includes the details about the internal structure of the
halo. The halo model expressions for the p-point correlation functions are written as
(p(rl, r) = ( '") + ( 2h) p Ih)
On smrall scalesi the one-halo termr, ('") domrinatesi. Th'le onle-h~alo termr inl thle halo
model involve terms containing the convolution of halo profiles. For the distribution
of discrete objects the one-halo term is
,,, S d2($nd7/dm) (N[?'] (m~)) f dclEr~s~l 91 "" 8
~~" [ f dmr(dll/dm)l (N~(m)l) f~ d37/1y)P@ (55
where yi = |ri lr'|r,, and r' is the position of the halo center.
Averaged over the cell volume it becomes,
(R) = [J` dml(dlb/dmr)(NE' (m))] J;F d37 ` d3 / (5-46)
f, dmRd/dm (N())f 37 3
The halo expressions apply equally to mass as it is to the discrete objects (galaxies).
One can get the halo expressions for mass simply by the replacement (NW61) m".
Under this replacement Eq. 5-46 becomes
[Jdmdndmm d37 [~ 3 ./ 9~y
(R) = l~s/cmm' J (5-47)
51, [S dm(dn/dmn)m rfo d~r 3 pl (,]
5.4.1 Point Cluster Limit Of The Full Halo Model
The point cluster expressions are derived for the condition when the cell volume
is large enough to include many halos and so were independent of the internal
structure of the halo. In the point cluster regimes, the one-halo expression of the
halo model should reduce to the one-halo term of the point cluster model. So in the
point cluster regime (R > r,), the integral over the halo position r' becomes unity
and so the integral over the cell volume simply returns the volume, V of the cell. So
Eq. 5-46 reduces to the following form
Elh f dm);(ln/d~) (NB 1 (1()) V
~" [J dmrl(dll/dm)l (N~(m)l) V]p
The integral in the denominator becomes
The integral in the numerator becomes
jdmgidu/dm) (N(m))V = na(N )V = NaN'pp (5-49)
la Na~y, p .(5-50)
S (N i~)" N
Thus we recover the 1-halo term of the point cluster expression.
Now we are ready to evaluate the hierarchical amplitudes in the intermediate regime
where the one-halo term of the point cluster expression is dominant. In this regfime
the hierarchical amplitudes are scale independent.
16r~l) p~-1 p1
f. dmn(dn/dm)]"- [J dm(dn/dm)m ]
JS dmn(dn/dmjm '] [J dm(d~n/dm)m]p-2
[J dm(dn/dmjm2] p-1
[JI dm(dn/dm)]"- [JI dmr(dn/dmn) (NE 1l)]
[J` dmr(drll n/d (N~)]p
J~I dmr(d~l n/dm (NE [1)] [J' dmr~(drl/dmr) (N)]p-2
[J dm71(d?2/dm)2 (N11 p-
Thus, on intermediate scales, the hierarchical amplitudes are independent of the cell
5.4.2 Resolved Halo Expressions And Form Factor
The point cluster model fails on scales where the halos are resolved (R < 1 Mpc).
This can be fixed if calculations are carried out for a resolved halo. The scale
dependence of S, on small scales comes from the integrals over the halo profile. Since
the halo profile integrals are not coupled to mass dependent part, the results of a
resolved halo calculation will be some form factor, F(R) times the point cluster
S,(R)= )() F ()Sp,(5-55)
JOb d3 0"~ 3 p / ~)2]
Here y = |r r'|/rs and S is the value of hierarchical amplitudes in the scale
independent limit given by Eqs. 5-54 & 5-52.
5.5 Numerical Simulations
We apply the halo model results to galaxies and halos identified in the numerical
simulation. The numerical simulations are performed with the Adaptive Mesh
Refinement (AMR) code RAMSES which is an N-body hydrodynamical code developed
by Remain Teyssier to study structure formation in the universe at a high resolution.
The code is based on Adaptive Mesh Refinement(AMR) technique with a tree based
data structure that allows for recursive grid refinements on a cell-to-cell basis. For
details about the algorithm refer to Teyssier (2002) 
Simulations are performed for a LCDM cosmology ( Om = 0.3, RA = 0.7) with
Ho -- 100h = 70 km s- Mpc. The amplitude of initial power spectrum is normalized
such that o-s = 0.93, when extrapolated linearly to the present time. o-s is the
variance of the density fluctuations in a sphere of radius 8 h-1Mpc. Particles are
initially distributed in a regular grid pattern and then the initial conditions are
set up by perturbing them to generate a Gaussian fluctuations with a CDM power
spectrum using the Zel'dovich approximation. This is achieved with the COSMICS
package developed by Bertschinger. The simulation involves 5123 dark matter
particles distributed on the AMR grid initially regular of size 5123, in a periodic
cube of size Lboz = 200 h-1Mpc. Mass of individual particles is mo = 6.9881 x 109Me-
Additional refinements are carried out during run time using the standard AMR
techniques described in Teyssier (2002) .
Front the simulation results a mock halo catalog, Eix and a mock galaxy catalog,
E, are generated using the public domain software called adaptaHOP described
in Aubert, Pichon & Colombi (2004) . In preparing the halo catalog, halos
are identified as connected regions with density contrast larger than 80 relying
on a softening of the particle distribution with standard SPH technique using
NVSPH = 64 neighbors. In preparing the galaxy catalog, it is assumed that the
galaxy distribution is probed by substructures. K~ravatsov et al. (2004)  show
that the halo occupation distribution of substructures in the N-hody hydrodynantical
simulation results have many of the same features of galaxy distribution in the
Semi Analytic (SA)siniulations. This is the justification for identifying galaxies with
substructures. Berlind et al. (2003)  too found a remarkable similarities in the
predictions of P(N|MI) front the SPH and SA simulations, provided one chooses
mass thresholds that fields the same galaxy number density n,.
5.6 Galaxy Surveys
The purpose of galaxy surveys is to nmap the Universe in our neighborhood and see
how the galaxies are distributed. The clustering properties of the distribution are
described by the n-point correlation functions, 6,(rl, r,) and the goal of every
galaxy survey is to determine the n-point correlation functions more accurately
over a wide range of scales. But the estimation of n-point correlation function front
galaxy catalogs become increasingly difficult with increasing n. Though the n-point
correlation functions contain complete information about the statistical properties
of the distribution, one may not necessarily want all the information. A volume
averaged n-point correlation function (,(R) is a compromise that simplifies the
calculation and carries useful information. In the hierarchical model the volume
averaged n-point correlation functions are expressed in terms of the volume averaged
two point correlation function and this hierarchy is written as
,2(R) = S,zi"- (5-57)
So an important quantity that is to be extracted from the galaxy catalogs is the
hierarchical amplitudes S,2. Galaxy surveys are either two dimensional angular
surveys or three dimensional redshift sure 1-i For the purpose of this study I
consider an angular survey (APhi catalog) and a redshift survey (2dFGR S catalog).
5.6.1 Angular Surveys
Cosmography of early d 7i-< located only the angular positions of the galaxies and
so lacked a three dimensional picture. Extracting the spatial two point correlation
functions from the angular surveys requires calculating the angular correlation
function, w(0) and then under the assumption that galaxies are correlated only
over small distances, deproject it to find the spatial correlation function, ((r). This
technique was first developed by Limber (1953, 1954) and it has, since then, been
used to extract the two point correlation function from various angular surveys
and the results are consistent with each other. Though the angular surveys are two
dimensional and lack information about the third coordinate, they contain more
galaxies than the three dimensional redshift sury, -i- In addition to this the angular
surveys are free from distortions such as the fingers of God effect, Kaiser effect etc.
APhi galaxy survey is derived from the scans of 185 contiguous UK( Schmidt plates,
each covering an area of 6o x 60 on the sky which amounts to approximately 1(1' of
the whole sky. The plates were scanned with the Cambridge APhi (Automated Plate
Measuring) laser scanner. Since the image is centered on the South Galactic Pole,
effects of galactic obscuration due to dust is negligible. The plate contains about 10
million objects with apparent magnitudes in the range 17 < bj < 20.5 of which about
2 million are identified as galaxies. APhi survey demonstrated, for the first time,
that the Universe has more large scale structures than are predicted by the standard
CDM structure formation model.
APhi survey was conducted for the purpose of determining the angular correlation
functions accurately. Gaztanaga (1994)  estimated the averaged angular
correlation function, Lc,(0) for the galaxies in the APhi catalog and calculated
the angullar hierarchical amplitutdes 8;>(0) = ",(0)/ '- (0). H~e found that s,, is
roughly constant up to p = 9 between 8OD ~ 0.5 h-thipc and O~D ~ 2h-1 ipc and
the decreases slowly. ~D ~ 400 h-thipc being the survey depth. He then derives an
expression to invert the angular hierarchical amplitude, s,(0) to spatial hierarchical
amplitude SI,(R). The results are shown in Figure 5-1. The variance of the galaxy
distribution in the APhi catalog is shown in Figure 5-2.
5.6.2 Redshift Surveys
To determine the three dimensional distribution of galaxies, it is important to
find the radial coordinate. This is done by measuring the redshift of the galaxies.
By 1980 the astronomers at the Harvard Smithsonian Center for A-r ~l e pi--;cs
(CfA) produced the first three dimensional map of the Universe in our immediate
neighborhood. Geller and Huchra published the second OfA catalog in 1995 which
contained 18, 000 galaxies up to a depth of 150h-1 hpe . The OfA catalog
explicitly showed the presence of large scale structures such as galaxy clusters,
voids, great walls and filaments. With improved technology it became possible to
spectral as~ llh-. -- several objects simultaneously which led to massive redshift survey
projects such as the Two degree Field Galaxy redshift survey (2dFGR S) and the
Sloan Digital Sky Survey (SDSS). These redshift surveys can probe deeper into the
Universe. The 2dFGR S survey covered an area of about 500 x 500 and contains in
total 232, 155 galaxies.
Baugh et al.(2005)  used a volume limited sample of the 2dFGR S data to test the
hierarchical ansatz. They calculated the higher order correlation functions and found
that they are in good agreement with the hierarchical ansatz. They observed that
on scales larger than about 4h 1\1pe the hierarchical amplitudes start increasing
unexpectedly which, they demonstrate, is because of the presence of two large
super clusters. When the two super clusters are masked out, they recovered the
expected behavior. The authors cite Szapudi & Gaztanaga (1998)  who report
similar problem in another volume limited angular catalog called Edinburgfh-Durham
Southern Galaxy Catalog (EDSGC) which covers a similar part of the sky. The
results are shown in Figure 5-3
R (Mpc h-l )
I II 111111 I II 111111 I I
I 1 1111111 1 1 1111111
Figure 5-1. Hierarchical amplitudes S, for p = 3, 4, 9 are plotted as a function of cell
radius, R for the galaxies in the APM catalog (Gaztanaga 1994 ).
R (Mpc h- )
lo ,,ll /h-1Mpc)
Figure 5-2. Variance in the distribution of galaxies in the APM survey.
4 ; 2t.~t ~~t;r_~: t t)t)--c)
,go(R ~h- M1 )
Hierarchical amplitudes, S, are plotted as a function of cell radius, R for a
volume limited sample of galaxies in the 2dFGRS catalog. The red curves
are for S3, gre6H for S4 and blue for Sg. The filled symbols connected by
solid lines are the results for the full volume whereas the open symbols
connected by short dashed lines are the results with the two large super
clusters masked.(Baugh et al. 2005).
RESITLTS AND DISCUSSION
The scale dependence of hierarchical aniplitudes, S, of the cell count moments have
been calculated in the halo model in both the continuum (for mass) and discrete
limits (galaxy number counts). Three distinct regimes are recognized in the expected
behavior of the hierarchical aniplitudes. On large scales, the cell includes many halos
and one can ignore the halos on the cell boundary and the internal structure of the
halos too can he ignored. In this regime the statistics are well explained by the point
cluster model. On intermediate scales the one halo term becomes dominant but the
internal structure of the halos are not resolved. In this regime the one halo term
of the halo model reduces to the one halo term of the point cluster model and the
hierarchical aniplitudes in this regime is locally scale independent. On small scales
(~ 1 Apc) the internal structure of the halo becomes important and a full halo
model with the halo internal structures resolved is required.
In a work that is in the process of publication Fry et al.  have derived a halo
model expression for the reduced moments of the total counts. As a first step I apply
the halo model in its point cluster limits to study the hierarchical aniplitudes for
a scale free power spectra and compare the results with the numerical simulation
results the prediction based on the HEPT. The results of this study are discussed at
length in section 6.1 of this OsI Ilpter.
Next I apply the halo model in its point cluster limits to study the the statistics of
mass and galaxy number counts on large and intermediate scales for the CDM power
spectrum. The results are then compared with the results of numerical simulations.
For galaxy number counts the results are compared with the numerical simulation
results and with the real galaxy catalogs generated front the APhi angular survey
and the 2dFGRS redshift survey. The results of this study are discussed at length in
section 6.3 of this OsI I pter.
6.1 Hierarchical Amplitudes For Scale Free Power Spectra
The hierarchical amplitudes are studied for a scale free power spectrum (P(k) ~
k's) using the halo model expression on intermediate scales where they are
scale independent. The results are summarized in a plot where the hierarchical
amplitudes, SI, are plotted as a function of the spectral index n. The hierarchical
amplitudes for mass and galaxies in the point cluster limits are given by the
expressions 5-52 & 5-54. These equations can he re written in terms of the
multiplicity function f(il) as follows. For mass,
For a scale-free spectrum,
d In o.2 (3+;z)/
d In R o- ni
where mi is the mass at which v(mi) = 1. Substituting v(m) = (m/m1)(3+')/6 in
Eq. 6-1 yields an analytical solution for Sk, in terms of gamma functions for both the
Press-Schecter and Sheth-Tormen mass functions. The solutions can he written in
:3(kr -111 3(k 1) 1 ~(3
,,, :3+~-13n 2a 21> ;3+n 2
The results are shown in Figure 6-1 along with the results from the numerical
simulations of Colombi, Bouchet & Hernquist (1996)  and the predictions based
on the Hyper Extended Perturbation Theory (Scoccimarro & Frieman 1999 [:34]).
The Press-Schecter mass function is in good agreement with the simulation results
as compared to the Sheth-Tormen mass function. This is because the Sheth-Tormen
mass function is relatively weighted towards small mass and in the numerical
simulations there is ah-liws a threshold mass. The results with lower mass cutoffs of
10-2m1 and 10-4m1 arT Shown as dotted and short-dashed curves. Introduction of
a threshold mass changes the behavior for Sheth-Tormen mass function significantly
while the behavior for Press-Schecter mass function is not changed much. Large
deviations from the simulation results are observed as a becomes more negative. It is
likely that the simulation results are at error because of the difficulties encountered
in simulating spectra with negative n (Jain & Bertschinger 1998 ).
For galaxy number counts we can find similar analytical solutions if we assume the
halo occupation number to be a power-law in mass (NV(m) ~ m ). Eq. 5-54 can be
rewritten in terms of the multiplicity function as
SI. = (6-4)
For a power-law halo occupation number, NV(m) ~ mP with no mass cutoff, there is
an analytical solution for Eq. 6-4 which can be written as
Ip (k) [Ip (1)]k-2
Sk [Ip(2)]k-1 65
Ipk I3P(k 1) 2113(k-)1
3+n 2 1, 29 3Pk-3+n 2 (6
The behavior of the solution is similar the behavior of Eq. 6-2
6.2 Numerical Simulation Results
Before discussing the results of applying the halo model to CDM power spectrum,
it is important to see the results of the numerical simulations. Details about the
numerical simulations are given in section 5.5 of C'!s Ilter 5. In brief, Colombi,
S., Teyssier, R. and Fosalba, P. performed numerical simulations called RAMSES
which uses an Adaptive Mesh Refinement technique. The simulation contained
in total 5123 partlCleS in a periodic cubic box of size Lboz = 200h-1 Mpc. The
chosen cosmology was ACDM. From the simulation results they have generated a
mock halo catalog, Eh and a mock galaxy catalog, E,. In generating the galaxy
catalog, galaxies are identified with the halo substructure. Without a selection
criteria, the catalog produces "galaxies" in abundance. To control the number of
galaxies, additional mass thresholds are introduced. A catalog of halos with mass
greater than 5 x 1011Me and another catalog of massive halos with halos of mass
greater than 4 x 1012Me0 were generated from the parent catalog. Some details
about the distribution of objects (halos and substructures) in these catalogs are
tabulated in Table 6-1. The first column shows the muss cutoff used in identifying
the halos, the second and third columns show the total number of halos and galaxies
in the corresponding catalogs. The third column shows the average number of
substructures per halo. The fourth and fifth columns show the number density of
substructures and halos respectively. The first row is for the parent catalog which is
generated without imposing additional selection criteria. The subsequent rows are
subsets of the parent catalog that are generated with mass thresholds given in the
first column of the corresponding rows.
Figure 6-2 shows a plot of the mass spectrum of the objects identified in the RAMSES
simulation. The data plotted there is for a subset of the parent catalog selected with
a threshold mass of 5 x 1011Me. The red histogram represents the distribution of
halo mass. The solid green curve is the Press-Schecter mass function and the long
dashed green curve is the Sheth-Tormen mass function. The black dotted curves are
the Press-Schecter and Sheth-Tormen mass functions with an exponential low mass
cutoff [ exp(-M/2/m2)] at the threshold mass M l. The shaded blue histogfram uses
the top scale and represents the halo occupation number. It is worth noting that the
mass spectrum of halos identified in the simulation closely follows the Press-Schecter
and Sheth-Tormen mass functions.
The quantities of interest are the volume averaged correlation functions. The first
in the hierarchy is the volume averaged two point correlation function which is also
the variance of the corresponding field (mass/substructure) Figure 6-3 shows a
plot of the variance in the distribution of mass observed in the simulation. In the
hierarchical models, the higher order correlation functions are related to the variance
through the hierarchical amplitudes, S,. Figure 6-4 shows a plot of the hierarchical
amplitudes S, = (p/"-1 for p = 3, 4, 5 calculated for the distribution of mass in the
RAMSES simulation. S3, S4 & S5 are shown as green, red and blue curves respectively.
It is to these results that we will compare the model results with.
6.3 Hierarchical Amplitudes For A CDM Power Spectrum
The results of the application of the halo model to CDM power spectrum is
discussed in this section. The results for the statistics of mass and the statistics
of galaxy number counts are discussed separately. For the case of mass distribution,
we only have simulation results to compare with the model calculation whereas
for the case of galaxy number count, in addition to simulation result, we have the
data from galaxy survey. Though we have data from both the APM survey and the
2dFGRS survey, only the APM data is reliable. 2dFGRS data is volume limited and
so has large uncertainties.
The model calculations have been performed for large and intermediate scales.
Calculations for the small scales are deferred to future. So the model calculations
are valid only for scales above 1 Mpc. On intermediate scales only the one-halo term
is dominant and on large scales we need multiple halo terms. Up to 10h-1 Mpc we
only need to consider the one-halo and two-halo terms. The naive point cluster
calculations required correction for bias. This issue was discussed in (I Ilpter 5. The
bias corrected point cluster expressions are given by Eqs. 5-33 5-36 and the bias
parameters are given by Eq. 5-31, for mass, and Eq. 5-32 for galaxy number counts.
I calculate the bias parameters for ACDM cosmology. The value of bias parameters,
(b), did not change significantly across the Mo-White form and the Jing form of bias
factors, b(u). But they differed slightly with the choice of mass function.A sample of
the results of these calculations is tabulated in Table 6-2, for mass, and in Table 6-3
for galaxy number counts.
6.3.1 Statistics Of The Mass Distribution
To study the statistics of mass distribution on intermediate and large scales I use
the one-halo and two-halo terms of the bias corrected point cluster expressions
Eqs. 5-3:3 5-36. The bias parameters, (b);, are calculated using Eq. 5-31 and the
halo occupation moments #:,3 are calculated using Eq. 5-51. The first thing to
calculate is the variance, (2(R) which can he resolved into a one-halo term and a
two halo term. Figure 6-5 shows the results of the model calculation of the variance
in the distribution of mass. Here the variance ( is plotted as a function of cell
radius, R. The dotted red line is the one-halo term, the long dashed red curve is
the two-halo term and the sunt of both is shown as the blue curve. Observe that,
around R = 4h-1 hpc the one-halo term crosses over the two-halo term and becomes
dominant. The halo model result is in good agreement with the simulation result on
large scales and on intermediate scales. It fails to match the simulation results in the
region where the halo internal structure becomes important. Figure 6-6 compares the
model calculations with the results of numerical simulation. The green curve shows
the variance of mass distribution in the numerical simulation. The red and blue
curves are the model calculations for the Press-Schecter and Sheth-Tornien mass
The higher order correlation functions are related to variance through the hierarchical
aniplitudes, SI,. The halo model calculations of hierarchical moments are compared
with the simulation results in Figure 6-7. Refer to the caption for the details about
the plots. Observe that without a low mass cutoff the Press-Schecter mass function
seems to fit better than the Sheth-Tornien and with a low mass cutoff of 5 x 10"Aln
the Sheth-Tornien mass function gives a better fit to the simulation results. The
data is not reliable on scales smaller than 1h-l hipc where we need a resolved halo
model. This is deferred to a future work. On the large and intermediate scales
however Sheth-Tormen mass function with a low mass cutoff of 5 x 10"A/. seems to
provide a better fit for the simulation data.
6.3.2 Statistics Of The Galaxy Number Counts
To study the statistics of galaxy number counts on intermediate and large scales, I
use the one-halo and two-halo terms of the bias corrected point cluster expressions
(5-33 5-36). The bias parameters, (b), are calculated using Eq. 5-32 and the halo
occupation moments py,s are calculated using Eq. 5-53. The first in the correlation
function hierarchy is the variance, a(2R) Which can be resolved into a one-halo term
and a two halo term. Figure 6-8 shows a comparison between APM survey results
and the model calculation. It is clear that the variance in galaxy distribution in
the APM survey results match well with the results of model calculations for a
Press-Schecter mass function with a low mass cutoff of Me = 5 x 1011Me-
The higher order correlation functions are related to variance through the hierarchical
amplitudes, S,. The halo model calculations of hierarchical moments are compared
with the simulation results in Figure 6-9 and with the results of APM survey
in Figure 6-10. In general, the model calculations match poorly with both the
simulation and APM survey data. Refer to the caption for the details of the plots.
Observe that the hierarchical amplitudes for the galaxy number counts fall more
gradually with the cell radius than that for mass distribution. For galaxies it is
almost flat on large scales.
Next let us compare the simulation results with the APM survey results. Figure
6-11 show the comparison. Simulation results with Ml = 0 and Ml = 5 x 1011Me0
differ only slightly on small scales and have a better match with the results of APM
catalog. This is because halos lighter than 5 x 1011Me0 have few substructures
and so their exclusion does not affect the statistics much. Simulation results with
Ml = 4 x 1012Me0 deviate largely from the APM results. By considering only
the massive halos we are under counting the substructures and so our sampling is
incomplete. The simulation results and the APM survey results match qualitatively
and not exactly.
Though simulation and model are in very good agreement with the value of variance
of galaxy distribution calculated from the APM catalog, they all fail to produce
results that are consistent when it comes to higher order correlation functions.
There seems to be large uncertainties in the determination of the higher order
correlation functions for galaxies. Refer to Figure 6-12 to see the differences in
the estimate of hierarchical amplitudes as measured from the APM survey data
and the 2dFGRS survey data. The results from galaxy surve7i- simulation and
the halo model calculations are mutually inconsistent, so----_ :-r;~! a serious gap
in our understanding of the galaxy distribution. Identification of substructures
with galaxies is still questionable and requires more investigation. One of the most
important ingredients used in determining the higher order galaxy correlation
functions is the Halo occupation moment. These are again determined through SPH
simulations where the substructures are identified as galaxies. So this assumption
requires to be studied properly.
(b), b(u) Me=0.0 Me=5 x1011
PS ST PS ST
(b)l MW 1.0109 1.0396 1.3387 1.3520
JING 1.0653 1.3582
(b)2 MW 3.5120 3.3929 3.5122 3.3930
JING 3.5159 3.51631
(b)3 MW 5.4696 5.2518 5.4696 5.2518
JING 5.4715 5.4715
(b)4 MW 7.2855 6.98411 7.2856 6.9841
JING 7.2868 7. 2869
(b)5 MW 9.0210 8.63445 9.0210 8.63445
JING 9.0220 9.0220
Table 6-1. Details of the distribution of objects (halos & substructures) in the catalogs
generated from the simulation data.
Mass cutoff Nvautos
5 x1011 43482
4 x1012 11934
1 x1014 494
Table 6-2. Bias
parameters for mass in LCDM
(b), b(v) PS ST PS ST
(b), MW 0.9768 1.0499 0.9992 1.0645
JING 1.0020 1.0240
(b)2 AlW :3.36;20 :3.317:3 :3.1678 :3.1107
JING :3.3667 :3.1728
(b), MW 5.5124 5.3178 5.1765 4.9957
JING 5.514:3 5.1786
(b)4 AlW 7.388:3 7.0964 6;.96;20 6.6887
JING 7.3895 63.9:334
(b), MW 9.16:34 8.7910 8.6499 8.2991
JING 9.164:3 8.6509
Table 6-:3. Bias parameters for
galaxies in LCDM
Hierarchical amplitudes, S, for p = 3, 4, 5 are plotted as a function of spectral
index n for both the Press-Schecter and Sheth-Tormen mass functions.
The red curves are for Press-Schecter mass function and the blue ones are
for Sheth-Tormen mass function. Solid lines show the results without any
mass cutoff. The dotted curves are for a lower mass cutoff of 10-2m1. The
short-dashed curves are for a lower mass cutoff of 10-4m1. The green curves
show the predictions of Hyper Extended Perturbation Theory (Scoccimarro
& Friemann 1999 ). The symbols with error bars show the results from
numerical simulations (Colombi, Bouchet & Hernquist 1996 ).
10'2 1013 1014 1016
M/ bottom], N [top]
Mass distribution of the objects (halos and substructures) in the catalog
generated using a threshold mass of Mi1 5 x 1011Me0. The red histogram
represents the distribution of halo mass. The solid green curve is the
Press-Schecter mass function and the long dashed green curve is the
Sheth-Tormen mass function. The black dotted curves are the Press-Schecter
and Sheth-Tormen mass functions with an exponential low mass cutoff at
the threshold mass M l. The shaded blue histogfram uses the top scale and
represents the halo occupation number.