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CELL COUNT MOMENTS IN THE HALO MODEL By ANAND BALARAMAN 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 2007 S2007 Anand Balaraman To my parents and to all my teachers ACKENOWLED GMENTS 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 page ACK(NOWLEDGMENTS ......... . .. .. 4 LIST OF TABLES ......... .... .. 8 LIST OF FIGURES ......... .. . 9 ABSTRACT ......... ..... 10 CHAPTER 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 RobertsonWalker Aletric . ...... .. 12 1.1.4 Friedman 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 Superhorizon And Subhorizon Modes ... ... .. 21 2.2.3 Composition Of The Cosmic Fluid And Growth Dynamics .. .. 21 2.2.4 Statistical Properties Of Inhomogeneities ... ... .. 2:3 2.2.4.1 Ergodicity . ..... .... 2:3 2.2.4.2 Auto correlation function and Power spectrum .. .. .. 24 2.2.5 Primordial Perturbations . ..... .. 25 2.2.5.1 Primordial spectrum .... .... . 25 2.2.5.2 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 2.3.3.1 Baryonic perturbations before recombination .. .. .. 29 2.3.3.2 Baryonic perturbations after recombination .. .. .. :30 2.:3.:3.3 Silk damping . ...... ... .. :31 2.3.4 Dark Matter Theories Of Structure Formation .. .. .. .. :32 2.3.4.1 Dark matter perturbations in radiationdominated era :32 5.3.1 Corrections To The Point Cluster Result 5.3.1.1 The bias corrected point cluster expressions 5.3.1.2 Backgfround particles and statistics of mass . 5.4 Full Halo Model. 2.3.4.2 Dark matter perturbations in matterdominated era. 2.3.4.3 Free Streamingf Paradigm 2.3.4.4 HDM and CDM scenarios 2.3.5 Processed Power Spectrum And Transfer Function. 2.3.6 Problems In SCDM And The Emergence Of ACDM 2.3.6.1 OCDM models 2.3.6.2 Mixed Dark Matter (ill)M) models. 2.3.6.3 TrCDM model. 2.3.6.4 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 ......__ Introduction. Statistical Definitions. 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 Table page 21 HOD parameters from SPH and SA simulations ... .. .. 48 61 Distribution of halos and substructures in the simulation .. .. .. 97 62 Bias parameters for mass in LCDM . ..... .. 97 63 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 Figu: 21 22 23 51 52 53 61 62 6;3 64 6i5 66 67 6;8 6;9 610 611 612 page 48 49 . 50 . 86; . 87 . 88 99 .. 100 . . 101 .. 102 . 10:3 . 104 .. 105 .. 106 Simulation ) .107 Model) .. .. 108 simulation) .. 109 dFGR S). .. 110 S 2 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 By Anand Balaranian May 2007 Cl.! I!1: Prof. James N Fry Major: Physics 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 nonlinear 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 Lynianalpha forest and C \lli SunyaevZel'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. CHAPTER 1 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, Hubble expansion, 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 spacetime 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 spacetime to the energy content of spacetime. To write down the spacetime metric of the Universe we need two more assumptions. One is called as the Cosmological principle, which determines the form of the spacetime 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 spacetime 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 ahrlw 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 RobertsonWalker 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 value . 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 RobertsonWalker metric. The RobertsonWalker metric is written as, 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)(12) 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 spacetime and the energymomentum tensor of the cosmic fluid. While the Cosmological principle determines the form of the metric, Weyl's postulate simplifies the form of the energymomentum 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 energymomentum tensor can be written as, T"" = (p + p)u~u" py"'' (13) 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 ). (15) a 3 c Friedman 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 a3(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 nonrelativistic 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 time. 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, up 1 50 1. (19) H2a2 P where R p/p,. is called the density parameter. Looking at Eq. 19, 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 nonzero 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/ (110) 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 ahrl . 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 hl factor and the value of h is left to the choice of the readers. For example distances are indicated in units of h1 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 hl 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 (111) 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 Pc Pc = 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 102. This following are the estimates of WMAP (Wilkinson Microwave Anisotropy Probe) for the present dwi composition of the Universe. Or 2 = 2.56 x 105 Ob 2 = 0.0224 R,7h2 = 0.135 RA = 0.73 CHAPTER 2 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 badlly. Structure formation can begin as soon as the Universe become matterdominated. So if there were any primordial inhomogeneities when the Universe became matterdominated, 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 105 which is consistent with primordial adiahatic perturbations as large as 104 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) (21) p(t) 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 Superhorizon And Subhorizon 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 linear regime. 611 At any epoch the modes 6k, can be categorized into superhorizon modes or subhorizon modes depending on whether they are bigger than or smaller than the horizon scale at that epoch. To study the evolution of superhorizon modes we need a general relativistic treatment of the problem. For subhorizon modes, a Newtonian treatment is sufficient. All the modes that correspond to interesting structures tod w are subhorizon modes. 2.2.3 Composition Of The Cosmic Fluid And Growth Dynamics The dynamics of growth of subhorizon 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/electromagnetic 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. O :CDM 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 a3(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 a4 and that in matter dilutes as a3, 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 spacetinte. 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. 2.2.4.1 Ergodicity 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 P 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 [28]. We assume that this criteria is met for our Universe. 2.2.4.2 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)). (22) The Power spectrum of the density field is the fourier transform of its two point correlation function. a(2x) i(r) = 6k~ 2 erxp(ik x)d~k. P(k) (Sk 2) is the power spectrum. The 2point 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 nonlinearity is around 10h Mpc. The density field when smoothed out on scales much larger than 10h1 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 [6] =(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 HarrisonZel'dovich spectrum (6k 2 ock). 2.2.5.1 Primordial spectrum Primordial power spectrum is the quantification of primordial inhomogeneities. Various surveys of galactic distributions have shown that the observed twopoint 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". (23) Based on certain physical requirements we can constrain the range of values that a can take. 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 HarrisonZel'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 HarrisonZel'dovich spectrum naturally. 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 (os normalization). 2.2.5.2 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). (24) 6rpr + 6mpm = 0 (isocurvature mode) (25) 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 grow algebraically. 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 subhorizon modes. 6, + 2 bk ( ~2 4a k=0 26 a a ~~), ,(6 2.3.2 Jean's Mass In Eq. 26 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 algebraic. 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 ) (27) One can define a Jean's mass which is the total mass enclosed inside the Jean's wavelength. My = p (28) 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. 2.3.3.1 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 subhorizon 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 = Hl is a measure of horizon size. 3 3 dH = H1 (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 2H1 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. 2.3.3.2 Baryonic perturbations after recombination Since' l'i10.10 remain coupled to radiation until decoupling, they continue to get the pressure support from radiation even after the epoch of matterradiation 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 .(210) 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. (211) For an EinsteindeSitter 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. 2.3.3.3 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 photonbaryon 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 damping. The photon mean free path for Thompson scattering is given as neXeo, or = 6.665 x 1029 2 1S the Thompson crosssection. 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 (212) 3 teech (taec) (213) 5 a2 tdec Taking zaec ~ 1100 and Xe = 0.1, A2 il =3. 1 2 m23/4 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(102) 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 nonbaryonic 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. 2.3.4.1 Dark matter perturbations in radiationdominated 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 subhorizon 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. (214) 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. 2.3.4.2 Dark matter perturbations in matterdominated 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 EinsteindeSitter Universe (Rm = 1; is = 0). The EinsteindeSitter 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 EinsteindeSitter 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 decoupling. 2.3.4.3 Free Streaming Free 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 nonrelativistic. The epoch at which the dark matter particles become nonrelativistic determines the free streaming length scale. The conioving distance that the free streaming particles can travel by the epoch t is r7 = dt' (216) 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 nonrelativistic. = 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) 2.3.4.4 HDM and CDM scenarios If we take the idea of nonbaryonic 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 nonrelativistic 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 topdown 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 bottomup 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 written as k61 =0=D2X) Tk2 k rmz (217) Here x is the primordial redshift and D(x) is the linear growth factor given by Eq. 215. 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 matterradiation equality. All the modes that entered the horizon in the matterdominated epoch grow at the same rate whereas those that entered the horizon when the Universe was radiation dominated froze until the Universe became matterdominated when it started growing again at a different rate. So corresponding to the scale of horizon at the epoch of matterradiation 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 publicdomain 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/ (218) 2.34q k Rmb2 MpC1 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 matterradiation 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. 2.3.6.1 OCDM models Sacrificing the inflationary prediction of flat spatial geometry if one accepts Rm = 0.3, then it increases the horizon size at matterradiation equality. This enhances the power on large scales. 2.3.6.2 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. 2.3.6.3 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. 2.3.6.4 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 favored model. 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. f dr4xrr26r Using the Friedmann equation the background density can be written as p = Ope II ( 3H2 Total mass inside the overdense region is H2r2 r. 0 [1 + a(r)] 2G, GM H2r2 O6[1 + a(r)] r 2 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). 2E (2E)3/2 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 EinsteindeSitter (a = 0, Rm = 1) background, rm= E hon=(2E)3/2 3xr 1 32Gt2 6;Gt2 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 turnaround. In an EinsteindeSitter 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.. PressSchecter 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) = xp [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 (220) 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, 1 1 F(0) =[1 erf (0)] 2 2 So Eq. 220 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 PressSchecter analysis ignores this fact. This is called "cloudincloud" 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 Nbody simulations of hierarchical clustering can provide us the mass functions for CDM halo distributions. The mass spectrum of bound objects as predicted by PressSchecter 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 Nbody simulation. dn p dln u (M~) = VfsT (u) (222) v fs() = 2A[1+ (qu2)' p 2 Oq p qu2/2) (2 23) PressShecter form : The ShethTormen form of mass function includes the PressSchecter form as a special case. The PressSchecter form is recovered through the following choice of parameters. q = 1, p = 0, A = 1/2. ShethTormen 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 EinsteindeSitter 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 EinsteindeSitter Universe II _ rh I1 powerlaw profiles. The simplest model for the density profile of virialized system is that of an Isothermal sphere (p OC T2). However Efstathion et al. (1988) and Frenk et al. (1988) observed significant departures from the powerlaw in the halos of CDM simulations. High resolution Nbody 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) (224) (r rs [ +(r /rs)]2 p(r) (:\!')9) (225) (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 Also, .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)ip (NW) ps 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/r2a) 1]); a~ ~ 1.7 (226) P2 r2 is defined as the distance from the halo center at which the density profile locally resembles an isothermal profile, i.e. p(r2) c 2 lOCally. Refer to Figure 21 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 1halo 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 PressSchecter 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 ;>2 b(m) = (1 +). Jing (1998) found some disagreement between the MoWhite 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. v2 b(m) = (1 + ) (1 + )0.060.02/1 6,. 2 174 Refer to Figure 22 for a comparison of the values of MoWhite and Jing hias parameters for the case of PressSchecter and ShethTormen mass functions. There is no Jingf correction available for ShethTormen 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 antibiased. 2.4.5 Halo Occupation Moments Studies by Berlind et al.(2003) [4] and K~ravatsov et al.(2004) [17] showed that the description of P(NMI) 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. [45] parameterized (NV)M foT galaXieS brighter than a threshold luminosity as, (N)Mn/ = (nc>)M s1V)M (227) 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. [4] 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 powerlaw extrapolation of (Ns,)M for higher masses, leading to the following expression for (Ns,)M/ (Nis) M M (229) 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) [46] extract these parameters from the SPH and SA simulations for various values of galactic number densities(n,). Refer to Table 21 for these values. The other halo occupation moments written down in terms of (Nc,) and (Ns,), (NV[2]) __ [12]) 2(Nc,1s) + (N B2]) (Ni[4]) __ [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,[3] [~2]) [12]1[3]) [t(i5]) If we assume that the satellite galaxies are Poisson distributed, then these expressions simplify to, (NE*) = (Ns)" + p(Ne,) (Ns,)"l (230) This is a very usefkil result which we will make use of later. Refer to Figure 23 for a plot of (NV)M/ RS a function of M~ 1 Qd101 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. Figure 21. Table 21. HOD parameters from SPH and SA simulations it, Model 0.02 SPH SA 0.01 SPH SA 0.005 SPH SA 0.0025 SPH SA 11.68 11.73 12.07 12.02 12.36 12.36 12.69 12.60 log ii t, 11.86 12.09 12.28 12.28 12.63 12.28 12.94 12.77 log M( 13.00 12.87 13.19 13.32 13.45 13.632 13.82 13.86 a 1.02 0.96 0.94 1.07 1.00 1.04 1.08 1.03 Flog M 0.15 0.32 0.18 0.26 0.15 0.42 0.15 0.28 101 1 101 10" 103 104 105 106 107 108 109 1010 101 1012 10'3 1014 1016 Figure 22. Comparison of the Halo hias parameters. The blue curves are for PressSchecter mass function and the red curves are for the ShethTornien mass function. The solid lines are for MoWhite model of hias and the dashed curves are for Jingf corrections. There is no Jingf correction available for the ShethTornien mass function. 3 1011 10"' 1013 1()14 M, Figure 23. 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. CHAPTER :3 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 lognormal. 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 larger. Cosmographical studies of Shapley and his associates showed a complex distribution of galaxies. Cosmography of the early d .14 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 twopoint 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 Npoint 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 functions . I(()= (31) (123 = (12 13 + 21 23 31 32). (32) y ~ 1.77 + 0.04; ro ~ 5 hlrMpc; 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]. (33) 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 nonlinear. An analytical understanding of NonLinear Gravitational Clustering(NLGC) is difficult to achieve. High resolution Nbody 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 alrws 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 nonideal 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~irkwoodlike 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 nonlinear integrodifferential 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 ChalI 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 EinsteindeSitter Universe (Rm = 1, a = 0) under the Stable clustering limit, the logarithmic slope of the nonlinear two point correlation function is related to the spectral index n as 3(3 +n) Y= >11 5+n The implication of this result is that if stable clustering hypothesis holds true, then the nonlinear 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;11lp II by it. Since gravity is scale free, gravity driven clustering should display simple scaling properties. This would manifest in the behavior of Npoint 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(N1)YN (x1, , xN) (35) 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 twopoint correlation function follows a power law I oc rY which is 0.1 h1Mpc < r < 100 h1Mpc. 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 EinsteindeSitter 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). tp p1] E(Xl.. (xX, ,x) = Q,,,p C AB. (36) a=1 labels edges So the ppoint 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 ptrees 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 ptree has tp = pp2 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 ppoint distribution function along with other symmetry assumptions he arrived at the following relation for Q,~, = Q, Q, = (37) 2 1p1 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. (39) 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 count Moments. 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 Countincells 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 Also, ((ni nl)(R2 82 83 nl)(n2 82 n> na2 3 3 > ns3 4 4 > (r1234 12 34 ~ n2 VibV2 12 n3 VibV2 V3 123 1 4 + 13 24 + 14 23  ((ni ((1 n1 82 (311) If 5%/' is the volume of element i, then (n ) = ab%~. adV = nV = N. (N) = (~ us) S(nL) a b% The first five moments are = V (NV) V3 (N2 _ V5 Here p, and (p are the follows NV, 2v 12~, 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). (317) It can be verified from Eqs. 313 316, that the factorial moments are related to the volume averaged moments, py, of the continuous number density field as (NW1 = 1~pp.(318) CHAPTER 4 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) [31]. 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) = nArt. (41) 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(21) = 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 27 Averaging this over a spherical shell of radiuS T12 and thickness 6r12, where r, = r~ + rf2 2rlrlalp. On substituting Eq. 41 for the density profile in Eq. 43 we get 26NV,(rl2) 8 r2A2n2r~' r~ 5e71 where loo 1l If NV is the total number of objects in the cluster, N= nrR d 4xurAr3e Now the correlation function is given by 26NV,(rl2) I(1~1 (3 )Ar 2e e3 6) In the limit r ((r) OC T32t _n7 Now one can choose e to be 0.6 to match the behavior of observed galaxy correlation function on small scales (~(r) oc rls), 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 hindsight 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) [6] 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) (48) 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~ bD0mi)) =(mn) (49) a(x) = d3Zim" 6 XXi)6D 0 i) p 0, X Xi (410) The 2point 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 1M) :b6x X2 X)bnD 2 m,) It can be resolved into a 1halo term (i = j terms) and a 2halo term (i / terms). dm 1n dm 2 ) t h ) (11 1 41 2; X 1 X 2) dm d dm\I1/VD X1 X2)bD 01 m2) (60D "")) (2h) (60D ))(1&) 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 1halo term plus a 2halo term. W1 3 1 W1) ; l 1 X1) Wl, 72 X1) (412) dm dm d dm d x d31 32 0) E 1 0) E 2 d) 0.1T"1, MX1, X2) 13 l(rl, r2 l 2h) (r1, r2 Similarly the expression for the ppoint correlation function involves terms that can be resolved clearly into 1halo term, 2halo term, ... etc. upto the phalo term. The 1halo term corresponds to the case when all the p points are inside a single halo, the 2halo terms correspond to the case when all the p points are distributed between 2 halos and so on up to the phalo term which corresponds to the case when all the p points are distributed in p separate halos. On the smallest scale the 1halo term is dominant and on the largest scale the phalo term is dominant. It is obvious why it should be so. On the smallest scale all the ppoints 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 ppoint correlation function in the halo model is, (p( x ,x ) I1.. (414) q=1 ji=1 jp=1 l= 1 i= 1 4.4 Model Ingredients From these expressions it is clear that the halo model requires the following quantities as inputs. 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 Nbody simulations provide us the halo mass function while the high resolution Nbody 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 1halo term is dominant . 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 PressSchecter 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 PressSchecter fornialism to triaxial system and their formula provides a better fit for the simulation results. The PressSchecter and ShethTornien mass functions are written in the general form, dn p d 1n u ( Af) = v f (u) ShethTormen form : The following choice of parameters provide a good fit for the simulation results v fS() = 2A [1 +(qu2D p 2 Op ( qua2/2 (415) A(p) = 1 + 2 m 0.3222, p a 0.3, qm 0.707. PressShecter form : The ShethTornien form of mass function includes the PressSchecter form as a special case. The PressSchecter 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, Frenk & 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 radius r,. NFW and M99 II rh I1 the following forms for the density profiles respectively, =p rs' 1 NFW u~) (1+2)2 (416) 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 + )0.060.02n 4.4.4 Halo Occupation Distribution For the Halo occupation distribution I use the five parameter model developed by Berlind et al. [4] 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 powerlaw 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 nonlinear 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) [14] and later generalized by Peacock & Dodds (PD) [29]. But this technique had the stable clustering hypothesis inbuilt. The validity of Stable clustering hypothesis has been questioned in the recent years. Ma & Fry (2000) [19] 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 [37] to study the evolution of two point correlation function in the non linear regime. Similar studies have been done on power spectra ([36], [30] & [19]). 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 [21] have demonstrated that the Halo model can retrace the power spectrum and bispectrum generated by cosmological simulations for various models. CHAPTER 5 CELL COUNT MOMENTS IN THE HALO MODEL 5.1 Introduction 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 [5] and with the results of calculations based on the Hyper Extended Perturbation Theory (HEPT) [34]. 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) [18] 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) [12] 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),(52) This gives the moments of discrete counts as derived in Peebles (1980) [31] (NV) = NV, (53) ((NV 1V)2) __ V2 2, (54) ((NV NV) ) = NV + 3NV2 2 13 3, (55) ((NV 1V)4) _(1 V2) __ 2 1 2~ + 6V3~ 3 V4 4. (56) 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 NV! NWB1 I = N(N1) (NV p +1). (NV p)! It can be verified from Eqs. 53 56, 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,(57) 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, (58) p=0 K((t) =i NW,.~ (59) p= 1 where K(t) = log(M~(t)). For discrete counts with probabilities PN Szapudi & CI I1 li (1993) [40] use the generating function G~z) = CPz". (510) N=0 G(z) is related to the exponential generating function through the relation M~(t) = G(z)zt+i. (511) In the hierarchical models the ppoint correlation functions are related to the 2point correlation function. The hierarchical amplitudes, S, are then defined by the relation (p = A' .(512) 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.(513) i= 1 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 (514) Pi = plqg (515) P2 =2 p9192~ (516) P3 P3ql + 2q192 + 193 (517) Pq4 p449 3p34 92 + 2(2qlq3 + 42" 19pl4 (518) Ps = psts + 4p49 92 93 p3(3 3 341422 + 2(2qlq4 + 2q243) p195. (519) Szapudi & CI. I1 li (1993) [40] 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, d M~(t) NV"Ip, = =0 (520) d K (t) NVp( = =0 (521) 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(522) 2a 2 ,h 2~,i (523) 3p2,i 2,h (524) 6p2,i 3 ,h (13,i , ~4 = 4,h +++(525) 10pi ,h (10p~3,i + 15p ~,)(3,h (1p2i3, 2h 5,i s ~ V =V sV" +V + (526) The expression for (p has contributions due to all moments of occupation number Il,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 ~ rCP1). 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.(527) 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. 5.3.1.1 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 halobias parameters. The clustering of ppoint correlation functions of halos up to the linear term is 6,a(xi, mi) = b(mi) b(m,)&(xl, x) (528) where b(m) are called the bias factors. The linear bias factors for halos of mass m are given by Mo & White [25] as Jing [16] corrected the MoWhite formula to find a better fit for the simulation results and the corrected bias factor is b (u) v (s 0.060.02n 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 S1m4 L w(m)2)b (m2) dm~ (b) =(529) 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, (530) (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) (b), (531) f, dm (dn/dm) (NE 1)b(m) (b) (532) 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, (533) g=(b)" 3(b)L(b)2 2,~iG 2 "; 3,i E3 (b):t (3jh 2 (534)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 I5,i 1V"1+ (536) Observe that the onehalo terms are independent of bias parameters. 5.3.1.2 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. 513 is applicable only for the number count of galaxies/substructures and not for mass. With this modification the cell count becomes N~b =V Ns +Nbb. (537) i= 1 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) (538) K((t) = log(G(z)z=1+t), =log(gh(gi(1 + t))) + log(gbl + :) = K'(t) + Kb t). Using 59 we find the following result for the irreducible factorial moments. d K (t) d K' (t) d Kb~t <=0 = =0 + <=0, (539) dtp dtp dtp niP(, = I'"( + R~~bpb (540) If we further assume that NVb is POiSSon distributed, then (p~b = 0, p > 2. z)' = ', (541) 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 p2lrv S'. (542) 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. (543) 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) [44], 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.(544) 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 ppoint correlation functions are written as (p(rl, r) = ( '") + ( 2h) p Ih) On smrall scalesi the onehalo termr, ('") domrinatesi. Th'le onleh~alo termr inl thle halo model involve terms containing the convolution of halo profiles. For the distribution of discrete objects the onehalo 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 / (546) 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. 546 becomes [Jdmdndmm d37 [~ 3 ./ 9~y (R) = l~s/cmm' J (547) 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 onehalo expression of the halo model should reduce to the onehalo 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. 546 reduces to the following form Elh f dm);(ln/d~) (NB 1 (1()) V i. (548) ~" [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 (549) la Na~y, p .(550) S (N i~)" N Thus we recover the 1halo term of the point cluster expression. Now we are ready to evaluate the hierarchical amplitudes in the intermediate regime where the onehalo term of the point cluster expression is dominant. In this regfime the hierarchical amplitudes are scale independent. (15) 16r~l) p~1 p1 For mass, f. dmn(dn/dm)]" [J dm(dn/dm)m ] (551) [JI dmn(dln/dm7rm] P5; p1, JS dmn(dn/dmjm '] [J dm(d~n/dm)m]p2 (552) [J dm(dn/dmjm2] p1 For galaxies, [JI dm(dn/dm)]" [JI dmr(dn/dmn) (NE 1l)] (553) [J` dmr(drll n/d (N~)]p S l J~I dmr(d~l n/dm (NE [1)] [J' dmr~(drl/dmr) (N)]p2 p1 (554) [J dm71(d?2/dm)2 (N11[2] p Thus, on intermediate scales, the hierarchical amplitudes are independent of the cell S1Ze. 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 results. S,(R)= )() F ()Sp,(555) F,(R) =_(556) 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. 554 & 552. 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 Nbody 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 celltocell basis. For details about the algorithm refer to Teyssier (2002) [43] 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 os = 0.93, when extrapolated linearly to the present time. os is the variance of the density fluctuations in a sphere of radius 8 h1Mpc. 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 h1Mpc. 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) [43]. 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) [1]. 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) [17] show that the halo occupation distribution of substructures in the Nhody 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) [4] too found a remarkable similarities in the predictions of P(NMI) 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 npoint correlation functions, 6,(rl, r,) and the goal of every galaxy survey is to determine the npoint correlation functions more accurately over a wide range of scales. But the estimation of npoint correlation function front galaxy catalogs become increasingly difficult with increasing n. Though the npoint correlation functions contain complete information about the statistical properties of the distribution, one may not necessarily want all the information. A volume averaged npoint correlation function (,(R) is a compromise that simplifies the calculation and carries useful information. In the hierarchical model the volume averaged npoint correlation functions are expressed in terms of the volume averaged two point correlation function and this hierarchy is written as ,2(R) = S,zi" (557) 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 1i 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) [10] 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 hthipc and O~D ~ 2h1 ipc and the decreases slowly. ~D ~ 400 hthipc 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 51. The variance of the galaxy distribution in the APhi catalog is shown in Figure 52. 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 Ar ~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 150h1 hpe [24]. 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) [2] 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) [42] who report similar problem in another volume limited angular catalog called EdinburgfhDurham Southern Galaxy Catalog (EDSGC) which covers a similar part of the sky. The results are shown in Figure 53 1 10 R (Mpc hl ) I II 111111 I II 111111 I I 108 104 103 102 101 107 106 I S, Iiii ,  106 3S I 1 1111111 1 1 1111111 I I 104 Figure 51. 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 [10]). 1 R (Mpc h ) lo ,,ll /h1Mpc) Figure 52. Variance in the distribution of galaxies in the APM survey. ; i 4 ; 2t.~t ~~t;r_~: t t)t)c) ,go(R ~h M1 ) Figure 53. 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[2]). CHAPTER 6 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. [13] 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 552 & 554. These equations can he re written in terms of the multiplicity function f(il) as follows. For mass, For a scalefree 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. 61 yields an analytical solution for Sk, in terms of gamma functions for both the PressSchecter and ShethTormen mass functions. The solutions can he written in the form Sk, =(62 [I(2)]k"1 (2 where , :3(kr 111 3(k 1) 1 ~(3 ,,, :3+~13n 2a 21> ;3+n 2 The results are shown in Figure 61 along with the results from the numerical simulations of Colombi, Bouchet & Hernquist (1996) [5] and the predictions based on the Hyper Extended Perturbation Theory (Scoccimarro & Frieman 1999 [:34]). The PressSchecter mass function is in good agreement with the simulation results as compared to the ShethTormen mass function. This is because the ShethTormen mass function is relatively weighted towards small mass and in the numerical simulations there is ahliws a threshold mass. The results with lower mass cutoffs of 102m1 and 104m1 arT Shown as dotted and shortdashed curves. Introduction of a threshold mass changes the behavior for ShethTormen mass function significantly while the behavior for PressSchecter 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 [15]). For galaxy number counts we can find similar analytical solutions if we assume the halo occupation number to be a powerlaw in mass (NV(m) ~ m ). Eq. 554 can be rewritten in terms of the multiplicity function as SI. = (64) For a powerlaw halo occupation number, NV(m) ~ mP with no mass cutoff, there is an analytical solution for Eq. 64 which can be written as Ip (k) [Ip (1)]k2 Sk [Ip(2)]k1 65 where , Ipk I3P(k 1) 2113(k)1 3+n 2 1, 29 3Pk3+n 2 (6 The behavior of the solution is similar the behavior of Eq. 62 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 = 200h1 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 61. 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 62 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 PressSchecter mass function and the long dashed green curve is the ShethTormen mass function. The black dotted curves are the PressSchecter and ShethTormen 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 PressSchecter and ShethTormen 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 63 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 64 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 onehalo term is dominant and on large scales we need multiple halo terms. Up to 10h1 Mpc we only need to consider the onehalo and twohalo 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. 533 536 and the bias parameters are given by Eq. 531, for mass, and Eq. 532 for galaxy number counts. I calculate the bias parameters for ACDM cosmology. The value of bias parameters, (b), did not change significantly across the MoWhite 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 62, for mass, and in Table 63 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 onehalo and twohalo terms of the bias corrected point cluster expressions Eqs. 53:3 536. The bias parameters, (b);, are calculated using Eq. 531 and the halo occupation moments #:,3 are calculated using Eq. 551. The first thing to calculate is the variance, (2(R) which can he resolved into a onehalo term and a two halo term. Figure 65 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 onehalo term, the long dashed red curve is the twohalo term and the sunt of both is shown as the blue curve. Observe that, around R = 4h1 hpc the onehalo term crosses over the twohalo 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 66 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 PressSchecter and ShethTornien mass functions respectively. 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 67. Refer to the caption for the details about the plots. Observe that without a low mass cutoff the PressSchecter mass function seems to fit better than the ShethTornien and with a low mass cutoff of 5 x 10"Aln the ShethTornien mass function gives a better fit to the simulation results. The data is not reliable on scales smaller than 1hl hipc where we need a resolved halo model. This is deferred to a future work. On the large and intermediate scales however ShethTormen 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 onehalo and twohalo terms of the bias corrected point cluster expressions (533 536). The bias parameters, (b), are calculated using Eq. 532 and the halo occupation moments py,s are calculated using Eq. 553. The first in the correlation function hierarchy is the variance, a(2R) Which can be resolved into a onehalo term and a two halo term. Figure 68 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 PressSchecter 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 69 and with the results of APM survey in Figure 610. 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 611 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 612 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 61. Details of the distribution of objects (halos & substructures) in the catalogs generated from the simulation data. Mass cutoff Nvautos 0 50234 5 x1011 43482 4 x1012 11934 1 x1014 494 Nvsubs 64316 57564 24918 5113 Ni ~Subs) 1.28 1.32 2.09 10.35 n, 0.008 0.007 0.003 0.0006 nh 0.006 0.005 0.001 0.00006; Table 62. Bias parameters for mass in LCDM SPH SA (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 103 Figure 61. Hierarchical amplitudes, S, for p = 3, 4, 5 are plotted as a function of spectral index n for both the PressSchecter and ShethTormen mass functions. The red curves are for PressSchecter mass function and the blue ones are for ShethTormen mass function. Solid lines show the results without any mass cutoff. The dotted curves are for a lower mass cutoff of 102m1. The shortdashed curves are for a lower mass cutoff of 104m1. The green curves show the predictions of Hyper Extended Perturbation Theory (Scoccimarro & Friemann 1999 [34]). The symbols with error bars show the results from numerical simulations (Colombi, Bouchet & Hernquist 1996 [5]). 10 ~103 104 105 10"6 10 10'2 1013 1014 1016 M/ bottom], N [top] 1016 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 PressSchecter mass function and the long dashed green curve is the ShethTormen mass function. The black dotted curves are the PressSchecter and ShethTormen 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. Figure 62. 