Astronomía Extragaláctica y Cosmología Observacional

Astronomía Extragaláctica y Cosmología ObservacionalDepto. de Astronomía (UGto)

Lecture 20 Structure Formation IV – Numerical Simulations

• Power Spectra of Fluctuations & Origin of Inhomogeneities

• Linear Evolution of Perturbations

• Non-Linear Evolution of Perturbations

• Simulations of Structure Formation N-body DM Simulations DM Simulations + Semi-Analytical Modeling Hydrodynamical Simulations (SPH)

N-body DM Simulations: Why DM? Baryonic matter alone can not account for structure formation:

• since baryonic matter and radiation were coupled before recombination, and their modes only oscillate in this phase, their fluctuations alone (printed in the CMBR) do not have enough intensity on small angular scales for structure formation in the required time

• masses of structures we expect in a baryonic Univ are ~ 1014–1016 M for adiabatic fluctuations and about 106 M for isothermal fluctuations, both far from the galaxy masses (107–1012 M). Or superclusters formed first and subsequently fragmented…

• primordial nucleosynthesis of light elements and measurements of the second peak in CMBR power spectra (and also clusters and Lyα forests) provide concordant and strict values for baryonic density parameter (Ωb h2 ≈ 0.019 0.002 or Ωb ~ 0.04 for h = 0.7), which is very low to account for matter density parameter (ΩM ~ 0.2–0.3, from several different measurements)

• extra matter beyond the baryonic one is claimed for explaining the formation of galaxy discs (increasing of spin parameter during dissipative collapse) and its stability, and spheroidal galaxies in the merging scenario (dynamical friction of DM halos to compensate the high cluster dispersion velocities) [last lecture]

• rotation curves of spirals and hot gas in ellipticals suggest that there is at least 5 times more mass on them than the visible “luminous” mass, probably non-baryonic (searches for MACHOs, prediction of brown dwarf and BH numbers all failed to account for baryonic DM)

N-body DM Simulations

Aarseth, Turner & Gott (1979) – general properties of nonlinear gravitational clustering Efstathiou & Eastwood (1981) – evidence that scale-free conditions evolve in a self-similar way White, Frenk & Davis (1983) – rejection of HDM Davis et al (1985) – flat CDM (SCDM) alone also discarded (strong bias?), but OCDM (Ω0 ~ 0.2) OK Frenk [1991, Ph.S. 36, 70] – review Gelb & Bertschinger (1994), Zurek et al. (1994) and Cole et al. (1997) – confirmation of previous results with larger N simulations Jenkins et al. [1998, ApJ 499, 20] – Virgo intermediate scale simulation (also a good small review) Kauffman et al. [1999, MNRAS 303, 188] – GIF simulation (inclusion of semi-analytic modeling) Jenkins et al. [2001, MNRAS 321, 372] – Hubble Volume simulation (the largest one in volume) Springel et al. [2005, Nature 435, 629] and Croton et al. [2006, MNRAS 365, 11] – Millennium simulation (the first 1010 particles simulation)

Brief history of N-body DM simulations:

VIRGO CONS. GIF Interm. Millennium Hubble Vol.L (Mpc) 141.3 239.5 500 3000Figure (deep) 8 10 15 30N 17106 17106 10109 1109

(2563) (2563) (21603)m(MSol) 1.41010 6.91010 8.6108 2.21012

Ref. Kauffmann 99 Jenkins 98 Springel 05 Jenkins 01

N-body DM Simulations: Examples of Large Current Simulations

GIF

Semi-analytical modelling

Intermediate scale

Millennium

HubbleVolume

N-body DM Simulations: Initial Conditions

The initial conditions for the simulations are set by the PS of fluctuations after decoupling (initial PS + transfer function) From the PS of temperature anisotropies in CMBR we can settle the shape of DM PS

The normalization of the PS (amplitudes) is usually determined from the current rms mass fluctuations in spheres of radius conventionally chosen as 8 h–1 Mpc (σ8 ~ 0.8–0.9), extrapolated back to the starting z with the growth function D(z)

P(k) = A k . {1 + [aq + (bq)3/2 + (cq)2]d}2/d

q = k/Γ Γ = Ωh2/θ2 = 0.21 d = 1.13a = 6.4 b = 3.0 c = 1.7 (h–1 Mpc)

See, e.g., Caretta et al. 2008, A&A 487, 445

N-body DM Simulations: Initial Conditions

The initial unperturbed density field is usually set by a “glass” distribution of the particles

then one assigns random amplitudes and phases for individual modes in Fourrier space, defined by the PS

using the ZA, the density fluctuations are converted to displacements of the particles

δk = Bk eiφk <|δk|2> = P(k)

N-body DM Simulations: Force Calculation

The DM particles are, by definition, acollisional (the particles just respond to their collective gravitational field)

At each step of the simulation, the gravitational potential that each particle is subjected is calculated, and then, its acceleration

where ε is the softening constant, which is needed for• compensate for the use of point particles (without volume)• prevent the formation of bound particle pairs and large-angle particle scattering• ensure that two-body relaxation time is sufficiently high• allow the system to be integrated with low order integration schemes

There is a compromise between maximizing resolution one wants to achieve with the simulation (to correctly resolves small structures and the desired underlying physics), and maximizing volume (necessary for having some statistics and for permitting rare objects to form), which will depend on the number of particles (N) that the available computational capabilities make possible for running in a reasonable amount of time. The naïve computation of the forces is an t N2 task.

Φ(xi) = –G j=1→N mj . (xi – xj)2 + ε2

xi´´ = –Φ(xi)


Many techniques have been proposed for optimizing the force calculation, making possible to increase N in a simulation and, consequently, to improve the resolution and/or the simulated volume:

Particle-Mesh (PM) method → the Poisson equation can be solved in real-space by a convolution of the density field with a Green’s function

that, in Fourrier space, becomes a simple multiplication

and so, one can solve the potential in 3 steps• FFT forward of the density field• multiplication with the Green’s function• FFT backwards to obtain the potential

For calculation the density field one usually uses a mesh (grid) in which the number of particles in each cell give the mass density of the cell

After calculating the potential, the same interpolation kernel used for the obtaining the density field is used for the assignment of mesh forces to the particle locations

Φ(x) = g (x – y) ρ(x) dy

Φ(k) = g (k) . ρ(k)


AP3M method → the PM method is very simple and fast, but it lacks force resolution at the same time for cosmological scales and internal structure of haloes (the needed dynamical range is very large). The P3M was proposed to solves this problem by supplementing the PM force calculation with direct summation for short range. So, P3M comes for P-M (particle-mesh) plus P-P (particle-particle). However, this direct summation slows down very much the processing, and a refinement is done by placing the direct summation only in clustered regions. This is called AP3M.


Tree algorithms → use of a hierarchical multipole expansion to account for distant particle groups

where M is the monopole and Q is the quadrupole (the dipole term vanishes when summed over all particles in a group)

The advantages of the Tree method are:• no intrinsic restrictions for its dynamic range• force accuracy can be adjusted to the desired level• the speed depends weakly on clustering state• its geometrically flexible

although, for high-z its very expensive to obtain accurate forces with the tree-algorithm alone – an hybrid Tree-PM is the solution, where the tree is used for short range

1 = 1 = 1 .|xi – xj| |(xi – s) – (xj – s)| y

M = mj

Qij = mk [3(xk – s)i (xk – s)j – δij (xk – s)2]

Φ(x) = –G [(M / |y|) + ½ (yT Q y) / |y|5]

Structure/object Mass (MSol) Diameter Intensity MorfologySupercluster > 11015 > 10 Mpc double—rich (>8) filaments and wallsCluster 31012 —11015 1—10 Mpc group—rich reg. (virial.), irreg. (subestr.)Galaxy 1108—31012 < 0.5 Mpc dwarf—bright E, Sp, Irr

N-body DM Simulations: DM “Halos”

Perc

olat

ion

Anal

ysis

N-body DM Simulations: DM Halos – Algorithms for Identification

Also called “Friends-of-Friends” algorithm (FoF) Originally proposed by Geller & Huchra (1982) and Einasto et al. (1984), applied first to simulations by Davies et al. (1985)

The algorithm groups together particles that have pair separations smaller than a chosen linking length, l . This l is frequently referred as b times the mean interparticle separation. The resulting “groups” are bounded by a surface of approximately constant density:

Assuming that the density profile of these groups can be approximated by an isothermal spheres, the average density contrast internal to this surface is given by 3 times the surface density. In general, the value of b is set to give a mean overdensity close to the value expected for a virialized object in the framework of the spherical collapse model, ~ 179, which gives b = 0.2.

The main advantages of the FoF are the simplicity (only one free parameter), the reproducibility (an unique group catalog for any chosen value of b), and the capability of detecting halos of any shape. An improved version of FoF is the HFoF (Klypin et al. 1999)

n/<n> = 2 . 3/(4l 3) . 1/<n> = 3/(2l 3) . <l > 3 = (3/2) . (1/b3) ~ 1/(2b3)

See, e.g., Caretta et al. 2008, A&A 487, 445

Galaxies

Clusters

Superclusters

Sphe

rical

Over

dens

ities

N-body DM Simulations: DM Halos – Algorithms for IdentificationDE

NMAX

Originally proposed by Lacey & Cole (1994)

The algorithm identifies density peaks and puts spheres around them, increasing the radius of the spheres until the average density contrast reaches a chosen value. The main parameter, in this case, is the threshold overdensity, .

The disadvantage is that it tends to loose the outer portions of ellipsoidal halos due to assumption of spherical symmetry.

An improved version of SO is the BDM (Bound Density Maxima, Klypin et al. 1999)

Originally proposed by Gelb & Bertschinger 1994

The algorithm “moves” the particles along local density gradients toward density maxima, separating halo candidates by 3D density valleys. The improvement of this method is the application of a self-boundness check by eliminating particles with positive total energy.

A popular advanced version of DENMAX is the HOP (Eiseinstein & Hut 1998)

Δ

N-body DM Simulations: Results – Correlation Functions

0.00.10.30.51.01.52.03.05.0


0.00.10.30.51.01.52.03.05.0

Evolution of CF parameterswith z

Caretta et al. 2008, A&A 487, 445


Springel et al. 2005, Nature 435, 629

Millennium simulation(CF at z = 0)

- - DM (mass) Millennium galaxies 2dFGRS galaxies

N-body DM Simulations: Results – Evolution of Halo Masses

N-body DM Simulations: Results – Mass Functions

Springel et al. 2005, Nature 435, 629

N-body DM Simulations: DM Halos

N-body DM Simulations: Results – Satellites Number Problem

Moore et al. 1999, ApJL 524, L19

N-body DM Simulations: Results – Mass Function Problem

Somerville & Primack 2001

Resimulation Technique

Semi-Analytical Modelling of Gastrophysics

In these techniques#, each of the complicated and interacting physical processes involved in galaxy formation is approximated using a simplified, physically based model.

These processes include:

• growth through accretion and merging of DM haloes (merging history trees)

• shock heating and virialization of gas within these haloes• radiative cooling of gas • settling of gas to rotational supported discs

• star formation

• feedback from SNe, stellar winds, and AGN

• evolution of stellar populations

• absorption and reradiation by dust

• galaxy merging with its accompanying SB and morphological transformations

# originally laid out by White & Frenk 1991, ApJ 379, 52; Cole 1991, ApJ 367, 45; and Lacey & Silk 1991, ApJ 381, 14.

Semi-Analytical Modelling of Gastrophysics

Merging history trees can be computed together with the simulation [e.g. Kauffman et al. 1999, MNRAS 303, 188; Springel et al. 2001, MNRAS 328, 726], with the expenses of requiring frequent data dumps and storage capacity

The results of these modeling have allowed detailed photometric comparisons with observations, of local and high-z galaxies (LFs, colors, Tully-Fisher and Faber-Jackson relations,...), and also predictions of SF histories, background radiation contributions from the UV to the far-IR, observed relations between AGNs and their host galaxies, etc

However, the adopted oversimplification of physical models is often shown up as inconsistencies with observations in other areas.

Semi-Analytical Modelling of Gastrophysics – LF




Croton et al. 2006, MNRAS 365, 11

Hydrodynamical Simulations (SPH)

The real gastrophysics is poorly understood

• gas cooling and condensation – sensitive to metal content, multiphase structure, UV background

• star formation – efficiency of SF, IMF are not known a priori

• stellar feedback (SNe, stellar winds) – SF regulation, metal enrichments are poorly known

• feedback from AGNs – physics of BH formation, feeding, etc not understood yet

• evolution of stellar populations – depend on population synthesis...

• absorption and reradiation by dust – also poorly known

• environmental interactions – mergers, accretion, tidal effects, ram-pressure, SB, morphological transformations not known

Hydrodynamical Simulations: Simulations with Gastrophysics

Smoothed Particle Hydrodynamics (SPH) are simulations that model the particles as a continuous fluid, by applying a kernel interpolation, allowing a hydrodynamical approach for their evolution the kernel is used to build continuous fluid quantities from discrete tracer particles this makes possible to include different types of particles to represent the gas of baryons

The treatment is Lagrangean an artificial viscosity needs to be introduced to capture shocks

The basic adaptive SPH schemes do not conserve energy and entropy simultaneously Variational derivation is able to do both conservations [Springel & Hernquist 2002] with this improvement, non standard physics like thermal conduction and cosmic-rays can be treated

Hydrodynamical Simulations

Hydrodynamical Simulations

Hydrodynamical Simulations: comparisons

Frenk et al. 1999: the Santa Barbara Cluster Comparison Project

Hydrodynamical Simulations: Mergers

Hydrodynamical Simulations: Mergers

Documents

Astronomía Extragaláctica y Cosmología Observacional