numerical cosmology & galaxy formation€¦ · numerical cosmology & galaxy formation 1...

Numerical Cosmology& Galaxy Formation

1

Benjamin Moster

Lecture 3: Generating initial conditions

Outline of the lecture course

• Lecture 1: Motivation & Historical Overview

• Lecture 2: Review of Cosmology

• Lecture 3: Generating initial conditions

• Lecture 4: Gravity algorithms

• Lecture 5: Time integration & parallelization

• Lecture 6: Hydro schemes - Grid codes

• Lecture 7: Hydro schemes - Particle codes

• Lecture 8: Radiative cooling, photo heating

• Lecture 9: Subresolution physics

• Lecture 10: Halo and subhalo finders

• Lecture 11: Semi-analytic models

• Lecture 12: Example simulations: cosmological box & mergers

• Lecture 13: Presentations of test simulations

2 Numerical Cosmology & Galaxy Formation 3 04.05.2016

Outline of this lecture

• Initial conditions for cosmological simulations

‣ Cosmological principle

‣ Power spectrum and transfer function

‣ Zel’dovich Approximation

‣ Putting particles in a simulation box

‣ Limitations

‣ Zoom simulations

• Initial conditions for disc galaxy simulations

‣ Properties of disc galaxies

‣ Creating a particle representation of a disc galaxy

‣ Merger orbits






‣ Limitations



The World EggIn many mythologies: universe comes into existence

by hatching from World Egg


• Egyptian mythology: Ra was contained in egg laid by celestial bird.

• Phoenician mythology: Môt burst forth from egg and heavens were created.

• Polynesian mythology: Primordial goddess created man in coconut shell

• Greek mythology: From the Orphic egg hatched the primordial deity Phanes who created other gods.

• Chinese mythology: Universe began as egg from which Pangu was born. Two halves of egg became sky and earth.

The World EggIn many mythologies: universe comes into existence

by hatching from World Egg


• Egyptian mythology: Ra was contained in egg laid by celestial bird.

• Phoenician mythology: Môt burst forth from egg and heavens were created.

• Polynesian mythology: Primordial goddess created man in coconut shell

• Greek mythology: From the Orphic egg hatched the primordial deity Phanes who created other gods.

• Chinese mythology: Universe began as egg from which Pangu was born. Two halves of egg became sky and earth.

Modern Cosmology:Universe expanded from hot dense state in Big Bang

Cosmic egg?

Cosmology

Numerical Cosmology


This is where we start

Computational Cosmology

• Cosmological model + initial conditions + simulation code = galaxies

SDSSCMB

generation theinitial conditions

running the simulation

analyzingthe data


Cosmological Principle

• On large scales, the properties

of the Universe are the same to

all observers

• Homogeneity: Universe looks

the same at every location

• Isotropy: Universe looks the

same in every direction

• Discretize density field into

particleshomogeneous & isotropic


Cosmological Principle

• On large scales, the properties

of the Universe are the same to

all observers

• Homogeneity: Universe looks

the same at every location

• Isotropy: Universe looks the

same in every direction

• Discretize density field into

particlesinfinite

(periodic boundary conditions)


Perturbations

• Universe shows small density

fluctuations already at high z(see CMB)

• Convert CMB fluctuations to

density perturbations

homogeneous & isotropic


Perturbations

• Universe shows small density

fluctuations already at high z(see CMB)

• Convert CMB fluctuations to

density perturbations

How?

perturbed density field


The power spectrum

• Use power spectrum to describe density fluctuations

• Temporal evolution (in the linear regime):

• How do we obtain P(k)?

P(k)

P (k, t) = P0(k)D2(t)

P (k) = h|�(k)|2i


The power spectrum


The transfer function

• From inflation: with n=1 called Harrison-Zel’dovich

• Many complex physical processes transfer P(k) during recombination due to coupling of radiation and matter:

- Silk damping for baryons

- Free-streaming damping for DM

- Different growth during radiation and matter dominated eras

• Initial power spectrum istransferred to post-recombinationpower spectrum as:

Pi(k) = Akn

P (k) = T 2(k)Pi(k)


Computing the power spectrum with CAMB

• Power spectra & transfer functions can be computed with CAMB Free code by Lewis & Challinor available at http://camb.info

• Either download and install Fortran 90 source codeor use web interface at lambda: http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm


http://camb.info

http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm

Computing the power spectrum with CAMB

• Output files are then generated that can be downloaded

• Cosmological parameters are printed

• Normalization must be chosen to get !8 right!


Perturbations

• How to impose a spectrum of

fluctuations on a particle

distribution?

P(k)

perturbed density field


• Using cosmological perturbation theory we have derived

• And the growth equation is solved by the growth function D(a)

• With the quantities:

The Zel’dovich Approximation

� +~r · ~ua

= 0

~u+H~u = �~r�P

a⇢�

~r��

a

�� = 4⇡G⇢a2�

¨� + 2H ˙

� = �

✓4⇡G⇢+

c2sk2

a2

◆

~r = a(t)~x

~v = ~r = a~x+ a~x = H~r + ~u

� =�⇢

⇢

�(~x, t) =

Zd3k

(2⇡)3�(~k, t)e�i

~

k·~x

c2s =P

⇢


• Growth function satisfies the growth equation: or rewritten:


• In a matter dominated universe:

• Integrate continuity equation or

�(a) = �0D(a)

~r2� = 4⇡Ga2⇢� = 4⇡Ga2⇢0a3

D(a)�0 =D(a)

a~r2�0

�(a) =D(a)

a�0

D(a) ⇠ a ! �(a) = �

@~u

@t+

a

a~u = �1

a~r�

• Substitute into Poisson equation (with ):

• So the potential also grows as

D + 2a

aD = 4⇡G

⇢ca3

D 1

a2@(a2D)

@t= 4⇡G

⇢ca3

D

⇢ = ⇢0/a3

@(a~u)

@t= �~r�

~u = �~r�0

a

ZD

adt


• Integrate this yields

• Use this with the integrated continuity equation to get


a2D

4⇡G⇢c=

ZD

adt

• Integrate this to get the Zel’dovich approximation:

• This formulation can be used to extrapolate the evolution of

structures into the regime where displacements are no longer small

~x� ~x0 = �D

~r�0

4⇡G⇢c= �a

~r�(a)

4⇡G⇢cwith the unperturbed position ~x0

~x =~u

a

= �~r�0

a

2

a

2D

4⇡G⇢c= � D

~r�0

4⇡G⇢c


• How apply the Zel’dovich approximation to a grid?

• Define and its Fourier transform

• With Poisson’s equation we get

Displacing particles on a grid

~

= ~x� ~x0

• Use the power spectrum : with where and are drawn from

Gaussian with standard deviation of 1 (P is only ensemble average)

• Initial velocities are given by

�~k =p

P (k)R~kei�~k

R~kei�~k = R1 + iR2 R1 R2

~x =D

D

~

! ~v~k = a

D

D

~

~k

P (k) = h|�(k)|2i

~ ~k = �a(i~k)�~k

4⇡G⇢0

�k2�~k =4⇡G⇢0�~k

a~ ~k = i~k

�~kk2


• There are a number of parameters that have to be chosen:

‣ Cosmology ΛCDM (?)

‣ Box size

‣ Number of particles

‣ Starting redshift

Limitations of this method

• Choices are not completely free, but interwoven…

B

N

zi


• N/B3 has to be chosen large enough to model interesting features

Wavenumber limitation

B

2B3pN B = 100h�1Mpc N = 10243




B

2B3pN B = 100h�1Mpc N = 2563




B

2B3pN B = 100h�1Mpc N = 323


• Largest mode (2"/B) has to stay linear or MF will be suppressed

• Iteratively computeknl using

• Set

• Results in typically B > 30 Mpc/h

Linearity constraint

1 =1

2⇡2

Z knl

0P (k, z = 0)k2dk

B � 2⇡

knl


• Initial redshift: not too early and not too late!

• Too early: integration of numerical noise

• Too late: shell crossing not taken into account

Initial redshift constraints

~x = ~x0 �D(a)~r�0

4⇡G⇢0


• Distances and masses have to be divided by h to get physical values!

Why is h used everywhere???

⇢ =Nmp

B3= ⌦0⇢c,0 = ⌦0

3H20

8⇡G= ⌦0

3 · (100h)2

8⇡G

mp = ⌦030000h2

8⇡G

B3

N

hmp = ⌦030000

8⇡G

(hB)3

N

mp = ⌦030000

8⇡G

B3

N


• Alternative to Grid ICs are Glass ICs

‣ Start with random positions for particles in the box

‣ Evolve box forward in time under gravity BUT with reversed sign

‣ Use resulting particle positions for Zel’dovich approximation

• Just cosmetics?

Grid vs. Glass ICs

GlassGrid


• Increasing the resolution of the box becomes very expensive

• Alternative: just increase the resolution in the area of interest:

1) Run low resolution simulation and identify interesting object(s)

2) trace back particles of that object to initial positions in ICs

3) resample this area with more particles and rerun the simulation

• Disadvantage: only one system simulated - no statistics

Zoom simulations

ICs z=0 zoom on halo


Nmesh 128 % This is the size of the FFT grid

Nsample 128 % sets the maximum k that the code uses, % Ntot = Nsample^3, where Ntot is the

Box 150000.0 % Periodic box size of simulation

FileBase ics % Base-filename of output files OutputDir ./ICs/ % Directory for output

GlassFile glass.dat % File with unperturbed glass/grid (163 particles)

TileFac 8 % Number of times the glass file is tiled

Omega 0.3 % Total matter density (at z=0) OmegaLambda 0.7 % Cosmological constant (at z=0) OmegaBaryon 0.0 % Baryon density (at z=0) HubbleParam 0.7 % Hubble paramater Redshift 63 % Starting redshift Sigma8 0.9 % power spectrum normalization

N-GenIC


SphereMode 1 % if "1" only modes with |k| < k_Nyquist are used WhichSpectrum 2 % "1" selects Eisenstein & Hu spectrum, % "2" selects a tabulated power spectrum % otherwise, Efstathiou parametrization is used

FileWithInputSpectrum spectrum.txt % tabulated input spectrum InputSpectrum_UnitLength_in_cm 3.085678e24 % defines length unit (Mpc) ReNormalizeInputSpectrum 1 % if zero, spectrum assumed to be normalized

ShapeGamma 0.21 % only needed for Efstathiou power spectrum PrimordialIndex 1.0 % may be used to tilt the primordial index Seed 123456 % seed for IC-generator

NumFilesWrittenInParallel 2 % limits the number of files that are written

UnitLength_in_cm 3.085678e21 % defines length unit (in cm/h) kpc UnitMass_in_g 1.989e43 % defines mass unit (in g/cm) 1010M⦿ UnitVelocity_in_cm_per_s 1e5 % defines velocity unit (in cm/sec) km/s

N-GenIC


Outline of this lecture






‣ Limitations





‣ Merger orbits




‣ Merger orbits


• Set up individual system(s) and simulate/study those

• Advantage: much higher resolution (more particles per system) ‘Nice’ discs can be studied (often hard in cosmological runs)

• Disadvantage: Cosmological background is neglected (no infall, etc)

Why simulate isolated disc systems?

Stars Gas29 Numerical Cosmology & Galaxy Formation 3 04.05.2016

• What does a disc galaxy system consist of?

Disc Galaxies

• Stellar disc

• Gaseous disc

• Stellar bulge

• Hot gaseous halo

• Dark matter halo


• Size of the dark matter halo is given by the virial radius (with mean

overdensity 200#crit) containing the virial mass:

• The virial velocity is and we have:

and

The dark matter halo

M200 = 200⇢crit4⇡

3R3

200

v2200 =GM200

R200

M200 =v3200

10GH(z)R200 =

v20010H(z)

• The density profile is the NFW profile:

• Often the Hernquist profile is used:

⇢(r) =�c

(r/rs)(1 + r/rs)2

⇢(r) =Mdm

2⇡

a

r(r + a)3 Spri

ngel

et

al. (

2005

)


• Disc has exponential radial profile:

Disc scale length is fixed by assuming the specific angular momentum of disc and dark matter halo are equal

• Vertical profile of the disc given by:

The stellar components

⌃(R) =

Md

2⇡R2d

exp

✓� R

Rd

◆

Rd

⇢(z) =Md

2z0sech2

✓z

2z0

◆

• For the bulge the spherical Hernquist profile is assumed:

• The bulge is typically assumed to be non-rotating

⇢b(r) =Mb

2⇡

rbr(rb + r)3


• Like the stellar disc, the gaseous disc has exponential radial profile

• The vertical profile of the gas disc cannot be chosen freelybecause it depends on the temperature of the gas

• For a given surface density, the vertical structure of the gas disc arises

as a result of self-gravity and pressure:

The gaseous components

• Hot gaseous halo follows a beta-profile:

• The gas temperature is determined using hydrostatic equilibrium:

� 1

⇢g

@P

@z=

@�

@z

�(r) = �0

"1 +

✓r

rc

◆2#� 3

2�

T (r) =µmp

kB

1

�hot

(r)

Z 1

r�hot

(r)GM(r)

r2dr


• We know the density profiles of each component. How can we put the particles such that these are reproduced?

• First step compute mass profile:

• Invert M(r), possibly analytically, otherwise numerically

• Draw random number q from uniform distribution between 0 and 1

• Radius is given by:

• For spherical distribution: (with random and )

Creating a particle representation of a galaxy

M(r) =

Z r

0⇢(r0)d3r0

r = r(q Mtot

)

0

@x

y

z

1

A= r

0

@sin ✓ cos�

sin ✓ sin�

cos ✓

1

A✓ �


• To get the velocities: solve the collisionless Boltzmann equation

• Alternative: assume that velocity distribution can be approximated by

a multivariate Gaussian ➙ only 1st and 2nd moments needed

• For static + axisymmetric system: E and Lz are conserved along orbitsIf distribution function depends only on E and Lz the moments are:

Velocity structure

vR = vz = vRvz = vz v� = vRv� = 0

v2R = v2z =1

⇢

Z 1

zdz0⇢(R, z0)

@�

@z0(R, z0)

v2� = v2R +R

⇢

@

@R(⇢v2R) +R

@�

@R

• Must be solved numerically


• For simulating galaxy mergers: put 2 systems on orbit.

• Parameters that specify the merger orbit:

‣ Initial separation Rstart (typically around the virial radius R200)

‣ Pericentric distance dmin (if galaxies were point masses)

‣ Orbital eccentricity e (usually parabolic orbits, i.e. e=1)

‣ Orientation of discs with respect to orbital plane:

The orbit of galaxy mergers

�1, ✓1, �2, ✓2

Rstart

dmin

e�1, ✓1

�2, ✓2


Final notes

• Text Books:

‣ Cosmology: Galaxy Formation and Evolution (Mo, vdBosch, White)

‣ Galactic Structure: Galactic Dynamics (Binney, Tremaine)

• Papers:

‣ Bertschinger (2001), ApJS, 137, 1

‣ Springel & White (1999), MNRAS, 307, 162

‣ Springel et al. (2005), MNRAS, 62, 79

• Gadget and N-GenIC website: http://www.mpa-garching.mpg.de/gadget/


Up next

• Lecture 1: Motivation & Historical Overview

• Lecture 2: Review of Cosmology

• Lecture 3: Generating initial conditions

• Lecture 4: Gravity algorithms

• Lecture 5: Time integration & parallelization

• Lecture 6: Hydro schemes - Grid codes

• Lecture 7: Hydro schemes - Particle codes

• Lecture 8: Radiative cooling, photo heating

• Lecture 9: Subresolution physics

• Lecture 10: Halo and subhalo finders

• Lecture 11: Semi-analytic models

• Lecture 12: Example simulations: cosmological box & mergers

• Lecture 13: Presentations of test simulations


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