Numerical Cosmology & Galaxy Formation

Transcription

1 Numerical Cosmology & Galaxy Formation Lecture 3: Generating initial conditions Benjamin Moster 1

2 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 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 3 Numerical Cosmology & Galaxy Formation

4 The World Egg In 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. 4 Numerical Cosmology & Galaxy Formation

5 Polynesian mythology: Primordial goddess created man in coconut shell The World Egg In many mythologies: universe comes into existence by hatching from World Egg Egyptian mythology: Ra was contained in egg laid by celestial bird. Modern Cosmology: Greek mythology: From the Orphic egg hatched the primordial deity Phanes who Phoenician mythology: Universe expanded from hot dense state in created Big Bang other gods. Môt burst forth from Cosmic egg? egg and heavens were created. Chinese mythology: Universe began as egg from which Pangu was born. Two halves of egg became sky and earth. 4 Numerical Cosmology & Galaxy Formation

6 Cosmology Numerical Cosmology This is where we start 5 Numerical Cosmology & Galaxy Formation

7 Computational Cosmology Cosmological model + initial conditions + simulation code = galaxies CMB SDSS generation the initial conditions running the simulation analyzing the data 6 Numerical Cosmology & Galaxy Formation

8 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 particles homogeneous & isotropic 7 Numerical Cosmology & Galaxy Formation

9 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 particles infinite (periodic boundary conditions) 7 Numerical Cosmology & Galaxy Formation

10 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 particles infinite (periodic boundary conditions) 7 Numerical Cosmology & Galaxy Formation

11 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 particles infinite (periodic boundary conditions) 7 Numerical Cosmology & Galaxy Formation

12 Perturbations Universe shows small density fluctuations already at high z (see CMB) Convert CMB fluctuations to density perturbations homogeneous & isotropic 8 Numerical Cosmology & Galaxy Formation

13 Perturbations Universe shows small density fluctuations already at high z (see CMB) Convert CMB fluctuations to density perturbations How? perturbed density field 8 Numerical Cosmology & Galaxy Formation

14 The power spectrum Use power spectrum to describe density fluctuations P (k) =h ˆ(k) 2 i Temporal evolution (in the linear regime): P (k, t) =P 0 (k)d 2 (t) How do we obtain P(k)? P(k) 9 Numerical Cosmology & Galaxy Formation

15 The power spectrum 10 Numerical Cosmology & Galaxy Formation

16 The transfer function From inflation: P i (k) =Ak n 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 is transferred to post-recombination power spectrum as: P (k) =T 2 (k)p i (k) 11 Numerical Cosmology & Galaxy Formation

17 Computing the power spectrum with CAMB Power spectra & transfer functions can be computed with CAMB Free code by Lewis & Challinor available at Either download and install Fortran 90 source code or use web interface at lambda: 12 Numerical Cosmology & Galaxy Formation

18 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 σ 8 right! 13 Numerical Cosmology & Galaxy Formation

19 Perturbations How to impose a spectrum of fluctuations on a particle distribution? P(k) perturbed density field 14 Numerical Cosmology & Galaxy Formation

20 The Zel dovich Approximation Using cosmological perturbation theory we have derived + ~ r ~u a ~u + H~u = =0 ~ r P a =4 G a 2 And the growth equation ˆ +2H ˆ = ˆ is solved by the growth function D(a) With the quantities: ~r = a(t)~x ~r a ~v = ~r =ȧ~x + a ~x = H~r + ~u 4 G + c2 sk 2 = (~x, t) = Z a 2 c 2 s = P d 3 k (2 ) ˆ( ~ 3 k, t)e i~ k ~x 15 Numerical Cosmology & Galaxy Formation

21 The Zel dovich Approximation Substitute (a) = 0 D(a) into Poisson equation (with = 0 /a 3 ): ~r 2 =4 Ga 2 =4 Ga 2 0 a 3 D(a) 0 = D(a) r a ~ 2 0 So the potential also grows as In a matter dominated universe: Integrate continuity equation or ~u = ~ r 0 a Z D a dt (a) = D(a) a D(a) a! + ȧ a ~u = 0 1 a = ~ r Growth function satisfies the growth equation: D +2ȧ or rewritten: aḋ =4 G c a 3 D 2 Ḋ) a =4 G c a 3 D 16 Numerical Cosmology & Galaxy Formation

22 The Zel dovich Approximation Integrate this yields a 2 Ḋ 4 G c = Z D a dt Use this with the integrated continuity equation to get ~x = ~u a = r ~ 0 a 2 Ḋ Ḋ r a 2 = ~ 0 4 G c 4 G c Integrate this to get the Zel dovich approximation: ~x ~x 0 = D~ r 0 4 G c = a~ r (a) 4 G c with the unperturbed position ~x 0 This formulation can be used to extrapolate the evolution of structures into the regime where displacements are no longer small 17 Numerical Cosmology & Galaxy Formation

23 Displacing particles on a grid How apply the Zel dovich approximation to a grid? Define ~ = ~x ~x0 and its Fourier transform ˆ~ ~k = a(i~ k)ˆ~ k 4 G 0 With Poisson s equation k 2 ˆ~k = 4 G 0ˆ~ k a we get ˆ~ ~k = i ~ k ˆ~ k k 2 Use the power spectrum P (k) =h ˆ(k) : ~ k = p 2 i P (k)r ~k e i ~ k with R ~k e i ~ k = R1 + ir 2 where R 1 and R 2 are drawn from Gaussian with standard deviation of 1 (P is only ensemble average) Initial velocities are given by ~x = Ḋ D ~! ~v ~k = aḋ D ~ ~ k 18 Numerical Cosmology & Galaxy Formation

24 Limitations of this method There are a number of parameters that have to be chosen: Cosmology ΛCDM (?) Box size Number of particles Starting redshift B N z i Choices are not completely free, but interwoven 19 Numerical Cosmology & Galaxy Formation

25 Wavenumber limitation B 2B 3p N B = 100h 1 Mpc N = N/B3 has to be chosen large enough to model interesting features 20 Numerical Cosmology & Galaxy Formation

26 Wavenumber limitation B 2B 3p N B = 100h 1 Mpc N = N/B3 has to be chosen large enough to model interesting features 20 Numerical Cosmology & Galaxy Formation

27 Wavenumber limitation B 2B 3p N B = 100h 1 Mpc N = 32 3 N/B3 has to be chosen large enough to model interesting features 20 Numerical Cosmology & Galaxy Formation

28 Largest mode (2π/B) has to stay linear or MF will be suppressed Iteratively compute knl using Linearity constraint 1= Z knl 0 P (k, z = 0)k 2 dk Set B 2 k nl Results in typically B > 30 Mpc/h 21 Numerical Cosmology & Galaxy Formation

29 Initial redshift: not too early and not too late! ~x = ~x 0 D(a) ~ r 0 4 G 0 Initial redshift constraints Too early: integration of numerical noise Too late: shell crossing not taken into account 22 Numerical Cosmology & Galaxy Formation

30 Why is h used everywhere??? = Nm p B 3 = 0 c,0 = 0 3H G = 0 3 (100h) 2 8 G m p = h 2 8 G B 3 N hm p = G m p = G (hb) 3 N B 3 N Distances and masses have to be divided by h to get physical values! 23 Numerical Cosmology & Galaxy Formation

31 Grid vs. Glass ICs 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 Glass 24 Numerical Cosmology & Galaxy Formation

32 Zoom simulations 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 ICs z=0 zoom on halo 25 Numerical Cosmology & Galaxy Formation

33 N-GenIC 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 % 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 (16 3 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 26 Numerical Cosmology & Galaxy Formation

34 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 e24 % 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 % seed for IC-generator NumFilesWrittenInParallel 2 % limits the number of files that are written UnitLength_in_cm e21 % defines length unit (in cm/h) kpc UnitMass_in_g 1.989e43 % defines mass unit (in g/cm) M UnitVelocity_in_cm_per_s 1e5 % defines velocity unit (in cm/sec) km/s 27 Numerical Cosmology & Galaxy Formation

35 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 28 Numerical Cosmology & Galaxy Formation

36 Why simulate isolated disc systems? 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) Stars Gas 29 Numerical Cosmology & Galaxy Formation

37 Disc Galaxies What does a disc galaxy system consist of? Stellar disc Gaseous disc Stellar bulge Hot gaseous halo Dark matter halo 30 Numerical Cosmology & Galaxy Formation

38 The dark matter halo Size of the dark matter halo is given by the virial radius (with mean 4 overdensity 200ρcrit) containing the virial mass: M 200 = 200 crit 3 R3 200 The virial velocity is M 200 = v GH(z) v = GM 200 R 200 and and we have: R 200 = v H(z) The density profile is the NFW profile: (r) = c (r/r s )(1 + r/r s ) 2 Often the Hernquist profile is used: (r) = M dm a 2 r(r + a) 3 Springel et al. (2005) 31 Numerical Cosmology & Galaxy Formation

39 Disc has exponential radial profile: Disc scale length The stellar components R d is fixed by assuming the specific angular momentum of disc and dark matter halo are equal Vertical profile of the disc given by: (R) = M d 2 R 2 d (z) = M d 2z 0 sech 2 exp For the bulge the spherical Hernquist profile is assumed: z 2z 0 R R d b (r) = M b 2 r b r(r b + r) 3 The bulge is typically assumed to be non-rotating 32 Numerical Cosmology & Galaxy Formation

40 The gaseous components Like the stellar disc, the gaseous disc has exponential radial profile The vertical profile of the gas disc cannot be chosen freely because 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: = Hot gaseous halo follows a beta-profile: (r) = 0 "1+ r r c 2 # 3 2 The gas temperature is determined using hydrostatic equilibrium: T (r) = µm Z p 1 1 hot(r) GM(r) k B hot(r) r 2 dr r 33 Numerical Cosmology & Galaxy Formation

41 Creating a particle representation of a galaxy We know the density profiles of each component. How can we put the particles such that these are reproduced? First step compute mass profile: M(r) = Invert M(r), possibly analytically, otherwise numerically Draw random number q from uniform distribution between 0 and 1 Z r 0 (r 0 )d 3 r 0 Radius is given by: For spherical distribution: (with random and ) r = r(q M tot ) 0 1 ya = r z 0 sin sin sin cos 1 A 34 Numerical Cosmology & Galaxy Formation

42 Velocity structure 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 orbits If distribution function depends only on E and Lz the moments are: v R = v z = v R v z = v z v v 2 R = v 2 z = 1 v 2 = v R 2 + R Z 1 z = v R v =0 dz 0 (R, z 0 (R, ( v 2 Must be solved numerically 35 Numerical Cosmology & Galaxy Formation

43 The orbit of galaxy mergers 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: 1, 1, 2, 2 dmin 2, 2 e 1, 1 Rstart 36 Numerical Cosmology & Galaxy Formation

44 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: 37 Numerical Cosmology & Galaxy Formation

45 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 38 Numerical Cosmology & Galaxy Formation