C l i l i l ti fCosmological simulations ofthe dark matterthe dark matter

Françoise Combes

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Galaxy cartographiesV CDMVersus CDMSimulations

Simulations reproduce very well large-scale structures:

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Cosmic web, filaments, walls, great walls,the structure of voids, the granularity of super-clusters.

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N-body simulationsCompute the interaction between N bodiesDirect method: time increases as N2

Possible onmy with N=104-5

To go f rther p to N 1010-11To go further, up to N = 1010-11

Judicious tricks: Fast Fourier Transforms (FFT), or Tree-codeJudicious tricks: Fast Fourier Transforms (FFT), or Tree codeComputation time in N logN (Hohl 1975)The potential is the convolution of 1/r and the densityAt each dt, one computes the TF of density, then one multipliesin Fourier space the TF(1/r) and the TF(ρ) inverse TFin Fourier space, the TF(1/r) and the TF(ρ) inverse TF

Softening 1/(r2 + a2), to avoid the 2-body relaxation

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Softening 1/(r a ), to avoid the 2 body relaxation Gives an idea of the spatial resolution

Analogic simulationAnalogic simulationResponse in cosinusin cosinus

Eric Holmberg (1941)Analogy between gravitation and lightAnalogy between gravitation and light Interaction between 2 galaxies composed of 37 points37 light bulbs with photo-electric cellsg p

The light flux varies in 1/r2, each bulbi t th l flreceives on two orthogonal axes a flux

proportionnal to the components Fx & Fy (dV/dt)

The new position of each particle (bulb) is deduced from velocity vectors

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Methods: Tree-codeApprox: monopole +quadrupole accordingquadrupole, accordingopening criterium

Advantage: no gridVariable resolution

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Barnes &Hut (1983)

Hydrodynamics: collisions, SPH, AMRFor gas hydrodynamics, the essential is a weak dissipation

Hydrodynamics: collisions, SPH, AMR

Collisions between particles ("sticky-particles")or finite differences (fluid code, on a grid)or finite differences (fluid code, on a grid)

Or variable spatial resolution: SPH"Smoothed Particles Hydrodynamics" (Lucy & Monaghan 1977)

Principle: a kernel fonction (or weight W( r ))Principle: a kernel fonction (or weight W( r ))with a variable size, which contains a fixed number of neighbors

Density is computed by averaging on neighbors (30-50 neighbors)

and all other quantities derived similarly9

SPH Technics: convolution

With the kernel W( r ) normalised to 1, and with bound support

E l ti f titEvaluation of any quantity:

Or derivative

Symmetrisation of pressure terms, etc…

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AMR: Adaptive Mesh RefinementAMR: Adaptive Mesh RefinementAMR: Methode on a fixed grid EulerianAMR: Methode on a fixed grid, EulerianDo not follow particlesGravity: Fast Fourier Transforms, PM (Particle-Mesh)y , ( )Or else Multi-grid codeVariable resolution, adapting to dense regions

Hydro: Jump conditions to satisfy for all shocksFollows much more finely shocks wavesFollows much more finely shocks waves

Difficult to anticipate supersonic motionsp p(no galilean invariance)

11ParallelisationPeano-Hilbert

The various refinement levels

Up to 25 levels, 225= 3 107

Large dynamical range in scale

12Teyssier, 2013

Possible zooms, first stars

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Advantages (and problems) comparedd a tages (a d p ob e s) co pa edSPH: Lagrangian method, follows particles, no gridSPH: Lagrangian method, follows particles, no gridArtificial viscosity (to spread shocks over h= spatial resolution)Problems of calefaction, surface tension, make impossible theexchanges over a size of the order of h hKelvin-Helmoltz instabilities: shear at the interface between 2 fluidsF t i tiFrequent supersonic motions

Agertz et al

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ge t et a2007

Moving meshMoving mesh

or fixed grid (AMR)Rayleigh-Taylor InstabilitiesDenser fluid above (gravity force)Fl id i h l i VFluid with a velocity Vx

Vx=0 1 10 and 100Vx=0, 1, 10 and 100

Galilean invariance: the result should not depend on Vx

Wi h fi d h (b )With a fixed mesh (bottom)Instabilities are lost for V>1

15Fixed meshSpringel 2010

Residual problems SPH AMRResidual problems SPH, AMRSPH: surface tension forces not so well evaluated spread shocksSPH: surface tension, forces not so well evaluated, spread shocksAMR: Preferential directions, may yield artefacts, nogalilean invariancegAbrupt transitions (jumps) at the refinement boundariesMoving objects require more and more refinementThe tree structure (division by 8) is heavy to manage

How to cumulate the advantages of Lagrangian and Eulerian?How to cumulate the advantages of Lagrangian and Eulerian?

Unstructured grid, moving mesh, as in AREPOg , g ,Volker Springel (2010) Voronoi tesselation

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The cells adapt automatically to the problem

Details of the AREPO methodM i id i t i 1970Moving grids exist since 1970But the evolution distorts the cells, which wrap upNecessary to regrid regularly or stopNecessary to regrid regularly, or stop

Delaunay triangulation DT(P)such that no point of the plane P falls inside the circle sourrounding the triangles of DT(P)Th id l t d t i lThus avoids elongated triangles

Tesselation of Voronoi (center of circlesTesselation of Voronoi (center of circles and mediators) N=64

17Voronoi

Update of the tesselation

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Regularisation of cellsegu a sat o o ce sN=625, random Poisson distribution

Algorithm of Lloyd (1982), applied 50 timesAnd the Voronoi tesselation is rebuiltE h i t f ll i t d t th t

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Each point of a cell is recentred at the mass center Rounder cells, structure in honeycomb

Springel 2010

Details of the operationsDetails of the operations

Flow of the fluid & allquantities transfered

The fluid flow of each cell (associated to each particle) iscomputed in the reference frame which moves with the face,with a velocity w = (wi+wj)/2

20 Generalisation 2D -3D

Springel 2010

Variable time-stepInterfaces with3 different dt

The exchanges are occuringon the smalleston the smallestdt

Yellow 2dtBlue dtGrey dt/2Grey dt/2

21Springel 2010

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Millennium Run Simulation(Volker Springel et al. 2005)

• In 2005, the biggest simulation of cold dark matter (> 10 billions of particles)

8 6 108 M• mp = 8.6 x 108 M⊙

• Box of 500 h-1 Mpc x 500 h-1 Mpc x 500 h-1 MpcS i l l i (f ) f 5 k• Spatial resolution (force) of 5 kpc

• More than 20 millions galaxies• The data are made public• Films available on

http://www.mpa-garching.mpg.de/galform/virgo/millennium/

Random gaussian fieldThe field of initial fluctuations , coming from inflationis assumed gaussian

Compatible with Planck results (2014)Co p b e w c esu s ( )

Advantage: All N-points correlations P(1,2, …n) can be writtenas a function of the 2-points correlation (r) = <12> Lack of structures at large scale

2= <2>= (0) variance

LCDM: isotropy, gaussianity,scale invariance for the

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fluctuations coming from inflationCopi et al 2010, 2013

Semi-analytic approachCosmological modelFl i (Pl k)

y pp

Fluctuations (Planck)

N b d Si l ti

Catalogs of galaxies:Comparison with the observed galaxiesN-body Simulation observed galaxies

St ll l tiDM Evolution MillenniumFor example

Stellar population synthesis, extinction etc

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FOF “Friend of FriendGalaxy evolution

Halos merger trees Model of galaxy formationHalos merger trees Model of galaxy formation

Baryon physicsBaryon physics

Cold gas(ISM)

Stars

Gas recycling

St f tiStars Star formation Heatin cooling

ng , sho gocksBlack hole

Hot gas(galaxy clusters)

Winds, outflows(SN feedback) re-incorporation

Hierarchical formationFor the most massive galaxies50% of stars formed at z=5; Af 1 l lAfter z=1, almost mergers aloneto assemble mass z=0.5De Lucia & Blaizot 2007&

Mass accretion by galaxiesy g

(1) Mergers of galaxies (*, gas)(1) Mergers of

galaxies (*, gas)

(2) Accretion of cold gas

(2) Accretion of cold gas

Dekel et al 2007gg

(3) Accretion of hot gas,

for M>M

(3) Accretion of hot gas,

for M>M

Ocvirk et al 2008

for M>Mcritfor M>Mcrit

Non-linear regimeNon-linear regimeThe growth of fluctuations is easy to follow g yin the linear regime << 1 grows as a(t)

f d i h li i l i l i l iAfterwards, in the non-linear regime >1, only numerical simulationscan follow the coupling of the various modesthe gaussianity is then brokenthe gaussianity is then broken

However one can have anidea of the evolutionin assuming a simple

t b ti fperturbation of« Top-hat » shapeSpherical symmetry

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Spherical symmetry

Collapse of the « Top-hat »Collapse of the « Top hat »Matter-dominated epoch after the equivalenceMatter dominated epoch, after the equivalence ~a(t) ~t2/3 <> ~1/t2

Dark

At this epoch, no dark energy effectAssuming only dark matter, without collision

energy

d2r/dt2 = - GM(r)/r2

z=0

Darkmatter

M(r) = 4/3 r3 <> (1+)z=0

Darkmatter

Solution of the spherical collapse withvarious concentric shells

31z=1000

Turning point, virialisationg p ,

The shell begins by continuing its expansion, untilg y g p ,a maximum point at tmax, where the mouvement turns downPoint of virialisation: 2 tmax

32Van den Bosch 13

Zeldovich approximation ppTo go a little beyond, in the non-linear regime,One can follow the particles, and their trajectory along their velocityp , j y g yX(t) = Xo(t) + b(t) f(x) Vector f(x) direction of velocity

Exact treatment at 1D, =ro

3/r3 = a(t)-3 Vo/VV comoving volumeV comoving volume

The gravitationnal collapse accelerates: an initial over-density grows fasterg p y gthe density increases, and the collapse time-scale in -1/2 is shorter

Formation of pancakes and filamentsFormation of pancakes and filaments

33Van den Bosch 13

Semi-analytical approachSemi-analytical approachFor large structures remaining quasi linear can we extrapolateFor large structures, remaining quasi-linear, can we extrapolatethe mass spectrum?Press-Schechter formula: gravity is scale-independentg y pMerger trees

The fluctuations (x)The fluctuations (x)grow linearly (x,t) = R(t) 0(x)( ,t) (t) 0( )

Those above the criticalthreshold cCollapse in a halo

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Semi-analytic approach

N EPS

MM

For a random gaussian field of fluctuations,Formula of Press-Schecter

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NFW profile of dark matter halospThe results of CDM simulations show the existence of an universal density law for dark matter halos Profiles NFW (Navarro, Frenk & White 1997)

Two power-laws ~r-1 at centre, then ~r-3 in the outer parts

The small mass halos are denser in the centerThis is due to their earlier formation in the Universe

The average density of a halo is proportionnal to the averaged i f h i i f idensity of the universe at its formation

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Universal profileU e sa p o e2 parameters MvirC t ti /P(k) ~kn Concentration c= rvir/rs

Comparison of light and massive halosmassive halos

Small halos form earlier (denser)earlier (denser)

Arrows: resolution atVirial radius= 100 x a

37Navarro, Frenk & White (1997)

Rotation velocity of the universal haloRotation velocity of the universal halo

r1/2

f(x) = ln(1+x) +x/(1+x)Rmax= 2.163 rs

38Vc = circular velocity

max . 63 sc=10, Vmax= 1.2 Vvir

Anti correlation Mass ConcentrationAnti-correlation Mass-Concentration

Small halos are more concentrated

With a strong scatter

At start, violent relaxation NFW c~4

Then slower accretion in whichmass and size of halos increaseinside outC=rvir/rs increases

39Maccio et al 2007

Fundamental questionsWhy do galaxies exist, with the sizes and masses observed?Why are they gathered in clusters and super-clusters?y y g pWhat is the origin of this hierarchy ?

The answer is in the main part in the nature of dark matter,the number and mass of its particles.Hot DM neutrinos: the masses formed are of 1015Mo thenHot DM, neutrinos: the masses formed are of 10 Mo, then they fragment and produce galaxies but not enough small scalestructuresCold DM, more massive particles: decouple much earlier thanneutrinos, are much less numerous in numberB i h b iBetter agreement with observations

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But this is for dark halos, not for galaxies!

Influence of the cooling gRelative values of tcool ~T/(n(T)) and tdyn ~n-1/2

Full curvetcool=tdynexpansion

Gas H, HeJeans mass

tcool>t0tcool<tdyn

Jeans massMJ curvesT~n1/3Gas free fall

Isothermal 104KLoci of cloudsin equilibrium

Isothermal 10 K

B quasi-staticclouds

Neutral gas

41Rees & Ostriker 1977

clouds

5.5 ncrit

Sizes and masses of structuresSizes and masses of structuresRadii defined byM 4/3 3

Influence of elementary processesM= 4/3 r3

rbc ~75kpctcool>t0 Masses 1010-1012 Mo

Collapse at theVirialisation epochC ld l i thtcool<tdynCould explain the range of masses ofgalaxiesAfter 10% of star formation galaxies

Small galaxies might

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form later

The collapse of baryons

Mvir= 4/3 R3kT ~mV2

V2= GMvir/Rvirtcool>tdyn

H

Groups,Clusters

He

Curves offluctuations of

Galaxies

Density at x

Galaxiestcool<tdyn

43Blumenthal et al 1984Viial temperature of the structure

Heating-Cooling of the gasg g gDepends on density, Temperature, and also metallicity

Complex processesStar formationStar formationEjection of heavy elements Formation of dusto o o dus

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Equation of state of the gasq g

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Artificial pressure, floor temperaturep pTo prevent the Jeans l h f ll b llength to fall below < hAddition of TminAddition of Tminor pressure min

This is equivalentto create a minimumJ l thJeans length

And also MJ minAnd also MJ min

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4 « phases »p

4 Zoom levels from 20 to 2.5 Mpc.p

z = 3. (from z=10.)

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Technique Multi-zoom

Objective:Objective:

Evolution of a galaxy(0 1 to 10 kpc)(0.1 to 10 kpc)

Accretion of gas (10 Mpc)

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Galaxies & Filaments

Multi-zoom

49(Semelin & Combes 2003)

Aquarius: Try to reproduce the Milky WayAquarius: Try to reproduce the Milky WayWith Tree-SPH (Gadget) Simulations of dark matter onlyS i l l 2008Springel et al 2008

But with a much larger resolution and a much smaller box thanBut with a much larger resolution, and a much smaller box thanthe Millenium (100Mpc)Focused on a halo similar to that of the Milky Wayy y

Density contrast of 106 non-linearR l ti 20 60 1 100 illi ti lResolution 20-60pc, 1-100 millions particles

300 000 sub-halos inside the principal halo!300 000 sub halos, inside the principal halo!But the mass comprised in the sub-halos decreases with the levelAt the 4th level, less than 3% in 100 kpc for Mearth masses

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earth

Movie from z=50 to z=0 Aq-E-2

Re-simulations, at higher resolutionMore smaller structures, dark halos athigh resolutiong

Profile NFW cusp slope -1

MdN/dm

M

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Intensity= density2

Color= velocity dispersionSpringel et al 2008

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Simulation EAGLE, with Hydro New feedback processes, Supernovae & AGN, more realistic

Mass spectrum for stars108-1011Mo, well reproducedAfter calibration of SF & feedback

Mass-Z relation M> 109Mo

Gas fraction, T too high

Code GADGET3 (SPH) withimproved recipes

53Schaye et al 2015

Formation of spiral galaxies p g

Re treatment with dust effectsRe-treatment, with dust effects

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Stellar mass functionStellar mass function

Comparison betweenall simulationsall simulations

Stellar mass functionS e ss u c oreproduced

But not yet metallicitydistribution

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Baryon fraction vs MhaloBaryon fraction vs Mhalo

Corresponds to modelsof HAM=of HAM« Halo Abundance Matching »

However, the size of galaxies isstill too smallP bl f l tProblem of angular momentum

Heating/cooling processesHeating/cooling processesLack of resolution

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CONCLUSIONS• Dark matter: multi-scale numerical simulationsAdapted algorithms 80003 ~500 billions of particles

• Hierarchical model– Universeal radial profile: NFW– 3D-shape of halos

A l– Angular momentum

• S i l ti d l• Semi-analytic modelsMerger trees

Formalism EPS Extended Press SchecterFormalism EPS Extended Press-Schecter

• Comparison with observations: baryonsComparison with observations: baryons

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