Galaxy Formation

Galaxy formation

Mojtaba Raouf Galaxy Formation - Outline 2

̣ Cosmology & initial condition

̣ Structure Formation & Dark Matter

̣ Evolution of Gaseous halos

̣ Galaxy Groups/ Clusters

̣ Star Formation

̣ Feedback - Active Galactic Nuclei

̣ Models for Galaxy formation

̣ Our works

Lecture series School of Astronomy - IPM Galaxy Formation - Cosmology & initial condition 3

A classical, general relativistic description of cosmology is expected to break down at very early times when the Universe is so dense that quantum effects are expected to be important.

The standard cosmology has a number of conceptual problems when applied to the early Universe, and the solutions to these problems require an extension of the standard cosmology to incorporate quantum processes.

One generic consequence of such an extension is the generation of density perturbations by quantum ﬂuctuations at early times.

It is believed that these perturbations are responsible for the formation of the structures observed in today’s Universe.

Lecture series School of Astronomy - IPM Galaxy Formation - Cosmology & initial condition 4

One particularly successful extension of the standard cosmology is the inflationary theory, in which the Universe is assumed to have gone through a phase of rapid, exponential expansion (called inflation) driven by the vacuum energy of one or more quantum fields. In many, but not all, inflationary models, quantum fluctuations in this vacuum energy can produce density perturbations with properties consistent with the observed large scale structure.

Cosmological simulations

This part of galaxy formation theory is still partly phenomenological: typically initial conditions are speciﬁed by a set of parameters that are constrained by observational data, such as the pattern of ﬂuctuations in the microwave background or the present-day abundance of galaxy clusters.

Springel et al., Nature, 2006

30th Jerusalem Winter School Romain Teyssier4 Lecture series School of Astronomy - IPM Galaxy Formation - Structure Formation & Dark Matter 5

The mass inferred from the luminous stars is not sufﬁcient to explain the dynamics of the object so-called Dark Matter.

There must be additional material, whose presence is only revealed by its gravitational effects, as seen for example in the orbital motions of galaxies in clusters, or in the rotation of spiral galaxies.

The physical nature of the dark matter is still one of the greatest mysteries of cosmology. Dark matter candidates include as of yet unknown elementary particles, primordial black holes, and stellar remnants.

Lecture series School of Astronomy - IPM Galaxy Formation - Structure Formation & Dark Matter 6

Dark matter halos form through gravitational instability. density perturbations grow linearly until they reach a critical density, after which they turn around from the expansion of the Universe and collapse to form virialized dark matter halos.

These halos continue to grow in mass (and size), either by accreting material from their neighborhood or by merging with other halos. .

Lecture series School of Astronomy - IPM Galaxy Formation - Dark Matter Merger Tree 7

Understanding the structure and formation of dark matter halos plays a pivotal role in the understanding of the formation and evolution of galaxies.

Based on Press–Schechter algorithm the time increases from top to bottom, and the widths of the tree branches represent the masses of the individual halos. A horizontal slice through the tree, such as that at t = tf gives the distribution of the masses of the progenitor halos at a given time. Merger trees play a very important role in hierarchical models of galaxy formation. In Lacey & Cole (1993) particular, they are the backbone of the semi- analytical models of galaxy formation

Lecture series School of Astronomy - IPM Galaxy Formation - Structure Formation & Dark Matter 8

The density distribution of dark matter in a high-resolution N-body simulation.

Credit: Millennium Simulation

Lecture series School of Astronomy - IPM Galaxy Formation - Evolution of Gaseous halos 9

Basic Fluid Dynamics and Radiative Processes Euler, and energy equations

The gravitational potential Φ satisﬁes the Poisson equation

Cooling is a crucial ingredient of galaxy formation. Depending on temperature and density, a variety of cooling processes can affect gas.

Lecture series School of Astronomy - IPM Galaxy Formation - Evolution of Gaseous halos 10

In a spherically symmetric gaseous system, the cooling time at radius r can be deﬁned as

An estimate of the rate at which gas cools out in the halo then follows from

Evolution of Gaseous Halos with Energy Sources

Radiative Cooling Compton Cooling Photoionization Heating Gas can also be heated through non- Hydrostatic Equilibrium Gas Density Proﬁle gravitational processes, such as Virial Theorem Applied to a Gaseous Halo radiation from stars and AGN, stellar Accretion Shocks Radiative Cooling in Gaseous Halos explosions and stellar winds.

Lecture series School of Astronomy - IPM Galaxy Formation - Galaxy Groups/ Clusters 11

Ideal Laboratories for Studying the the galaxy formation and evolution

Lecture series School of Astronomy - IPM Galaxy Formation - Galaxies 12

H.Mo et al 2008

Lecture series School of Astronomy - IPM Galaxy Formation - Galaxies 13

H.Mo et al 2008

Lecture series School of Astronomy - IPM Galaxy Formation 11

Galaxy Formation - Star Formation 14 Galaxy Formation 11 Figure 3. Aschematicdiagramshowingthetransferofmassandmetalsbetween stars and the hot and cold gas phases during a single timestep. The solid lines indicate the routes and rates by which mass is transferred between the three reservoirs, while the dashed lines refer only to the exchange of metals. The instantaneous rate of star formation is ψ and the cooling rate is M˙ cool.Themetallicitiesof the cold gas, stars and hot halo gas are Zcold, Z∗ and Zhot respectively. The yield of the assumed IMF is p and the parameters β and e describe the eﬀect of SN feedback and the direct ejection of SNmetalsintothehothalogas.

Zcold the metallicity of the cold gas, and β the efficiency of the galaxy. The evolution of the stellar metallicity differs stellar feedback. Each of the arrows in Fig. 3 gives rise to from the closed-box model because it is affected by both the Cole et al. 2000 aterminthefollowingdifferential equations that describe ejection of reheated gas and the accretion of cold gas and Figure 3. Aschematicdiagramshowingthetransferofmassandmetalsbetween stars and the hot and cold gas phases during a single timestep. The solid lines indicate the routes and rates by which mass is transferred between the three reservoirs, while the dashed lines ˙ refer only to the exchange evolution of metals. The instantaneous of rate of starthe formation mass is ψ and the cooling and rate is M metalcool.Themetallicitiesof content of the three associated metals. the cold gas, stars and hot halo gas are Zcold, Z∗ and Zhot respectively. The yield of the assumed IMF is p and the parameters β and e describe the ereservoirs:ffect of SN feedback and the direct ejection of SNmetalsintothehothalogas. R: The fraction of mass recycled by stars (winds and SNe) Zcold the metallicity of the cold gas, and β the efficiency of the galaxy. The evolution of the stellar metallicity differs stellar feedback. Each of the arrows in Fig. 3 gives rise to from the closed-box model because it is affected by both the aterminthefollowingdi˙ fferential equations that describe ejection of reheated gas and the accretion of cold gas and the evolutionM of the⋆ mass and metal=(1 content of the threeR)associatedψ metals. the instantaneous star formation(4.6) rate 4.2.2 Star Formation Law and Feedback Parameterization reservoirs: − ˙ M⋆ =(1˙R)ψ ˙(4.6) M− hot = Mcool4.2.2+ Starβψ Formation Law and Feedback Parameterization (4.7) M˙ hot = M˙ cool + βψ (4.7) − In our previous work (e.g. Cole et al. 1994), we specified the ˙ ˙ − In our previous work (e.g. Cole et al. 1994), we specified the M = Mcool (1 R + β)ψ (4.8) cold − − star formation timescale and feedback efficiency in terms of β the efficiency of stellar feedback. ˙ Z ˙ ˙ M =(1 R)Zcoldψ (4.9) the circular velocity of the halo in which each galaxy formed, ⋆ M− cold = Mcool (1 R + β)ψ (4.8) ˙ Z ˙ VH.Therelationsweadoptedwere star formation timescale and feedback efficiency in terms of Mhot = McoolZhot +(pe + βZcold)ψ (4.10) − − − ′ ˙ Z ˙ 0′ −1 α⋆ Mcold = McoolZZhot τ = τ (VH/300 km s ) (4.12) p: The yield of the assumed IMF ˙ ⋆ ⋆ +(Mp(1 e) (1 + β R=(1)Zcold)ψ, (4.11)R)Zcoldψ (4.9) the circular velocity of the halo in which each galaxy formed, −⋆ − − and e: the fraction of newly produced metals Z Z ′ where Zcold = M /Mcold and Zhot = M /Mhot−.The ′ −α cold hot hot ejected directly from the stellar disk to the hot gas phase, Z β =(VH/Vhot) . (4.13) values of R and˙p in these equations are related to the˙ IMF, VH.Therelationsweadoptedwere M = M Z +(0′ pe + βZ )ψ (4.10) as discussed in Sectionhot 5.2. coolThe parameterhot τ⋆ ,wetreatedasafreeparameter,whilethecold ′ ′ ′ We assume that over one timestep the cooling rate, other three parameters, α⋆, Vhot and αhot,weconstrained ˙ − ′ Mcool,andthemetallicityofthehotgas,˙ Z Zhot,canbetaken˙ by comparing our models to the numerical simulations of 0′ −1 α⋆ to be constant.M Thiscold set of first-order,= coupledM differentialcoolZgalaxyhot formation of Navarro & White (1993). These simula- equations can be straightforwardly solved to give the change tions had only one free parameter, the fraction of SN energy τ⋆ = τ⋆ (VH/300 km s ) (4.12) in mass and metal content of cold gas, hot gas and stars since injected as kinetic energy into the interstellar medium. In Zcold the metallicity of the cold gas the start of the timestep (Appendix B). The model is quite order to suppress the formation of low luminosity galaxies, flexible: its behaviour is determined+( by specifyingp how(1 the eand) thus produce(1 a + galaxyβ luminosityR function)Z withcold a rea-)ψ, (4.11) functions τ⋆, β and e depend on the properties of the galaxy− sonably− shallow faint end slope,− as observed, we adopted and and its surrounding halo. We note that compared to the afiducialmodelwithverystrongfeedbackforlowcircular simple, “closed-box” chemical enrichment model, the yield Zvelocity halos, which we obtained by setting the parameter Z ′ ′ ′ −1 ′ is modified by the metal ejection and feedback to produce values α⋆ = 1.5, Vhot =140kms and αhot =5.5. ′ − where Zcold = Mcold/M− cold and Zhot = Mhot/Mhot.The αhot an effective yield peff =(1 e)p/(1 R + β)(equationB9), The more detailed modelling that we now perform of the Lecture− series− Schoolβ of=( AstronomyVH/Vhot) - IPM. (4.13) which is thereforevalues a function of of the potential-wellR and depthp of instructure these of our model equations galaxies allows us to specify are the star related to the IMF, ⃝c 0000 RAS, MNRAS 000,000–000 0′ as discussed in Section 5.2. The parameter τ⋆ ,wetreatedasafreeparameter,whilethe ′ ′ ′ We assume that over one timestep the cooling rate, other three parameters, α⋆, Vhot and αhot,weconstrained M˙ cool,andthemetallicityofthehotgas,Zhot,canbetaken by comparing our models to the numerical simulations of to be constant. This set of first-order, coupled differential galaxy formation of Navarro & White (1993). These simula- equations can be straightforwardly solved to give the change tions had only one free parameter, the fraction of SN energy in mass and metal content of cold gas, hot gas and stars since injected as kinetic energy into the interstellar medium. In the start of the timestep (Appendix B). The model is quite order to suppress the formation of low luminosity galaxies, flexible: its behaviour is determined by specifying how the and thus produce a galaxy luminosity function with a rea- functions τ⋆, β and e depend on the properties of the galaxy sonably shallow faint end slope, as observed, we adopted and its surrounding halo. We note that compared to the afiducialmodelwithverystrongfeedbackforlowcircular simple, “closed-box” chemical enrichment model, the yield velocity halos, which we obtained by setting the parameter is modified by the metal ejection and feedback to produce values α′ = 1.5, V ′ =140kms−1 and α′ =5.5. ⋆ − hot hot an effective yield peff =(1 e)p/(1 R + β)(equationB9), The more detailed modelling that we now perform of the − − which is therefore a function of the potential-well depth of structure of our model galaxies allows us to specify the star

⃝c 0000 RAS, MNRAS 000,000–000 Galaxy Formation - Feedback 15

Missing Satellite problem Down sizing problem

Small dense halos Pursues Cluster form at high redshift and cooling within Böhringer et al., 1991 them is predicted to Fabian et al., 2003 be very efﬁcient. This disagrees badly with observations, which show that only a relatively small f r a c t i o n o f a l l baryons are in cold gas or stars

Lecture series School of Astronomy - IPM Galaxy Formation - Active Galactic Nuclei (AGN) 16 Centaurus A Active Galactic Nuclei (AGN) play a key role in the formation and evolution of galaxies through so- called AGN feedback.

Virgo Cluster

Churazov et al., 2001 Forman et al., 2005

Credit by: NASA/CXC/CfA/R.Kraft et al.; Submillimeter: MPIfR/ESO/APEX/A.Weiss et al.; Optical: ESO/WFI Lecture series School of Astronomy - IPM

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K Öç Į¾Õ õñ K/ W = CòÿÖÙ ½ī ĥÖ¾ö¾ĵ ĥñÖĔß ĲÙ ĭò¿ûõ ¼ÕöÙ ĩ÷Ń ĭÖī÷ ĒğÕĵ õñ ĭÖī÷ õñ ĩöç Ē¾÷Ķß ŃĶĬĔī ñĶÿ ½ī ıñÖĜàûÕ ñĶçĶī òİī Ýõòğ#% ñĶÿ¼Öĳ ½ī ößĶ¿īÖ¶ ĲàïÕñö¬ ½ī ĵ ¼ñòē Öĳ ¼÷Öû ¼ÖĳĲ¿Úÿ Į¾Õ ýĵõ ĥĶăÕ ÷Õ ½ûõöÙ ÝŃñÖĔī ĲÙ þðÙ Į¾Õ Į¾Õ Ħë¼ ĲīÕñÕ ¼ÕöÙ õñ ½·¿īÖİ¾ñĵõò¿ĳ| | ¼Öĳ ¼÷Öû ñõÕôº ½ī òÿÖÙ ÖÙ ĩöç öĄİē öĳ Þ¶öë úû Ĳ·Úÿ µ¾ ¼ĵõ ĩöç ĭñö¶ þð¬ Ö¾ òÿÖÙ ½ī ÝÕõó ÝõĶă ĲÙ Ö¾ Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ N N Ĳ¿Úÿ ĵ Īüç ¼Öĳ ¼÷Öû Ĳ¿Úÿ òįĶÿ ½ī Ī¿üĠß Ĳàûñ ĲÙ ĭÖĀ·Ĵ¶ ¼ö¿º Ħ·ÿ ¼ñòē ¼Öĳ ¼÷Öû Ĳ¿ÚÿĪüç ÖīĶĬēN µ¿īÖİ¾ñ Ğ¿ğñ ½ßõĶă õñ Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ ññöº ½ī Į¿¿Ĕß ¼ñòē ¼Öĳ ýĵõ ĊûĶß CöăÖİē ö»¾ñ ÖÙ ĭÉ þİ·ĬĳöÙ ÖÙ ĲçĶß K = / mi vi | | #% ñĶÿ ½ī ĲàïÕñö¬ ½ī Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ ĥĶăÕ ½ûõöÙ ĲÙ þðÙ Į¾Õ ¼ ĲīÕñÕ õñ ½·¿īÖİ¾ñĵõò¿ĳ ¼Öĳ ¼÷Öû i= Öĳ Īàü¿û ıòİĳñ Ħ¿·Āß ÝÕõó ñÕòĔß Ĳ¶ Öç ĭÉ ÷Õ ½ħĵ½ĀįÕöº òİÿÖÙ þİ·ĬĳöÙ Į¾ñÖ¿İÙ Þìß ÝÕõó ĩÖüçÕ Þ¶öë Ö¾ Öĳ ÝŃñÖĔī ĪßÕ ¼÷Öû ¼ñòē Ĳ¿Úÿ Ħë ÷Õ ÝÕõó ÞûÕ ÝõÖÚē Ĳ¶ ñĶÙ ½ûÖİÿ òİĳÕĶï ĭÖĴ¿¶ Īüç N ¼Öĳ ¼÷Öû Ĳ¿Úÿ" Ĳ¿Úÿ ÝÕõó ĩöç CĦĬē õñ Þü¿į ö¾ô¬ ĭÖ·īÕ öīÕ Į¾Õ¼ò¿ÿõĶï òİÿÖÙ ¼Öĳ ½ī ĲīĶĐİī ĭĶßĵö¬ ¼÷Öû Ĳ¿Úÿ Į¿İ±Ĭĳ½ºõøÙ ĵ ĲÚßöī ½ûÖİÿ ÷Õ ĭÖĴ¿¶ Öĳ ĵ µ¾ø¿ěĭÖĀ·Ĵ¶ öàïÕ õñ ĭÖĴ¿¶ ¼Õ ıñöàüº õñ ÝõĶă Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ Ħ¿üįÖà¬ ¼²öįÕ W ĵ òįĶÿ ½ī Ĳàěöº õÖ¶ ĲÙ Öĳ ¼÷Öû Ĳ¿Úÿ Þìă öÙ ½ĴçĶß ĦÙÖğ Þ¾ñĵòìī Ø¾öĠß Į¾Õ ĵ òÿÖÙ ½ī öß ¸õøÙ ĪßÕ ĩöç ÷Õ ½ºõøÙ ĲÚßöī Į¾òİ° ¼÷Öû #% N W = / Gmimj/ ri rj ÝÕõó ĩÖĬß Þēöû ĵ ĭÖ·ī õÕñöÙ ÖÙ t ıÕĶðħñ ĭÖī÷ öĳ õñ Īàü¿ûĪüç ÞħÖë CòÿÖÙ ½īµ¿īÖİ¾ñ ĩöç Īĳ ıõó N ĦīÖÿ Īàü¿û− ò¿İ¶ ąöě | − | ñõÕôº ½ī i=j (ri, ui)(i = , ,...,N) "̸ N CòÿÖÙ ½ī ½ĀįÕöº Īàü¿û Į¾Õ õñ Öĳ þİ·ĬĳöÙ ĩÖĬß Ĳ¶ Öç ĭÉ ÷Õ Cññöº ½ī Ě¿ăĶß Īàü¿û Ĳ¿Úÿ ĵ Īüç ¼Öĳ ¼÷Öû Ĳ¿Úÿ òįĶÿ ½ī Ī¿üĠß Ĳàûñ ĲÙ ĭÖĀ·Ĵ¶ ¼ö¿º Ħ·ÿ ¼ñòē ¼Öĳñö¶ ¼÷Öû Ě¿ăĶß Ĳ¿Úÿ ÝõĶă ÖīĶĬē Į¾òÙ ĭÕĶß ½ī Õõ ĩÕ i ıõó Þ¶öë ½ĀįÕöº þİ·ĬĳöÙ Þìß ĩÖüçÕ Þ¶öë ÝŃñÖĔī ¼ñòē Ħë ÷Õ ÞûÕ ÝõÖÚē ½ûÖİÿÝÕõó ĭÖĴ¿¶ Į¿Ù ½īÕõÉ Īüç ĲÙ ¼²öįÕN ¼Öĳ ñĶÿ ½ī ¼÷Öû Ĳè¿àį ĩöç Ĳ¿Úÿ ĭò¿īõÉ ÷Õ Ĳ¶ CñĶÿ ½ī ĥĶìß ÷Öě ñõÕĵ Īàü¿û C½·¿īÖİ¾ñ ĥñÖĔß ÷Õ ú¬ #% ñĶÿ ½ī ĲàïÕñö¬ ½ī Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ ĥĶăÕ ½ûõöÙ ĲÙ þðÙ Į¾Õ ¼ ĲīÕñÕ õñ ½·¿īÖİ¾ñĵõò¿ĳTrel ¼Öĳ ¼÷Öû 17 Galaxy Formation - N-body SimulationÝÕõód ñÕòĔßr ĲÙ Īàü¿û ĭò¿īõÉ ½įÖī÷ ùÖ¿Ġī ñĵõ ½ī þ¿¬ ½·¿īÖİ¾ñĶīöß ĥñÖĔß ÞĬû ĲÙ Īàü¿û ĵ òÙÖ¾ ½ī ĥÖĠàįÕ i = u , ¼ò¿ÿõĶï ¼Öĳ ĲīĶĐİī ¼÷Öû Ĳ¿Úÿ Į¿İ±Ĭĳ ĵ ½ûÖİÿ ĭÖĴ¿¶ ĵ µ¾ø¿ě öàïÕ õñ ¼Õ ıñöàüºdt i ÝõĶă Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ ñõÕñ ½»àüÙ ĲĨÔüī ¼ Ĳûòİĳ ĵ ĵ òįĶÿ ½ī Ĳàěöº õÖ¶ ĲÙ PP (Particle–Particle) N Trel N/log( N)Tcr duĪüçi µ¿īÖİ¾ñ ∝ Algorithm = Fi = φ i, #% t Į¾Õdt ĵ ¸õøÙN−∇ ñÖĔÙÕ| ĵ ñÖ¾÷ ÝÕõó ñÕòĔß òįõÕñ ½¾ŃÖÙ ĭò¿īÕõÉ ĥĶìß ĭÖī÷ ½·¾õÖß ıñÖī ĵ ½įÖĀ·Ĵ¶ ¼Öĳ Īàü¿û #% ÝÕõó ĩÖĬß Þēöû ĵ ĭÖ·ī õÕñöÙ ÖÙ ıÕĶðħñ ĭÖī÷ öĳ õñ Īàü¿û ÞħÖë CòÿÖÙ ½ī ĩöç Īĳ ıõó NĦīÖÿ Īàü¿û ò¿İ¶ ąöě ½ĀįÕöº þİ·ĬĳöÙ Þìß ĩÖüçÕ Þ¶öë ÝŃñÖĔī ¼ñòēĵ òÿÖÙ ½ī Ħë ½ĀįÕöº ÷Õ ÞûÕ Ħ¿üįÖà¬ ÝõÖÚēφ òÿÖÙ ½ûÖİÿ ½ī ıõó Ħìī ĭÖĴ¿¶õñ õñ µ¾õÖß ×Öàÿ Īüç ıñÖī ĭÖĬĳ ñĶçĵ¼Öĳ Ö¾ ĩöç ęöă ¼÷Öû òëÕĵ ĊĠě ĲÙ Ĳ¿Úÿ ıòÿ òįĶÿ ñõÕĵ ¼ĵö¿į ½ī ąöěFi ŃÖÙ ¼ñõĶïöÙ ĲČÙÕõ õñ ö¿ė Öĳ Īàü¿û Į¾Õ Õøħ CòÿÖÙ ½ī ĭÖĴ¿¶ öĬē ÷Õ þ¿Ù ĭÖī÷ PM (Particle–Mesh) Algorithm(ri, u i)(i = , ,...,N) CòÿÖÙ ½ī¼ò¿ÿõĶï ½ĀįÕöº ¼Öĳ Īàü¿û ĲīĶĐİī Į¾Õ ¼÷Öû õñ Öĳ Ĳ¿Úÿ þİ·ĬĳöÙ Į¿İ±Ĭĳ ĩÖĬß ĵ ½ûÖİÿ Ĳ¶ Öç ĭÖĴ¿¶ ĭÉ ĵ ÷Õ µ¾ø¿ě öàïÕ õñ ¼Õ ıñöàüºú¾ÖĠī ÝõĶă ĵò¾É öàĬ¶ ½īCññöº Öĳ ÝÕõó Þûñ ¼÷Öû ÖÙ ĲÙ ½ī òįõÕñ ĭĶûÕĶ¬ Ĳ¿Úÿ Ě¿ăĶß ĲÿĶï ĲħñÖĔī Į¾Õ ñĶï Ğ¾öċ Īàü¿û öĬē ÷Õ Cĩöç ÷Õ öß Ē¾÷Ķß µ°Ķ¶ ĊûĶß ĭò¿īÕõÉ ĭÖī÷ Ĳ¶ ¼ĵö¶ ĵ ÷ÖÙ ¼Öĳ ĲÿĶï Þü¿į ĲĨÔüī i ñö¶ Ě¿ăĶß ÝõĶăĲÙ ĉĶÙöīφ Į¾òÙ= π Ħ¿üįÖà¬GòįĶÿρ ĭÕĶß(r,t) ½ī ÷Õ ½ī ½ºõøÙ Ĳàěöº Õõ ĪĴû ĩÕ õÖ¶ Ĳ¶ ĲÙıõó ÞûÕ Þ¶öë ½įÖī÷ ĩöç ö·¾ĵõ ÷Õ ıñÖàûÕ ö»¾ñ ñõÕĶīòįĶÿ ½ī ąöě ¼ñõĶïöÙ µ°Ķ¶ Tree Algorithm ∇ ñÕòĔß µ¾ñøį þİ·ĬĳöÙ òİàüĳ#% ĭĶī÷É ĩöç ĦĬē õñ Öĳ ıõÖ¿û Öĳ ĭÉ õñ Ĳ¶ ¼Õ ıõÖ¿û ¼Öĳ ĲīĶĐİī Ħäī òÿÖÙ ıõó µ¾ ½ī öß ØûÖİī ıÕöĬĳ ¼Öĳ òëÕĵ ÷Õ ıñÖĜàûÕ ĵõ Į¾Õ ÷Õ CòÙÖ¾ ½ī ĉÖüÚįÕ ĭÖī÷ õôº ÖÙ Ĳİ¿ī÷ ¼ÖĈě ½ûÖİÿ ĭÖĴ¿¶ ĦÓÖüī õñ ÝÕõó ĩÖĬß Þēöû ĵ ĭÖ·ī õÕñöÙ ÖÙ t ıÕĶðħñ ĭÖī÷ öĳ õñ Īàü¿ûv ÞħÖëdx/dt CòÿÖÙx ½ī ĩöç Īĳ ıõóĥĶìßN ñĶÿĦīÖÿ ½ī Īàü¿û öèİī ½¾Öß ò¿İ¶ ąöě ÖßòĬē a Öĳr Īàü¿ûx = ö¾÷r/a Ħ¿·Āß ĲÙ Ĳ·ĨÙ ÞûÕ ıÕöĬĳ ¼²öįÕ ĥÖĠàįÕ ÖÙ ÖĴİß Ĳį ıõó ¼öàĀ¿Ù P3MĪ¿ĳÕĶï (Particle–Particle–dri C ≡ ĵ Øüë öÙ ŃÖÙ ÝŃñÖĔī ½ü¾Ķį ÷ÖÙ ÖÙ òÿÖÙ ½ī ıÕöĬĳ ĥĶċ Ĳ¶ −→ òÿÖÙ CòÿÖÙ ½ī ½ĀįÕöº Īàü¿û Į¾Õ õñ Öĳ þİ·ĬĳöÙ ĩÖĬßParticle–Mesh) Ĳ¶ Öç ĭÉ= uAlgorithm ÷Õi, (ri, u i)(i = , ,...,NC½ĀįÕöº) Cññöº ñĶï ĦÓÖüī ½ī Ě¿ăĶß õñ Į¾Õ öÙ Īàü¿û ıĵńē ñĶÿ Ħë Īüç ¼ÞÿÕñ ıòÿ õÕĶĬĳ ö¿ė ĵ Ī¿Ġàüī ýĵõ ÷Õ ÞûÕ öàĴÙ ½¾Öß ¼Öĳ ĒĬèß dt ½·¿īÖİ¾ñĶīöß čÖìħ ÷Õ Īàü¿û ñĶÿ ½ī âēÖÙ ĭĶ° #% ñĶÿ Ĳàěöº öĐį õñ Þü¾ÖÙ ½ī Īĳ ýø¾õ ĵöě ÷Õ ĦăÖë ¼Öīöº ñö¶ Ě¿ăĶß ÝõĶă Į¾òÙdx ĭÕĶßi/dt = ½īvi Õõ ĩÕ i ıõó Þ¶öë òĳñ ½ī õÕöğ ö¿ãÖß Þìß Õõ ĥĶìß ĵ ñĶÿ õÕò¾Ö¬Öį

du dri i dvi/dt + H(t)vi = xΦ i = Fi==ui, φ i, −a ∇ | dt dt −∇ | ĮīøàĨÙ ¼ ĲħñÖĔīĭÖĴ¿¶ ĲČûÕĵ ùÖ¿Ġī ĭÕò¿ī ¸õøÙ ñö·¾ĵõ ¼÷Öû Ĳ¿Úÿ òÿÖÙ ½ī ½ħÖ»° ÝŃńàïÕ ĲÙ ĲçĶß ÖÙ ½ĀįÕöº Ħ¿üįÖà¬ Φ ĵ H(t)=˙a/a ÝŃñÖĔī Į¾Õ õñ ıõó öĳ ÝÖĄàðī ¼ÕöÙ òĔÙ µ¾ ĵ Þēöû ¼ÕöÙ òĔÙ ñõÕñ ¼òĔÙ N + ÷Öě ¼ÖĈě µ¾ ñõÕñ ıõó N Ĳ¶ ½Ĭàü¿û φ xΦ = πGa [ρ(x,t) ρ¯(t)]. Fi ĵ òÿÖÙ ½ī ½ĀįÕöº Ħ¿üįÖà¬ òÿÖÙ ½ī ıõó Ħìīdu õñi ×Öàÿ ĭÖĬĳ Ö¾ ĩöç òëÕĵ∇ ĲÙ¼Öĳ ıòÿıõÖàû ýĵõ ñõÕĵ ñõÖ¿Ĩ¿ī ÷Õ ĭÕĶß ¼ĵö¿į− ½ī òÿÖÙ ½ēĶį ĭÖĀ·Ĵ¶ öß µ°Ķ¶ŃÖÙ µ¾ ĲČÙÕõ öĐį õñ ñõĶī Ĳ¶ õñ ½ßõĶă ½įÖī÷ õñ ¼Öĳ CòİÿÖÙ ĩÖº ÷Õ½ī ĩöç ıõó ñõÖ¿Ĩ¿ī ĭò¿īÕõÉ Öß ĭÖī÷ ĭĶ¿Ĩ¿ī Ĳ¶ òÿÖÙ ĦīÖÿ ñÖ¾÷ ÖÚħÖė ÝÕõó Öĳ ¼÷Öû ñÕòĔß öºÕ = Fi = φ i, ùÖ¿Ġī ÖÙ Ĳü¾ÖĠī ĦÙÖğ Į¿¾Ö¬ ĭò¿īÕõÉ ĭÖī÷ ÖÙ ½Ĭàü¿û CòįõÕñ ñöÙõÖ¶ ø¿į ¼ñõĶïöÙ ¼Öĳ Īàü¿û õñ ÝŃñÖĔī Į¾Õ ñõÕñ dtöĐį ęöă ĉÖüÚįÕ−∇ ÷Õ ĵõ| þ¿¬ ÝÖÚûÖìī õñ Còİ¶ĲČûÕĵ ½Ĭį ½ăÖï ĭÕò¿ī ÝĵÖĜß Į¾Õ ÝŃñÖĔī ÞïÖû ½ĨăÕ Į¾Õ Ħë ıò¾Õ CĭÖĴ¿¶ ñÕñ ĉÖüÚįÕ þĳÖ¶ Įàěöº òĔÙ öĐį õñ ÖÙ ĲÙ Õõ ÷Öě ¼ÖĈě ĭÕĶß ½ī õÖ¶ Į¾Õ ÖÙ ñö¶ ıñÖĜàûÕ ¼õÖīÉ ò¾É ½ī Þûñ ĲÙ ĭĶûÕĶ¬ ĲħñÖĔī Ğ¾öċ ÷Õ Cĩöç Ē¾÷Ķß ĊûĶß òĳÕĶï ĲěÖćÕ ĭÖīøàħĶÙ ĲħñÖĔī ĲĨĬç µ¾ ÝõĶă Į¾Õ õñ Īàü¿û öĬē ½įÖī÷ Ħ¿üįÖà¬ µ¾ ıĵńĔÙ ıõó åĵ÷ öĳ Į¿Ù ½ßĶ¿į þįÕöºÕò¿¬ ½īĶĬē ĥÖĬàëÕ ĭĶįÖğ ĭÖĬħÕ ÷Õ ÞûÕf(x ÝõÖÚē, v,t) ½Ĩ¶dxd µ¾ø¿ěv Ĳ¶ ĦÓÖüī òÿÖÙ Į¾Õ ½ī õñf( ÞûÕx, v,t ıòÿ) ıõó µß µ¾ Ē¾÷Ķß ĒÙÖß µĬ¶ ĲÙ ½·¿īÖİ¾ñ Īàü¿û µ¾ ¼ φ Fi Df ∂f ½çõÖï∂f ∂φT ∂f ĵ òÿÖÙ ½ī ½ĀįÕöº Ħ¿üįÖà¬ òÿÖÙ ½ī ıõó Ħìī õñ ×Öàÿ ĭÖĬĳ Ö¾ ĩöç òëÕĵ ĲÙ ıòÿ ñõÕĵ ¼ĵö¿į ŃÖÙ ĲČÙÕõòÿÖÙ õñ ½ī t=ĭÖī÷+ õñv.v Þēöû õÕñöÙ. ĵ=r CĭÖ·ī[f] õÕñöÙ ĥĶë d xd v ĭÖĬħÕ õñ ıõó ĭñö¶ φ = πGρ(r,t) Dt ∂t ∂x − ∂x ∂v ∇ mimj (rñĶÿi rj ½ī) Ĳè¿àį Ħ¾ĵĶ¿ħ ¼ ĲħñÖĔī ÷Õ Ĳ¶ ñĶÿ ½ī Ě¿ăĶß ĮīøàħĶÙ ¼ñõĶïöÙ ö¿ė ¼ ĲħñÖĔī ĊûĶß µ¿īÖİ¾ñ ò¾É ½ī Þûñ ĲÙ ĭĶûÕĶ¬Fi = ĲħñÖĔīG Ğ¾öċ− ÷Õ Cĩöç.φ Ē¾÷Ķßext(ri) ĊûĶß − ri rj −∇ j=i ½ī ñõĶïöÙ ö»ĨĬē ¼Öĳ ñöÙõÖ¶ ÷Õ ÷Öě ¼ÖĈě õñ ½ĀįÕöº ñõĶïöÙ ÷Öě ĲÙ ıõó ñĵõĵ ĥÖĬàëÕ CòÿÖÙ ½ī ñõĶïöÙ ö»ĨĬē C[f] !̸ | − | Df ∂f ∂f ∂φT ∂f ½ī öß ØûÖİī ıÕöĬĳ ¼Öĳ òëÕĵ ÷Õ ıñÖĜàûÕ ĵõ Į¾Õ ÷Õ CòÙÖ¾φ = ½īπG ĉÖüÚįÕρ(r,t) ĭÖī÷ õôº ÖÙ Ĳİ¿ī÷ĭÕò¿ī ¼ÖĈě ¼Öĳ ÞĳÖÚÿ ½ûÖİÿ ĲÙ ĲçĶß ĭÖĴ¿¶ ÖÙ ½ěöċ ĦÓÖüī ÷Õ ñö¶= ıõÖÿÕ õñ ½ħÖĔĜįÕ+ v. ĵ ĦĔě ñĶï µ¾õÖß. = ıñÖī ĵ ¼ĵö¶ ¼Öĳ ĲÿĶï µ¿īÖİ¾ñ ĲÙ ĭÕĶß SPH Dt ∂t ∂x − ∂x ∂v Ī¿ĳÕĶï Cv dx/dt ĵ x Øüë öÙ ŃÖÙ ÝŃñÖĔī∇ ½ü¾Ķį ÷ÖÙ ÖÙ òÿÖÙ ½ī ıÕöĬĳ ĥĶċ aĲ¶ ĲÙ ĭÕĶßr ½ī õÕøÙÕ Į¾Õx ĪĴī= ¼Öĳr/a ĲįĶĬįòÿÖÙ ÷Õ ÞûÕ ıòÿ ĲÓÕõÕ ĭÉ ¼ÕöÙ ½ēĶİàī ¼Öĳ ñöÙõÖ¶ Öĳ ıõÖÿ µ¿īÖİ¾ñ ÖÙ ĲČûÕĵ ≡ −→ φ = φ (x,t)+φ(x,t) ñö¶ ıõÖÿÕ ½ī öß ØûÖİī ıÕöĬĳ ¼Öĳ òëÕĵ ÷Õ ıñÖĜàûÕ ĵõLecture Į¾Õ ÷Õ CòÙÖ¾ series ½ī ĉÖüÚįÕ ĭÖī÷ õôº ÖÙ Ĳİ¿ī÷ ½ĨïÕñ ¼ÖĈě õÖ¶÷ÖûñĶï ½ûÖİÿ ĭÖĴ¿¶ Ħ¿üįÖà¬ ĦÓÖüī ĵ ½çõÖï õñ Ħ¿üįÖà¬ ÞÿÕñSchool đĶĬèī of ÷Õ Astronomy ĲÙ ÞûÕ ÝõÖÚē -T IPMext Ħ¶ Ħ¿üįÖà¬ ĭÕò¿ī v dx/dt x Ī¾õÕña Ē¾÷Ķßr ĒÙÖß ¼ÕöÙx = ĵr ñÖ¿ī/a Þûñ ĲÙ ĲħñÖĔī ĭĶûÕĶ¬ ĲħñÖĔī Ħë ÖÙ ĵ ò¾É ½ī Þûñ ĲÙ Ē¾÷Ķß ĒÙÖß ÷Õ ½ĨïÕñ Ħ¿üįÖà¬ Ī¿ĳÕĶï C ≡ ĵ Øüë öÙ ŃÖÙ ÝŃñÖĔī ½ü¾Ķį ÷ÖÙ ÖÙ òÿÖÙ ½ī ıÕöĬĳ ĥĶċ Ĳ¶ −→ òÿÖÙ ρ(x,t)= f(x, v,t)dv dxi/dt = vi ÞÿÕñ ½·¾ø¿ě öàïÕ ¼Öĳ ĵöĬĨğ # öĳ ñĶÿ ½ī Ĳàěöº öĐį õñ Öĳ ¼÷ÖûN Ĳ¿Úÿ Į¾Õ ¼ÕöÙ ½·¾ø¿ěöàïÕ ĵöĬĨğ CĲĨÔüī Þ¾ñõĶïöÙ ĵ ½įÖī÷ ùÖ¿Ġī ĲÙ ĲçĶß ÖÙ dxi/dt = vi ¼ñòē ¼Öĳ ýĵõ ĲÙ ĦăÖë ĩöç Īàü¿û đÖçõÕ úû ĵ Ĳ¿ħĵÕ Ē¾÷Ķß ĒÙÖß ÷Õ ¼÷Öû ĲįĶĬį ÖÙ ÕòàÙÕ ıÖ»àûñ Į¾Õ Ħë òİİ¶ ½ī ıñÖĜàûÕ ×ĶĨČī ½¾ÕõÖ¶ ĵ Þğñ ĲÙ ĮàěÖ¾ Þûñ õĶĐİī ĲÙ CĲĨÔüī ÖÙ ØûÖİàī ¼ñòē ýĵõ ÷Õ Öĳ ĵöĬĨğ Į¾Õ ÷Õ ĩÕò¶ Ĳį ¼÷Öû õÕĶĬĳ ñĶÿ ½ī ĩÖèįÕ Īüç ñö·¾ĵõ ÖÙ µ°Ķ¶ ùÖ¿Ġī õñ ĵ ½ĔćĶī ÝõĶă ĲÙ ÝŃñÖĔī Į¾Õ ñĶÿ ½ī ĩÖèįÕ ĵ ÞûÕ ØħÖė Īàü¿û öÙ Īüç µ¾ þįÕöº ĵöĬĨğ Į¾Õ õñ ¼Õ ıõÖàû ĵ ¼Õ ıõÖ¿û ¼Öĳ ĲīĶĐİī ¼ĵÖĬû µ¿įÖ·ī ½¾Öß Ħ¿·Āß ÷Õ ÞĔįÖĬī ¼ÕöÙ āÖï õĶċ ĲÙ Ĳ·ĨÙ Cµ¾ñøį ¼Öĳ þİ·Ĭĳ öÙ õñ ĵö¿į ĵ Þēöû ĭñö¶ ñĵòìī ¼ÕöÙ ÖĴİß dvi/dt + H(t)vi = xΦ i ½ßÖÚûÖìī ¼Öĳ ÖČï ĭòÿ ØħÖė ÷Õ ¼ö¿ºĶĨç ¼ÕöÙ Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ õñ òÿÖÙ ½ī ĭĶī÷É ÝÕõó Ĳ¿Úÿ ÝÕõó ½ğÖÙ Þ¶öë Ĳ¿Úÿ òÿÖÙ ½ī ĪĴī Īàü¿û õñ ĥÖĔě ÝÕõó ñÕòĔß Öĳ ¼÷Öû Ĳ¿Úÿ Į¾Õ õñ Õö¾÷ CñĶÿ ½ī ÝŃñÖĔī ñõÕĵ ½ēĶİĄī ¼Öĳ −a ∇ | òÿÖÙ ½ī ½īÕøħÕ ñÖ¾÷ õÖ¿üÙ Þğñ CĲàüüº ĭÖī÷ ¼ÖČï Ħ¿Úğ ÷Õ dvi/dt + H(t)vi = xΦ i −a ∇ | òÿÖÙ ½ī ½ħÖ»° ÝŃńàïÕ ĲÙ ĲçĶß ÖÙ ½ĀįÕöº Ħ¿üįÖà¬ Φıõóĵ ¼ñÖ¾÷H(t)=˙ ñÕòĔß ĦīÖÿa/a Īàü¿ûÝŃñÖĔī ĦÓÖüī Į¾Õ Į¾Õ õñ õñ ÷ÖÙ ĵ ¼ĵö¶ ¼Õ ıõÖàû ¼Öĳ ĲÿĶï ĥÖ»° ¼Õ ıõÖàû ¼Öĳ Īàü¿û òÿÖÙ ½ī ½ħÖ»° ÝŃńàïÕ ĲÙ ĲçĶß ÖÙ ½ĀįÕöº Ħ¿üįÖà¬ Φ ĵ HĭÖī÷(t)=˙ ½įÖī÷a/a ùÖ¿ĠīÝŃñÖĔī õñ ¼÷Öû Į¾Õ Ĳ¿Úÿ õñ Þü¾ÖÙ ½ī Õôħ Cò¾É ½ī ×Öüë ĲÙ ñõĶïöÙ ö¬ ¼Öĳ Īàü¿û ĵøç ÷Õ ĵ òÿÖÙ ½ī ĩöç Īĳ ñö¶ ĥÖÚįñ ĥĶìß ö¿û õñ Õõ ÝÕõó µ¾ñøį Öĳ þİ·ĬĳöÙ ĭÕĶàÙ Öß ñĶÿ ÕöçÕ ĭò¿īõÉ xΦ = πGa [ρ(x,t) ρ¯(t)]. ∇ xΦ = πGa [ρ(x,t) −ρ¯(t)]. ∇ − ıÖ¿û öÙÕ þįÕöº CòÿÖÙ ½ī ½ĨÚğ ĵöĬĨğ Ø¿¶öß ĲĨÔüī Į¾Õ ½įÖĀ·Ĵ¶ µü¾ñ ĲħÖ° ıÖ¿û öÙÕ ö¿ãÖß Þìß ¼ ıö¶ öĐį ęöăöĐį ĉÖüÚįÕ ęöă ĉÖüÚįÕ ÷Õ ĵõ ÷Õ þ¿¬ ĵõ þ¿¬ ÝÖÚûÖìī ÝÖÚûÖìī õñ õñ Còİ¶ Còİ¶ ½Ĭį ½Ĭį ½ăÖï ½ăÖï ÝĵÖĜß ÝĵÖĜß ÝŃñÖĔī ÝŃñÖĔī Į¾Õ Į¾Õ Ħë Ħë CĭÖĴ¿¶áñÖë CĭÖĴ¿¶ ĉÖüÚįÕ ¼õÖ¿üÙ ĉÖüÚįÕ Įàěöº ½¾Öß µ¾ñøį öĐį Įàěöº õñ ¼Öĳ ÖÙ öĐį þİ·ĬĳöÙ õñ ÖÙÖĳ ıõÖàû ĥÖ»° ½ëÕĶį õñ ½ěöċ ÷Õ ĵ ÞûÕ ØħÖė Īàü¿û ÝÕõó Ħ¶ öÙ ĲħÖ° òÿÖÙ ½ī öß Ħ·Āī ĵ öß ıò¿±¿¬ ½ĨÚğ ¼Öĳ ĵöĬĨğ ÷Õ ÝŃñÖĔī Ħë Õôħ òįĶÿ ½ī Ħ¿üįÖà¬Ħ¿üįÖà¬ µ¾ ıĵńĔÙ µ¾ ıĵńĔÙ ıõó åĵ÷ ıõó åĵ÷ öĳ Į¿Ù öĳ Į¿Ù ½ßĶ¿į ½ßĶ¿į þįÕöº þįÕöº ½īĶĬē ½īĶĬē ĭĶįÖğ ĭĶįÖğ ÷Õ ÷Õ ÞûÕ ÞûÕ ÝõÖÚē ÝõÖÚē ½Ĩ¶ ½Ĩ¶ µ¾ø¿ě µ¾ø¿ě ĦÓÖüī ĦÓÖüī Į¾Õ õñ Į¾Õ ÞûÕ õñ ıòÿ ÞûÕ ıòÿ Cµ¾õÖß ıñÖī õĶĈë õñ½çõÖï āĶĄï ĲÙ ½įÖĀ·Ĵ¶ Õöě ĵ ½įÖĀ·Ĵ¶ ĥĶìß ½ûÖİÿ ĭÖĴ¿¶ ĵ ½įÖĀ·Ĵ¶ µ¿īÖİ¾ñ ÷Õ Õôħ CòįÕ ıòÿ Ħ¿·Āß ıõó ¼ñÖ¾÷ õÖ¿üÙ½çõÖï ñÕòĔß ÷Õ µ¾õÖß ıñÖī ¼Öĳ ĲħÖĳ āÖï õĶċ ĲÙ ĵ Öĳ ĭÖĀ·Ĵ¶ ½įÖĴ¿¶ µ¿īÖİ¾ñ m m (r r ) ĲÙ ¼÷Öû õÕĶĬĳ ÷Õ ŃĶĬĔī ĵ òÿÖÙ ½Ĭį ĪĴī µ¾ñøį ¼Öĳ ñõĶïöÙ ñö¶ Ě¿ăĶß ĲČûÕĵ ĭÕò¿ī ÖÙ ½àëÕõ ĲÙ Õõ Öĳ ĭÉ ĭÕĶß ½ī F = Gm imj (ri jr ) .φ (r ) i i j i− j ext i ÖĳWIMP µ¾õÖß ıñÖī ÝÕõó Ĳ¶ ½àħÖë õñ ñĶÿ ½ī Ĳàěöº õÖ¶ ĲÙ ½·¾ø¿ě ö¿ė ¼Öĳ ½¾Öß Ħ¿·Āß ÷Õ ¼ö¿ºĶĨç õĶĐİī F = − G ri rj −∇ .φ (r ) i j=i − ext i − ̸ |ri − rj| −∇ õĶċ ĲÙ ýĵõ Į¾Õ ñĶÿ éńăÕ ½ĀįÕöº ö¿ė þİ·ĬĳöÙ ñĶï ÖÙ Õõ ÝÕõó Į¾Õ µ¿īÖ¿įñ ĭÕĶß ½ī òįĶÿ ½ī Ĳàěöº öĐį õñ j=!i !̸ | − | ñõÕñ ½¾ÕõÖ¶ Öĳ ĲħÖĳ ø¶öī õñ āÖï Öĳ ýĵõ ½İßĶ¿į þįÕöº

Ē¾öû Öĳ Īà¾õĶ»ħÕ ĲĔûĶß ÖÙ #% ıõó ñĵòë ¼ÕöÙ ò¾ñöº đĵöÿ Ĳĳñ Ħ¾ÕĵÕ õñ ¼ößĶ¿īÖ¶ ¼Öĳ ¼÷Öû Ĳ¿Úÿ ĵ#% ¼Õ Ĳ·Úÿ¼Õ ıõó ¼Öĳ ò¶ òįò¿ûõ ĕĶĨÙ ĲÙ ¼÷Öû Ĳ¿Úÿ Į¾Õ C Ĳĳñ õññĶïöÙ ĭĵòÙ ĦÓÖüī Ħë ¼ÕöÙ òīÕõÖ¶ ĵ þİ·ĬĳöÙ Ħë ¼ÕöÙ ¼÷Öû Ĳ¿Úÿ ¼ÕöÙ ¼Öĳ Īà¾õĶ»ħÕ Þ¿ë Į¿Ĭĳ õñ òİÿÖÙ ½ī Öĳ ò¶ Į¾Õ ¼Öĳ ĲįĶĬį ÷Õ #% ½àïõñ ¼Öĳ Īàü¿û ¼Öĳ ¼÷Öû Ĳ¿Úÿ ĥÖû õñ #% òÿ ĲÓÕõÕ ¼ñõĶïöÙ ¼Öĳ Īàü¿û õñ Öĳ ½¾Ößĵñ µ¿īÖİ¾ñ ĵ µ¾ñøį Öß ¼ñõĶïöÙ ö¿ė ¼Öĳ Īàü¿û ĵ Īüç ĭò¿īõÉ ½įÖī÷ ùÖ¿Ġī ĵ Ī¿Ġàüī ýĵõ ÖÙ #% ıõó Öß ¼ñõĶïöÙ ñÕòĔß ½ºõøÙ ĲÚßöī ĵ ÞûÕ ıòÿ ÝÖÚûÖìī ñõÕĵ Millenium ¼÷Öû Ĳ¿Úÿ òİįÖī ½įÖĀ·Ĵ¶ µ¿īÖİ¾ñ ¼÷Öû Ĳ¿Úÿ õñ ıõó ĵ ıõó Öß CþįÕöº Þìß ĲĠĨë ĵ ¼Öĳ µü¾ñ C¼Õ ıõÖ¿û ¼Öĳ Īàü¿û Ĳ¿Úÿ ĦÓÖüī õñ òÙÖ¾ ½ī þ¾ÕøěÕ ĭÖİ±Ĭĳ ÝÕõó ¼ÕöÙ ¼ñÖ¾÷ ¼Öĳ ýńß ö¿ïÕ ¼Öĳ ıñ õñ #% òÿÖÙ ½ī ¼÷Öû Ĳ¿Úÿ ĦÙÖğ ½·¿īÖİ¾ñ ĭÖī÷ öÙÕöÙ òă òİ° ½įÖī÷ ùÖ¿Ġī ĭÉ ÷Õ #% òÿÖÙ ½ī GRAP E õÕøěÕ Þðû ĭÉ Ĳè¿àį ĵ ÞûÕ ıòÿ ĩÖèįÕ ø¿į ¼õÕøěÕ Þðû ½ßÖÚûÖìī ĭÕĶß þ¾ÕøěÕ Õöß ½ºõøÙ ĲÚßöī ĲÙ ½ßÖÚûÖìī Ýõòğ ÞûÕ ıòÿ ½ëÕöċ ½ĀįÕöº þİ·ĬĳöÙ ¼ÕöÙ āÖï õĶċ ĲÙ Öĳ õÕøěÕ Þðû Į¾Õ Ĳ¶ Öç òĳñ ½ī ĲÓÕõÕ ªńě

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What30th Jerusalem is the Winter ideal School N-body simulation? Romain Teyssier31

Lecture series School of Astronomy - IPM ĭÖĴ¿¶ ùÖ¿Ġī ¸õøÙ ¼÷Öû Ĳ¿Úÿ

Ī¾õÕñ ½įÖī÷ ĩÖº ´ńī ¼ÕöÙ Õôħ CòÿÖÙ ĩĵñ ĲĨĬç ÷Õ öß ¸õøÙ ĥĵÕ ĲĨĬç Ĳ¶ ñĶÿ ½ī ĦăÖë ½įÖī÷ ñÖĬàēÕ ĦÙÖğ Þğñ u ∆t = α | | tol F | | òİ¶ ½ī ĆĠį Õõ Īàü¿û ¼ö¾ô¬ ÞĀº÷ÖÙ ÝõĶă Į¾òÙ ¼÷Öû÷ÖÙ Ĳ¶ ÖèįÉ ÷Õ òÿÖÙ ½ī úįÕöħĶßĭÖĴ¿¶ ùÖ¿Ġī òĔÙ ĭĵòÙ ¸õøÙ öàīÕõÖ¬ ¼÷Öûαtol Ĳ¿ÚÿC ÷Õ ÞûÕ ÝõÖÚē ´ńī öß ØûÖİī ×ÖðàįÕ σ ĭÖĴ¿¶ ùÖ¿Ġī ¸õøÙ ¼÷Öû Ĳ¿Úÿ Ī¾õÕñ ½įÖī÷ ĩÖº ´ńī ¼ÕöÙ Õôħ CòÿÖÙ ĩĵñ ĲĨĬç ÷Õ∆t öß= ¸õøÙαtol ĥĵÕ ĲĨĬç Ĳ¶ ñĶÿ ½ī ĦăÖë ½įÖī÷ ñÖĬàēÕ ĦÙÖğ Þğñ F | | u Ī¾õÕñ ½įÖī÷ ĩÖº ´ńī ¼ÕöÙ Õôħ CòÿÖÙ ĩĵñ ĲĨĬç ÷Õ∆ ößt = ¸õøÙαtol ĥĵÕ| | ĲĨĬç Ĳ¶ ñĶÿ ½ī ĦăÖë ½įÖī÷ ñÖĬàēÕ ĦÙÖğ Þğñ σ #% òÿÖÙF ½ī ĲĨÔüī õñ ÝÕõó Þēöû ½ēĶį ½ºòİ¶Õö¬ Ĳ¶ C u| | ∆t = α | | tol F α òİ¶ ½ī ĆĠį Õõ Īàü¿û ¼ö¾ô¬ ÞĀº÷ÖÙ ÝõĶă Į¾òÙ ¼÷Öû÷ÖÙ| Ĳ¶| ÖèįÉ ÷Õ òÿÖÙ ½ī úįÕöħĶß òĔÙ ĭĵòÙ öàīÕõÖ¬ tol C òİ¶ ½ī ĆĠį Õõ Īàü¿û ¼ö¾ô¬ ÞĀº÷ÖÙ ÝõĶă Į¾òÙ ¼÷Öû÷ÖÙ Ĳ¶ ÖèįÉ ÷Õ òÿÖÙ÷Õ ½ī ÞûÕ úįÕöħĶß ÝõÖÚēµ¿īÖİ¾ñĵõò¿ĳ òĔÙ ´ńī ĭĵòÙ öàīÕõÖ¬ öß ØûÖİīαtol C ×ÖðàįÕ σ ÷Õ ÞûÕ ÝõÖÚē ´ńī öß ØûÖİī ×ÖðàįÕ ∆t = αtol σF ∆t = α | | ıòÿ õÕĶĬĳ ¼Õ ıõó µ¿īÖİ¾ñĵõò¿ĳ tol F | | #% òÿÖÙ ½ī ĲĨÔüī õñ ÝÕõó Þēöû ½ēĶį ½ºòİ¶Õö¬ σ Ĳ¶ C ĭÖĀ·Ĵ¶ ¼ö¿º Ħ·ÿ ĲĔħÖČī õñ µ¿īÖİ¾ñĵõò¿ĳ ĦÓÖüī Ħë# æ¾Õõ% òÿÖÙ ¼Öĳ ½ī µ¿İ·ß ĲĨÔüī õñ ÷Õ ½·¾ÝÕõó ıòÿ Þēöû õÕĶĬĳ ½ēĶį ¼Õ ½ºòİ¶Õö¬ ıõó µ¿īÖİ¾ñĵõò¿ĳσ Ĳ¶ C ¼³įÕöºŃ ĩöě ÷Õ ÞħĶĴû õĶĐİī ĲÙ Õôħ C#% ñĶÿ ½ī ĥÖÚįñ ÝÕõó µß µß Þ¶öë ýĵõ Į¾Õ õñ #% òÿÖÙ ½ī Öĳ òİİ¶ ½ī Þ¶öë ĥÖ¿û ÖÙ ÝÖĄàðī ñĶÿ ½ī ıñÖĜàûÕ Þ¶öë ÝŃñÖĔī µ¿īÖİ¾ñĵõò¿ĳ dρ µ¿īÖİ¾ñĵõò¿ĳ Galaxy Formation+ ρ .u = - Hydrodynamic Simulations 19 dt ∇ Smoothed-Particle Hydrodynamicsıòÿıòÿ (SPH) õÕĶĬĳ õÕĶĬĳ ¼Õ ¼Õ ıõó ıõó µ¿īÖİ¾ñĵõò¿ĳ µ¿īÖİ¾ñĵõò¿ĳ C Grid-Baseddu AlgorithmsP ĭÖĀ·Ĵ¶ĭÖĀ·Ĵ¶ ¼ö¿º ¼ö¿º Ħ·ÿ Ħ·ÿ ĲĔħÖČī ĲĔħÖČī õñ µ¿īÖİ¾ñĵõò¿ĳ õñ µ¿īÖİ¾ñĵõò¿ĳ ĦÓÖüī ĦÓÖüī= Ħë Ħë∇ æ¾Õõ æ¾Õõ ¼Öĳ ¼Öĳφ µ¿İ·ß µ¿İ·ß ÷Õ ½·¾ ÷Õ ½·¾ ıòÿ ıòÿ õÕĶĬĳ õÕĶĬĳ ¼Õ ıõó ¼Õ µ¿īÖİ¾ñĵõò¿ĳ ıõó µ¿īÖİ¾ñĵõò¿ĳ dt − ρ −∇ ¼³įÕöºŃ¼³įÕöºŃ ĩöě ÷Õ ĩöě ÞħĶĴû ÷Õ ÞħĶĴû õĶĐİī õĶĐİī ĲÙ ĲÙ Õôħ Õôħ C# C#%% ñĶÿ ñĶÿ ½ī ½ī ĥÖÚįñ ĥÖÚįñ ÝÕõó ÝÕõó µß µß µß µß Þ¶öë Þ¶öë ýĵõ ýĵõ Į¾Õ õñ Į¾Õ # õñ% # òÿÖÙ% ½ī òÿÖÙ Öĳ ½ī Öĳ òİİ¶ òİİ¶ ½ī ½ī Þ¶öë Þ¶öë ĥÖ¿û ĥÖ¿û ÖÙ ÖÙ ÝÖĄàðī ÝÖĄàðī ñĶÿ ñĶÿ ½ī ıñÖĜàûÕ ½ī ıñÖĜàûÕ Þ¶öë Þ¶öë ÝŃñÖĔī ÝŃñÖĔīC dρ dE dρ P+ ρ .u =L(E,ρ) = dt + ρ∇.u.u = dt dt− ρ ∇∇ − ρ C CĭĶûÕĶ¬ ĲħñÖĔī öÙ ıĵńē CÝŃñÖĔī Ħë ¼ÕöÙ òİÿÖÙd ½īu ĩöç òëÕĵP þ¾Öīöº ĵ þ¾Öīöû íöį H ĵ C ĵCL = C H Ĳ¶ CC = ∇ φ − ddtu − ρP −∇ ñĶÿ ½ī Ĳàěöº öĐį õñ ø¿į ÞħÖë ĲħñÖĔī = ∇ φ dt − ρ −∇ C P =(γ )ρE dE P − L(E,ρ) C = .u dt − ρ ∇ − ρ γ dE P L(E,ρ) The IllustrisòÿÖÙ simulations ½ī ĵõñ ½Ùwere ú¾òįÕ run on supercomputersĲ¶ C in France, Germany, and = .u SPH ĩöç ¼³įÕöºŃCĭĶûÕĶ¬ ĲħñÖĔī ĭĶ¿ûŃĶīöě öÙ ıĵńē õñ CÝŃñÖĔī òÿÖÙ ½ī Ħë ıõó ¼ÕöÙ ¼ñÕòĔßdt òİÿÖÙ ½ī− ÝõĶăρ ĩöç∇ òëÕĵ ĲÙ− ĵ þ¾Öīöº ĶħõÖ¶ρ ÞįĶī ĵ þ¾Öīöû ýĵõthe US. íöį ÖÙ TheH ĥÖ¿û ĵlargestC þ¾ÖĬįĵC Lwas= runC onH 8,192½ĨăÕĲ¶ C compute ıò¾Õ cores, and took 19 million CPU hours (the equivalent of one− computer CPU running for 19 million hours, or ĥÖäī ¼ÕöÙ ñö¶ Į¿¿Ĕß ĥÖ¿û ¼ÕöÙ Õõ ½įÕò¿ī öĳ ĭÕĶß ½ī ÝÕõó Į¾Õ ÷Õ ıñÖĜàûÕ ÖÙñĶÿ òİÿÖÙ ½ī Ĳàěöº ½ī öĐįĲàü¾Ö¬ õñ ø¿į ĥÖ¿û ÞħÖë öăÖİē ĲħñÖĔī ĵ ÝÕõó about 2,000 years).H C L = C H CĭĶûÕĶ¬ ĲħñÖĔī öÙ ıĵńē CÝŃñÖĔī Ħë ¼ÕöÙ òİÿÖÙ ½ī ĩöçĪ¾õÕñ òëÕĵ ½Ú¾öĠß þ¾Öīöº õĶċ ĵ ĲÙ þ¾Öīöû CòÿÖÙ r íöįĭÖ·ī õñĵ öĐįĵC ñõĶī ĭÕò¿ī− A Ĳ¶öºÕ C P =(γ )ρE − ñĶÿ ½ī Ĳàěöº öĐį õñ ø¿į ÞħÖë ĲħñÖĔī HydroN simulations establishing through directly accounting for the baryonic A γ A(r)=componentm j W (gas,(r r stars,; h) supermassiveòÿÖÙ ½ī ĵõñ ½Ù ú¾òįÕ black Ĳ¶ holes, C etc.) in cosmological P =(jγ )ρE j SPH ĩöç ¼³įÕöºŃ ĭĶ¿ûŃĶīöě õñ òÿÖÙ ½ī ıõó ¼ñÕòĔßsimulations ÝõĶăρ−j ĲÙ that ĵ ĶħõÖ¶ − calculate ÞįĶī ýĵõ ﬂuid ÖÙ ĥÖ¿û motion þ¾ÖĬį ("hydrodynamics")½ĨăÕ ıò¾Õ as well as gravity, in j!= ĥÖäī ¼ÕöÙ ñö¶ Į¿¿Ĕß ĥÖ¿û ¼ÕöÙ Õõ ½įÕò¿ī öĳprinciple ĭÕĶß ½ī ÝÕõó offering Į¾Õ ÷Õ ıñÖĜàûÕa self-consistent ÖÙ òİÿÖÙ ½īòÿÖÙ Ĳàü¾Ö¬ and ½ī ĥÖ¿û fully ĵõñ öăÖİē predictive ½Ù ú¾òįÕ ĵ ÝÕõó γ methodology.Ĳ¶ C Ī¾õÕñh ½Ú¾öĠß õĶċ ĲÙ CòÿÖÙ r ĭÖ·ī õñ öĐį ñõĶī ĭÕò¿īWA(r,öºÕ h) ÞÚüįĩöç ¼³įÕöºŃ öºÕ Ĳ¶ ñĶÿ ĭĶ¿ûŃĶīöě ½ī Į¿¿Ĕß õñ ¼Ķìį òÿÖÙ ĲÙ ½ī Ĳàüĳ ıõóLecture Į¾Õ ¼ñÕòĔß ĦĬē series ÝõĶă õñ đÖĔÿ ĲÙ ĵ õñĶħõÖ¶ ¼÷Öû ÞįĶī õÕĶĬĳ ýĵõ Ĳàüĳ ÖÙ ĥÖ¿û ÷Õ ÞûÕ þ¾ÖĬį ÝõÖÚē SPH ½ĨăÕ Ĳ¶ ıò¾Õ C School of Astronomy - IPM N η r/h ÝõĶăĥÖäī ¼ÕöÙ òİÿÖÙ ñö¶ öĐį Į¿¿Ĕß ñõĶī ĥÖ¿û ıõó ½»¾ÖüĬĳ ¼ÕöÙ Õõ ½įÕò¿ī õñ Ĳ¶ öĳ òÿÖÙ ĭÕĶß ½ī ½ī ½ßÕõó ÝÕõóAj ĒĬç Į¾Õ ú¬ ÷Õ ıñÖĜàûÕ òÿÖÙ öÙÕöÙ ÖÙ òİÿÖÙ òÿÖÙ ½ī öß ¸õøÙ Ĳàü¾Ö¬ ĥÖ¿ûÞÙÖã öăÖİē õÕòĠī ÷Õ ĵ ÝÕõó A(r)= mj W (r rj; h) r A ½ī ĩÖèįÕ ½ûòİĳ ÝõĶă ĲÙ ĵ ½àïõñ Īà¾ö»ħÕ ĊûĶß þİ·ĬĳöÙĪ¾õÕñρj Þü¿ħ ½Ú¾öĠß− Ĳ¿Ĵß õĶċ ¼ÕöÙ ĲÙ CòÿÖÙ Ķç ĵ ÞüçĭÖ·ī õñr öĐįrj ñõĶīηh ĭÕò¿īCñö¿º ½īöºÕ j!= | − | ≤ N ÖÙ ñĶÙ òĳÕĶï öÙÕöÙ ĭÕò¿ī ½¾ÖĈě ĞàĀī #% ñĶÿ ÞÚüį öºÕ Ĳ¶ ñĶÿ ½ī Į¿¿Ĕß ¼Ķìį ĲÙ Ĳàüĳ Į¾Õ ĦĬē õñ hAđÖĔÿj õñ ¼÷Öû õÕĶĬĳ Ĳàüĳ ÷Õ ÞûÕ ÝõÖÚē W (r, h) Ĳ¶ C A(r)= N mj W (r rj; h) Aρj − η r/h ÝõĶă òİÿÖÙ öĐį ñõĶī ıõó ½»¾ÖüĬĳ õñ Ĳ¶ òÿÖÙj ½ī= ½ßÕõój ĒĬç ú¬ òÿÖÙ öÙÕöÙ òÿÖÙ öß ¸õøÙ ÞÙÖã õÕòĠī ÷Õ A(r)= !mj W (r rj ; h) ½ī ĩÖèįÕ ½ûòİĳ ÝõĶă ĲÙ ĵ ½àïõñ∇ Īà¾ö»ħÕ ĊûĶß þİ·ĬĳöÙρj ∇ Þü¿ħ Ĳ¿Ĵß− ¼ÕöÙ Ķç ĵ Þüç r rj ηh Cñö¿º ½ī j!= | − | ≤ ÞÚüį öºÕ Ĳ¶ ñĶÿ ½ī Į¿¿Ĕß ¼Ķìį ĲÙ Ĳàüĳ Į¾Õ ĦĬē õñ h đÖĔÿÖÙ õñ ñĶÙ ¼÷Öû òĳÕĶï õÕĶĬĳ öÙÕöÙ ĭÕò¿ī Ĳàüĳ ½¾ÖĈě ÷Õ ÞûÕ ĞàĀī ÝõÖÚē #W%( ñĶÿr, h) Ĳ¶ C N η r/h ÝõĶă òİÿÖÙ öĐį ñõĶī ıõó ½»¾ÖüĬĳ õñ Ĳ¶ òÿÖÙ ½ī ½ßÕõóAj ĒĬç ú¬ òÿÖÙ öÙÕöÙ òÿÖÙ öß ¸õøÙ ÞÙÖã õÕòĠī ÷Õ A(r)= mj W (r rj ; h) ½ī ĩÖèįÕ ½ûòİĳ ÝõĶă ĲÙ ĵ ½àïõñ Īà¾ö»ħÕ∇ ĊûĶß þİ·ĬĳöÙρj ∇ Þü¿ħ− Ĳ¿Ĵß ¼ÕöÙ Ķç ĵ Þüç r rj ηh Cñö¿º ½ī j!= | − | ≤ ÖÙ ñĶÙ òĳÕĶï öÙÕöÙ ĭÕò¿ī ½¾ÖĈě ĞàĀī #% ñĶÿ

N Aj A(r)= mj W (r rj ; h) ∇ ρj ∇ − j!=

Galaxy Formation - Hydrodynamic Simulations 20

Hubble DEEP Hydro Simulation

Lecture series School of Astronomy - IPM Galaxy Formation - Semi-Analytic Model (SAM) 21

Indirect galaxy formation

Images of the Hubble Ultra Deep Field as predicted by the Millennium Run Observatory (left) and as actually observed by the Hubble Space Telescope (right).

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Lecture series School of Astronomy - IPM Galaxy Formation-Schematic overview of Galaxy formation 28

Cosmological Model

Dark matter halo merger tree

Gas Cooling & Disk Galaxy merger Density proﬁle formation spheroid formation Spheroid sizes Disk sizes

Star Formation & Feedback Feedback Bursts

AGN SN Chemical evolution

Jet model Stellar population

Dust extinction

Observable galaxy properties

Lecture series School of Astronomy - IPM Galaxy Formation- Our works 29 Physical consequences In the radio mode or low energy activity SN (∝SFR) some of hot gas accreted onto the central super massive black hole after cooling down and settled at the galaxy center. Hence, relative to the black hole accretion rate, the amount of energy add to the cooling gas, AGN instantly. 3 AGN (∝mBHσ ) Croton et al. 2006

AGN

Lecture series School of Astronomy - IPM Galaxy Formation- Our works 30

Physically Motivated AGN Feedback

Lecture series School of Astronomy - IPM Galaxy Formation- Our works 31

Lecture series School of Astronomy - IPM Galaxy Formation- Our works 32

Dynamically Old & young identiﬁcations

Lecture series School of Astronomy - IPM Galaxy Formation- Our works 33

Effect of environment; Old: evolved & young : evolving

6 / 0.96 Gyr] • O

4 /dt [M BH

Log dM 2

0 11.0 11.2 11.4 11.6 11.8 -1 Log M* [M O• h ]

Recent Major merger Accretion of SMBH

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AGN physic in galaxy formation Optimization of our model for galaxy formation Geometrical environment in the Simulation data Hydro-dynamical simulation and metallicity gradient Properties of galaxy groups in hydrodynamic and SAM

Lecture series School of Astronomy - IPM Galaxy Formation- Book 35

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Thanks for your attention

Lecture series School of Astronomy - IPM Galaxy Formation- Extra slide 36

Lecture series School of Astronomy - IPM