ASTR 610 Theory of Galaxy Formation
Summary Slides
Frank van den Bosch Yale University, Fall 2020 Lecture 1 Introduction Galaxy Formation in a Nutshell
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Galaxy Formation in a Nutshell
radiation-dominated matter-dominated Λ
λH super-horizon scales 15 M~10 M⦿
12 λMW M~10 M⦿
log[scale] Jeans-stable Jeans-unstable 6 M~10 M⦿ Silk bar damping λJ
teq tdec virialization shock +3 cooling AC ~200 virialization non-linear non-linear collapse
] 0
MW
δ
log[
linear growth linear -3 baryonic oscillations
dark matter -6 baryons stagnation Silk damping
-3.5 -3 log[scale-factor] -0.3 0
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 2
Overview of Cosmology I (Riemannian Geometry & FRW metric) Cosmology in a Nutshell
Lecture 2 Lecture 3
Cosmological Principle General Relativity Universe is homogeneous & Isotropic
Riemannian Geometry Einstein’s Field Equation 1 8 G R g R g = T µ 2 µ µ c4 µ
Friedmann-Robertson-Walker Metric ds2 = a2( ) d 2 d 2 f 2 ( ) d 2 + sin2 d 2 K Friedmann Equations
a˙ 2 8 G Kc2 c2 = ⇥ + a 3 a2 3 ✓ ◆
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key words & important facts
Key words Fundamental observer Cosmological Principle Proper time vs. conformal time FRW metric Comoving vs. proper distance Hubble parameter Angular diameter distance redshift Luminosity distance peculiar velocity
Physical laws can be made manifest invariant by writing them in tensor form.
The geometry of space-time is described by the metric gµ⇥ = gµ⇥ (x )
The FRW-metric is the most general metric consistent with the cosmological principle, that the Universe is homogeneous and isotropic (on large scales).
Due to the expansion, the peculiar velocities of particles that do not 1 experience an external force decay with time as v a pec / Since energy densities of baryons & dark matter evolve in the same way, it is sufficient to describe the (non-relativistic) matter as one component .
Since energy densities of radiation & relativistic matter (i.e., neutrinos) evolve in the same way, it is sufficient to describe them as one component .
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
dr2 ds2 = c2dt2 a2(t) + r2(d 2 +sin2 d⇥2) Two ways of writing 1 Kr2 the FRW-metric ds2 = a2( ) d 2 d 2 f 2 ( ) d 2 + sin2 d 2 K
d 3(1+w) Thermodynamics + 3(1 + w) =0 a da a non-relativistic matter (baryons & dark matter) w = 0 relativistic matter (radiation) w = 1/3 cosmological constant (dark energy) w = -1
Redshift, wavelength, scale-factor & obs em a(tobs) z = 1 peculiar velocity ⌘ em a(tem)
v =˙a + a ˙ v + v 1+z =(1+z )(1+z ) ⌘ exp pec obs cos pec
angular diameter distance a r d (z)= 0 d (z)=a r (1 + z) luminosity distance A 1+z L 0
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 3 Overview of Cosmology II (General Relativity & Friedmann Eqs) Cosmology in a Nutshell
Lecture 2 Lecture 3
Cosmological Principle General Relativity Universe is homogeneous & Isotropic
Riemannian Geometry Einstein’s Field Equation 1 8 G R g R g = T µ 2 µ µ c4 µ
Friedmann-Robertson-Walker Metric ds2 = a2( ) d 2 d 2 f 2 ( ) d 2 + sin2 d 2 K Friedmann Equations
a˙ 2 8 G Kc2 c2 = ⇥ + a 3 a2 3 ✓ ◆
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key words & important facts Key words Equivalence Principle Riemann tensor Christoffel symbols Ricci tensor covariant derivative Einstein tensor
Why Newtonian gravity only holds in inertial systems, is covariant under Galilean transformations, and moving mass has immediate effect all throughout space. we inertial systems do not exist (you can’t shield yourself from gravity) need but SR: inertial systems transform according to Lorentz transformations GR SR: universal speed limit; no information can propagate instantaneously
Since gravity is `permanent’ (can only be transformed away locally), it The must be related to an intrinsic property of space-time itself.
Key Space-time of freely falling observer (no gravity) is flat Minkowski space; to hence, gravity originates from curvature in space-time (Riemann space)
GR Einstein Field equation is the manifest covariant version of Poisson equation
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
Poisson Equation 2 =4 G⇥ r GR GR
1 8 G Einstein Field Equation R Rg g = T µ 2 µ µ c4 µ
a˙ 2 8 G Kc2 c2 The Friedmann Equation = ⇥ + a 3 a2 3 ✓ ◆
2 2 2 H (z)=H0 E (z) where
1/2 E(z)= + (1 )(1+z)2 + (1 + z)3 + (1 + z)4 ,0 0 m,0 r,0
(1 + z)2 Density Parameter (z) 1=( 1) 0 E2(z)
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 4
Newtonian Perturbation Theory I. Linearized Fluid Equations Summary: key words & important facts Key words Euler equations Hubble drag Equation of State Perturbation analysis Ideal Gas Isentropic perturbations Sound Speed Isocurvature perturbations
Dark matter can be described as a collisionless fluid as long as the velocity dispersion of the particles is sufficiently small that particle diffusion can be neglected on the scale of interest. This is true on scales larger than the free-streaming scale.
In the linear regime, all modes evolve independently (there is no `mode-coupling’)
If evolution is adiabatic, isentropic perturbations remain isentropic. If not, the non-adiabatic processes create non-zero S
Isentropic and isocurvature perturbations are orthogonal; any perturbation can be written as a linear combination of both.
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
D continuity equation + r ⇥u =0 k T 1 k T Dt · ideal gas P = B = B µmp 1 µmp D⇤u P Euler equations r = r r⇥ 1/2 Dt r sound speed cs =( P/ )S Poisson equation 2⇤ =4 G⇥ rr
Perturbation analysis in expanding space-time
2 a˙ c2 2 T¯ +2 =4 G ¯ + s 2 + 2S t2 a t a2 3 a2
Fourier Transform “Master equation”
d2 a˙ d k2c2 2 T¯ k +2 k = 4 G ¯ s k2 S dt2 a dt a2 k 3 a2 k
3 isentropic perturbations r =(4/3) m S = r m 4 isocurvature perturbations / = (a/a ) r m eq
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 5
Newtonian Perturbation Theory II. Baryonic Perturbations Summary: key words & important facts Key words Jeans criterion Linear growth rate Jeans length Silk damping Horizons (particle vs. event) Radiation drag
Perturbations below the Jeans mass do not grow, but cause acoustic oscillations.
At recombination photons decouple from baryons huge drop in the Jeans mass.
Hubble drag resists perturbation growth perturbations above the Jeans mass do not grow exponentially, but as a power-law: (t) ta k The index a depends on cosmology and EoS, as characterized by linear growth rate.
Growth of super-horizon density-perturbations is governed by conservation of the associated perturbations in the metric.
If matter is purely baryonic, at recombination Silk damping has erased all perturbations on 15 relevant scales (Md ~ 10 M⦿) structure formation proceeds in top-down fashion.
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
1/2 c 3 b(t) prior to recombination: relativistic photon-baryon fluid cs = +1 3 4 r(t)
after recombination: baryon fluid is `ideal gas’ c =( P/ )1/2 T 1/2 s
4 3 Jeans length & mass: prop = c M = ¯ J = ¯ 3 J s G ¯ J 3 2 6 J
t a radiation era c dt c da prop 2 ct comoving particle horizon: H(a)= = = a a a˙ H 3 ct matter era 0 0
Poisson equation (Fourier space) is constant implies that 2 1 k k (¯ a ) k2 =4 Ga2 ¯ k k
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 6
Newtonian Perturbation Theory III. Dark Matter Summary: key words & important facts Key words Thermal vs. Non-thernal relics Freeze-out Cold vs. Hot relics (CDM vs. HDM) Meszaros effect Collisionless Boltzmann equation Free-streaming damping Jeans equations ISW effect
A collisionless fluid with isotropic and homogeneous velocity dispersion is described by the same continuity and momentum equations as a collisional fluid, but with the 2 1/2 sound speed c s replaced by = v i A collisionless fluid does not have an EoS moment equations are not a closed set
Collisionless dark matter and baryonic matter have the same linear growth rate.
Collisional fluid: perturbations below Jeans mass undergo acoustic oscillations Collisionless fluid: perturbations below Jeans mass undergo free streaming
After recombination, baryons fall in DM potential wells, thereby un-doing Silk damping.
The integrated Sachs-Wolfe effect probes (linear) growth rate of structure. In an EdS cosmology D ( a ) a and the ISW effect is absent. ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
df f f f Collisionless Boltzmann Equation (CBE) = + vi =0 dt t xi xi vi
k Moment equations: multiply all terms by v i and integrate over all of velocity space
1 k =0 Continuity equation + [(1 + ) v ]=0 t a x j j j v a˙ 1 v 1 1 2 k =1 Jeans equations i + v + v i = ij t a i a j x a x ¯a(1 + ) x j j i j j
teq com v(t ) Free-streaming scale = dt fs a(t ) 0
EdS cosmology D(a) a Linear growth rate ΛCDM cosmology D(a) a ( < 1)
Poisson equation in Fourier space: k2 =4 G ¯a2 k k 3 k D(a)/a In matter dominated Universe: ¯ a ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 7
The Transfer Function & Cosmic Microwave Background Summary: key words & important facts Key words ergodic principle Power spectrum Gaussian random field recombination vs. decoupling two-point correlation function last scattering surface Harrison-Zeldovic spectrum diffusion damping
The power-spectrum is the Fourier Transform of the two-point correlation function
A Gaussian random field is completely specified (in statistical sense) by the power-spectrum. The phases of all modes are independent and random.
CMB dipole reflects our motion wrt last scattering surface (lss)
Location of first peak in CMB power spectrum curvature of Universe Ratio of first to second peak in CMB power spectrum baryon-to-dark matter ratio
Finite thickness of lss causes diffusion damping of CMB perturbations
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
ergodic principle: ensemble average = spatial average
first = ( )d = ( x )d3 x =0 moment P
1 1 Gaussian Q ( ) exp( Q) 2 i C ij j random ( , , ..., )= i,j P 1 2 N [(2 )N det( )]1/2 field = = (r ) C Cij i j 12
1 2 ( r12)= (r12) two-point correlation cosmological principle: isotropy n (r dr) function 1+ (r)= pair ± n (r dr) random ±
2 2 2 1 3 P (k, t)=Pi(k) T (k) D (t) (k) k P (k) Power 2 2 spectrum dimensionless power spectrum The transfer k(am) & T (k)= function T(k) is transfer k(ai) independent of am 4 2 function 4 k 4 k4 P (k) as long as Ω(am)≃1 2 | k, i| ,i Pi(k)= k(ai) = 2 4 = 2 4 | | 9 m,0 H0 9 m,0 H0
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function CMB Summary
Hu & Dodelson 2002 ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University CMB Summary Lecture 8 Non-linear Collapse & Virialization Summary: key words & important facts Key words Spherical/Ellipsoidal collapse Mode coupling Secondary Infall model Violent relaxation Zel’dovich approximation Phase Mixing critical overdensity Virial Theorem shell crossing Two-body relaxation
In the non-linear regime ( > 1 ) perturbation theory is no longer valid. Modes start to couple to each other, and one can no longer describe the evolution of the density field with a simple growth rate: in general, no analytic solutions exist...
Because of this mode-coupling, the density field looses its Gaussian properties, i.e., in the non-linear regime, density field cannot remain Gaussian.
Spherical Collapse (SC) model can be used to `identify’ when and where collapsed objects will appear. Ellipsoidal Collapse model improves upon SC by accounting for the impact of tides, which typically are more important for less massive objects
The Zel’dovich approximation is a Lagrangian treatment of the displacement field. It remains accurate in the quasi-linear regime, up to first shell crossing.
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key words & important facts Key words Spherical/Ellipsoidal collapse Mode coupling Secondary Infall model Violent relaxation Zel’dovich approximation Phase Mixing critical overdensity Virial Theorem shell crossing Two-body relaxation
There are four relaxation mechanisms for collisionless systems: - phase mixing - chaotic mixing - violent relaxation - Landau damping
The only way in which a particle’s energy can change in a collisionless system is by having a time-dependent potential.
Unlike collisional relaxation, violent relaxation does not cause mass segregation
Violent relaxation operates on the free-fall time, only mixes at the course-grain level of the distribution function, and is self-limiting.
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
linear non-linear = / ¯ 1 turn-around collapse SC model SC model 4.55 ∞
linear model 1.062 1.686
physicaldensity bound halo Spherical Collapse model x8 ⇢ 9 (✓ sin ✓)2 1+ = = ⇢¯ 2 (1 cos ✓)3 x5.5 x2.062 Linear theory x2.686 linear theory 2/3 WARNING background 3 2/3 t not to scale 3 lin = (6 ) density; a 20 t max aNL amax avir scale factor Virialization:
rvir = rta/2 Virial Theorem: 2K + W + =0 1+ = 18 2 178 200 vir Violent Relaxation: dE/dt = / t Zel’dovich Approximation
N D(a) two-body relaxation time: t t x (t)= x i i relax ' 10lnN cross 4 G ¯i
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 9 Press-Schechter Theory Summary: key words & important facts Key words (Extended) Press-Schechter Mass Variance Excursion Set Formalism Halo Mass Function Moving Collapse Barrier Multiplicity Function Markovian random walk Characteristic Halo Mass
Locations in linearly extrapolated density field where � > �c ≃ 1.686 correspond to collapsed objects (halos)
If �(x) is Gaussian, then so is the smoothed density field �(x;R)
Excursion sets are Markovian if, and only if, the density field is smoothed with a sharp-k space filter
The cosmological parameter σ8 is defined as the mass variance of the linearly extrapolated density field at z=0, smoothed with a Top-Hat filter of size R=8 h-1Mpc
The ellipsoidal collapse model gives rise to a moving barrier in excursion set formalism
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
3 Mass Smoothing (⇥x; R) (⇥x0) W (⇥x ⇥x0; R)d ⇥x0 ( k; R)= ( k) W (kR) ⌘ Z 1 2(M)= 2( x; R) = P (k) W 2(kR) k2 dk M = ¯R3 Mass Variance 2 2 f PS F (>M,t)=2 [ M > c(t)] (E)PS ansatz P EPS F (>M,t)=1 F (>S) S = 2(M) FU dn 1 dn ¯ F (>M,t) Halo Mass n(M,t) = = Function dM M d ln M M M
2 2 ¯ c c d ln M ¯ d ln EPS + Gaussian n(M,t)= exp = fPS( ) M 2 2 2 d ln M M 2 d ln M M M 2 /2 shorthand (t)/ (M) f ( )= 2/ e c PS 2(M) 0.615 (t) (t) 1+0.47 Ellipsoidal c c 2(t) Characteristic Mass c Collapse 2 Model 1 (M )= c(t) f ( ) f ( )=0.322 1+ f (0.84 ) PS EC (0.84 )0.6 PS ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 10
Merger Trees & Halo Bias
Summary: key words & important facts Key words Merger Tree Halo Formation time Progenitor Mass Function Halo Bias Mass Assembly History Assembly Bias
Construction of halo merger tree is subject to two conditions accurately samples progenitor mass function at all times (self-consistency) mass conservation (sum of progenitor masses = descendent mass) Different methods for constructing EPS merger trees differ in handling corresponding subtleties…
Even in the limit of infinitesimally small time-step there is a non-zero probability of having more than two progenitors binary merger tree method fails
Mass assembly histories of dark matter halos are universal, if scaled appropriately.
More massive halos assemble later, and are more strongly clustered (i.e., dbh/dM > 0)
Halos that assemble earlier are more strongly clustered than halos of the same mass that assemble later (= halo assembly bias) ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
Progenitor Mass Function: M dS n(M ,t M ,t )dM = 2 f (S , S , ) 1 dM 1 1| 2 2 1 M FU 1 1| 2 2 dM 1 1 1
Halo Bias
2 2 ⇥1 1 ⇥1 1 h(M1,z1 M0, 0)= (z1)+ 0 + 0 (z1) | 1 1
⇥2 1 in linear regime h(M1,z1 0) bh(M1,z1) (z1) with b (M,z)=1+ | h ✓ c ◆ (~x M)=b (M) (~x) h | h ⇠ (r M ,M )=b (M ) b (M ) ⇠ (r) hh | 1 2 h 1 h 2 mm
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 11
Structure of Dark Matter Halos Summary: key words & important facts Key words NFW/Einasto profile Halo Concentration Parameter Halo virial relations Halo Spin Parameter Cusp-Core controversy Linear Tidal Torque Theory
More massive haloes are less concentrated, are more aspherical, and have more substructure All these trends are mainly because more massive haloes assemble later
Both concentration and spin parameter follow log-normal distributions
The (median) spin parameter is independent of halo mass or redshift
Dark matter halos have a universal density profile, a universal angular momentum profile, and a universal assembly history
Subhalos reveal very little segregation by present-day mass, a weak segregation by accretion mass, and strong segregation by accretion redshift and retained mass fraction
Dark matter haloes acquire angular momentum in the linear regime due to tidal torques from neighboring overdensities...
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
Halo Virial Relations Subhalo Mass Function 1/3 1/3 1 Mvir vir 1/3 1 dn m rvir 163 h kpc 12 1 ⇥m,0 (1 + z) ' 10 h M 200 exp [ (m/ M)] d ln(m/M) M 1/3 1/6 Mvir vir 1/6 1/2 Vvir 163 km/s 12 1 ⇥m,0 (1 + z) ' 10 h M 200 0.9 0.1 0.3 ±
=3 s NFW (r)= concentration 2 parameter c = rvir/rs r 1+ r Halo rs rs Density Profiles 2 r d ln r Einasto (r)= 2 exp 1 = 2 r 2 d ln r r 2
2 Halo Spin Parameter Ji(t)=a (t) D˙ (t) ijkTjl Ilk J E 1/2 J = | | 0 = GM5/2 p2MVR F grav F grav
tta tta 2 ˙ Linear TTT Jvir = J(t)dt = ijk Tjl Ilk a (t) D(t)dt Z0 Z0 ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 12
Large Scale Structure Summary: key words & important facts Key words reduced/irreducible corr fnc projected correlation function Poisson sampling Redshift space distortions Wiener-Khinchin theorem Kaiser effect Limber equation Finger-of-God effect
The reduced (or irreducible) correlation functions express the part of the n-point correlation functions that cannot be obtained from lower-order correlation functions
For a Gaussian random field, all connected moments (=reduced correlation functions) of n> 2 are equal to zero (i.e., = =0 ). One can use and to test whether the density field is Gaussian or not...
If galaxy formation is a Poisson sampling of the density field, then all n-point correlation functions of the galaxy distribution are identical to those of the matter distribution This is not the case though; galaxies are biased tracers of the mass distribution
On large (linear) scales, redshift space distortions (RSDs) depend on linear growth rate. On small (non-linear) scales, RSDs reveal FoG indicative of virial motion within halos
Redder and more massive/luminous galaxies are more strongly clustered
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
n-point correlation function DD(r) r 2-pt function (discrete) (n) (r)= 1 1 2 ... n RR(r) r DD( )d n-point irreducible correlation function w( )= 1 angular 2-pt (discrete) (n) RR( )d red 1 2 ... n c 1 power spectrum P (k) V 2 = P (k)+ ⇥| k| ⇤ gg n¯ (discrete)
Limber equation projected correlation function
4 2 r dr w( )= dyy S (y) dx ( x2 + y2 2) wp(rp)= (rp,r )dr =2 (r) 0 2 2 1/2 r (r rp) p
1 1 dwp drp (r)= 2 2 1/2 ⇥ drp (r r ) Zr p
redshift space distortions 2 1 d ln D f( ) 0.6 P (s)(⇥k)= 1+ µ2 P (k) = = m m k b d ln a b b h i 1 1+ (r ,r )= [1 + (r ,r )] f(v r)dv p ⇡ lin p ⇡ 12| 12 Z 1
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 13
The Halo Model & Halo Occupation Statistics Summary: key words & important facts Key words Halo model 1-halo & 2-halo terms halo exclusion Halo Occupation Distribution (HOD) galaxy-galaxy lensing Conditional Luminosity Function (CLF)
The Halo model is an analytical model that describes dark matter density distribution in terms of its halo building blocks, under ansatz that all dark matter is partitioned over haloes.
In combination with a halo occupation model (HOD or CLF), the halo model can be used to compute galaxy-galaxy correlation function and galaxy-matter cross-correlation function. The latter is related to the excess surface density measured with galaxy-galaxy lensing.
HOD is mainly used to model clustering of luminosity threshold samples. CLF can be used to model clustering of galaxies of any luminosity (bin).
It is common to assume that satellite galaxies obey Poisson statistics, such that
ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions
halo model 1 P 1h(k)= dMM2 n(M) u˜(k M) 2 2 | | | P (k)=P 1h(k)+P 2h(k) 1 2 P 2h(k)=P lin(k) dMMb(M) n(M)˜u(k M) |
Galaxy-Galaxy lensing: tangential shear, excess surface density and galaxy-matter cross correlation
Ds (R) = (R)= ¯( CLF: the link between light and mass L 1 1 2 (L)= (L M) n(M)dM L = (L M) L dL N = (L M)dL | ⇥M | x⇥M x | Z0 Z0 ZL1 Characteristic examples of CLF and HOD for both centrals and satellites 2 1 ln(L/Lc) dL 1 log M log Mmin c(L M)dL = exp Nc M = 1 + erf | ⇤2 ⇥c ⇤2⇥c L h i 2 ⇤ ⇥ ⌅ ✓ log M ◆ L s M s 2 M if M>Mcut s(L M)dL = exp (L/Ls) dL Ns M = 1 | Ls Ls ⇤ ⌅ h i ( ⇣0if⌘ M ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 14 Galaxy Interactions Summary: key words & important facts Key words Impulse & tidal approximations dynamical friction distant encounter approximation gravitational capture tidal shock heating orbital decay tidal mass stripping negative heat capacity Gravitational encounter results in tidal distortion. If tidal distortion lags perturber, the resulting torque causes a transfer of orbital energy into internal energy of the objects involved. An impulsive encounter that results in an (internal) energy increase �E that is larger than the system’s binding energy does not necessarily result in the system’s disruption During re-virialization, following an impulsive encounter, the subject converts 2x�E from kinetic into potential energy, resulting in the system `puffing’ up. Dynamical friction is a global, rather than a local effect. Unlike hydrodynamical friction, the deceleration decreases with increasing velocity, at least at the high-velocity end. Dynamical friction is only important for subjects with a mass larger than a few percent of the host halo mass. For less massive subjects, tdf > tH Dynamical friction does not generally result in orbital circularization. More eccentric orbits decay faster. the halo bias function Summary: key equations & expressions 1 4 M 2 r2 Impulse Approximation E = v( r) 2 (r)d3 r = G2 M P S 2 | | 3 S v b4 P Tidal Radius Point masses + centrifugal force + extended mass distributions 1/3 1/3 m 1/3 m/M m(rt)/M (R0) rt = R rt = R rt 2 3 R0 R0 d ln M 2M 3+m/M 2+ R GM(R0) d ln R | 0 Impulse Approximation Chandrasekhar dynamical friction force Coulomb logarithm 2 d v S GMS v S bmax Mh F df = MS = 4 ln ( dynamical friction time scale (isothermal sphere) evolution of orbital eccentricity 2 2 1.17 ri Mh rh de de v dv tdf = = 1 ln rh MS Vc dt v d V dt " ✓ c ◆ # the halo bias function Lecture 15 Heating & Cooling Summary: key words & important facts Key words Hydro-static Equilibrium Overcooling Problem Accretion Shock Cold mode vs. Hot mode Virial Temperature Ionization equilibrium Cooling Function Photo-ionization heating Mass estimates based on the assumption of hydrostatic equilibrium need to correct for non-thermal pressure sources (turbulence, magnetic fields, cosmic rays) Gas infalling in a halo through an accretion shock is heated to the virial temperature at which the gas is in hydrostatic, virial equilibrium with the halo potential. 12 Low mass halos (Mh < 10 M☉) are predicted to experience cold mode accretion (via streams), as they can’t support an accretion shock. When ignoring photo-ionizations, it is typically assumed that the gas is in collisional ionization equilibrium (CIE) one uses CIE cooling functions In the presence of photo-ionization, the net heating/cooling rate, ( ) /n 2 , C H H is a function of both temperature and density. This arises because of the competition between photo-ionization & recombination. ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Summary: key equations & expressions Cooling Time Hydrostatic Equilibrium P (r)= gas (r) 3nkBT tcool = 2 Spherical Symmetry Ideal Gas 2 nH (T ) d GM(r) dP kB d = = P = = ( T) r dr r2 dr µmp dr Cooling Function C H= (T,n ) n2 H k T (r) r d ln d ln T H M(r)=M (r)+M (r)= B gas + gas DM µm G d ln r d ln r in presence of heating p virial temperature µm V 2 T = p V 2 3.6 105 K vir vir 2 k vir 100 km/s B Photo-ionization heating rate: = n + n + n + + H H0 H0 He0 He0 He He 1 4⇥ J( ) where ⌅ = ⇤ ( )[h h ]d i h i P P i Z i P ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University the halo bias function Lecture 16 Star Formation Summary 2 At high gas densities ( gas > 10 M pc ) conditions are such that self- shielding becomes efficient, and molecular gas forms. Various mechanisms trigger instabilities, creating GMCs supported by supersonic turbulence. Turbulent compression creates clumps and cores; the latter are Jeans unstable and collapse to form stars. Overall SFE per GMC is low 0.002 SF,GMC ⇠ 2 At low gas densities ( gas < 10 M pc ) star formation is suppressed, due to inability for gas to self-shield (i.e., form molecules), and due to reduced self-gravity, which enhances stability. Mergers and tidal interactions cause efficient transport of angular momentum out (funneling gas in). This boosts efficiency of creating GMCs, so that galaxy enters starburst phase. Energy and momentum injection due to star formation process itself is likely to be important regulator of star formation efficiency in GMCs. ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University Lecture 17 Supernova Feedback BlastwavesSummary SN feedback is essential ingredient of galaxy formation. It helps explain why overall SF efficiency is low, and is invoked to explain why galaxy formation is less efficient in lower mass haloes... Unless SN go off in hot, low density medium, almost all SN energy is radiated away during radiative phase of blastwave. Efficient SN feedback requires three-phase ISM envisioned by McKee & Ostriker (1977), with most volume (mass) being in hot (cold) phase. Radiation pressure, stellar winds, and photo-ionization seem to be crucial ingredients for paving the way for efficient SN feedback. Numerical simulations that include wide spectrum of stellar feedback processes yields mass-loading efficiencies that scale similar to momentum- driven winds. They predict multi-phase winds; winds properties depend on SFR rate; radiation pressure becomes more important in systems with higher SFRs. ASTR 610: Theory of Galaxy Formation © Frank van den Bosch, Yale University