Part III: Galaxies & Cosmology | Chapter 1

Galaxy Formation

How galaxies assemble from primordial density fluctuations: the Jeans instability, hierarchical merging, disk formation, and morphological classification

Overview

Galaxies are the fundamental building blocks of the observable Universe, each containing$10^8\text{--}10^{12}$ stars, interstellar gas, dust, and dark matter. The formation and evolution of galaxies involves the interplay of gravity, gas dynamics, star formation, and feedback processes across a vast range of scales. In the standard$\Lambda\text{CDM}$ cosmological model, galaxies form through the hierarchical merging of smaller structures, with dark matter halos providing the gravitational scaffolding for baryonic matter to cool and condense.

In this chapter we derive the Jeans instability criterion for gravitational collapse, analyze the hierarchical assembly process, derive the conditions for disk formation through angular momentum conservation, and explore the Hubble morphological classification and its physical underpinnings.

1. The Jeans Instability

The fundamental question of galaxy formation is: under what conditions does a self-gravitating gas cloud collapse? The answer, first derived by James Jeans in 1902, sets the minimum mass and length scale for gravitational fragmentation.

1.1 Linear Perturbation Analysis

Consider a uniform, infinite, self-gravitating gas with density $\rho_0$, pressure $P_0$, and sound speed $c_s = \sqrt{\gamma P_0/\rho_0}$. We apply small perturbations: $\rho = \rho_0 + \rho_1$, $\mathbf{v} = \mathbf{v}_1$,$\Phi = \Phi_0 + \Phi_1$. Linearizing the continuity, Euler, and Poisson equations and assuming plane-wave solutions $\propto e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)}$:

$$\boxed{\omega^2 = c_s^2 k^2 - 4\pi G\rho_0}$$

This is the Jeans dispersion relation. For $\omega^2 > 0$, perturbations oscillate as sound waves. For $\omega^2 < 0$, perturbations grow exponentially — gravitational instability occurs.

1.2 The Jeans Length and Mass

The critical wavenumber where $\omega^2 = 0$ defines the Jeans length:

$$\boxed{\lambda_J = c_s\sqrt{\frac{\pi}{G\rho_0}}}$$

Perturbations with wavelength $\lambda > \lambda_J$ are unstable. The corresponding Jeans mass (the mass contained within a sphere of diameter $\lambda_J$) is:

$$\boxed{M_J = \frac{4}{3}\pi\rho_0\left(\frac{\lambda_J}{2}\right)^3 = \frac{\pi^{5/2}}{6}\frac{c_s^3}{G^{3/2}\rho_0^{1/2}}}$$

1.3 Jeans Mass in the Expanding Universe

In a cosmological context, the Jeans analysis must be modified for an expanding background. Before recombination ($z > 1100$), the baryonic matter is coupled to radiation and the sound speed is $c_s \approx c/\sqrt{3}$, giving:

$$M_J^{\text{pre-rec}} \sim 10^{16}\,M_\odot$$

This exceeds galaxy cluster masses, so baryonic perturbations cannot grow before recombination. After recombination, the baryons decouple from photons and the sound speed drops to the thermal value $c_s \approx 6$ km/s at $T \approx 3000$ K:

$$M_J^{\text{post-rec}} \sim 10^5\,M_\odot$$

This corresponds to globular cluster scales. The resolution of the apparent contradiction (galaxies are much more massive) lies in dark matter: CDM perturbations begin growing at matter-radiation equality ($z \approx 3400$), building potential wells into which baryons subsequently fall after recombination.

2. Hierarchical Structure Formation

In the CDM paradigm, structure forms bottom-up: smaller halos collapse first and subsequently merge to form larger ones.

2.1 The Press-Schechter Mass Function

Press and Schechter (1974) developed an analytic model for the abundance of collapsed halos. The fraction of mass in halos above mass $M$ at redshift $z$is determined by the probability that the linearly-extrapolated density field, smoothed on scale $R(M)$, exceeds the critical overdensity for collapse $\delta_c \approx 1.686$:

$$\boxed{\frac{dn}{dM} = \sqrt{\frac{2}{\pi}}\frac{\bar{\rho}}{M^2}\frac{\delta_c}{\sigma(M)}\left|\frac{d\ln\sigma}{d\ln M}\right|\exp\left(-\frac{\delta_c^2}{2\sigma^2(M)}\right)\frac{1}{D(z)}}$$

where $\sigma(M)$ is the rms density fluctuation on mass scale $M$and $D(z)$ is the linear growth factor. The exponential cutoff at high masses produces a characteristic mass $M_*$ where $\sigma(M_*) = \delta_c/D(z)$; this mass grows with time as structure assembles hierarchically.

2.2 Merger Trees

The extended Press-Schechter formalism allows construction of merger trees that trace the assembly history of a halo. The average number of progenitors of mass $M_1$that merge to form a halo of mass $M_0$ follows from the conditional probability. A Milky Way-mass halo ($\sim 10^{12}\,M_\odot$) typically has $\sim 10$significant progenitors at $z = 2$ and $\sim 100$at $z = 6$. The last major merger (mass ratio $> 1:3$) typically occurred at $z \sim 1\text{--}2$.

3. Disk Formation Through Angular Momentum Conservation

Disk galaxies like the Milky Way are rotationally supported. The origin of their angular momentum and the formation of thin disks follows from tidal torque theory and cooling.

3.1 Tidal Torque Theory

Before collapse, proto-galactic regions acquire angular momentum through tidal interactions with the surrounding matter distribution. The dimensionless spin parameter quantifies the ratio of angular momentum to that needed for rotational support:

$$\boxed{\lambda = \frac{J|E|^{1/2}}{GM^{5/2}}}$$

N-body simulations show that $\lambda$ follows a log-normal distribution with median $\bar{\lambda} \approx 0.035$ and dispersion $\sigma_{\ln\lambda} \approx 0.5$, independent of halo mass, redshift, or cosmology.

3.2 Disk Size from Angular Momentum Conservation

If the baryonic component of a halo (fraction $m_d = M_{\text{disk}}/M_{\text{halo}}$) conserves its specific angular momentum during cooling and collapse, the resulting exponential disk has scale length:

$$\boxed{R_d = \frac{1}{\sqrt{2}}\lambda\,R_{\text{vir}} \approx 5\left(\frac{\lambda}{0.035}\right)\left(\frac{R_{\text{vir}}}{200\,\text{kpc}}\right)\;\text{kpc}}$$

This remarkably simple prediction reproduces the observed sizes of disk galaxies. The Milky Way has $R_d \approx 3$ kpc, consistent with a typical spin parameter in a $\sim 200$ kpc virial radius halo.

3.3 The Angular Momentum Problem

Early numerical simulations of galaxy formation suffered from the "angular momentum catastrophe": disks formed too small because gas lost angular momentum to the dark matter during mergers. Modern simulations with improved resolution and realistic stellar feedback (supernovae, stellar winds, radiation pressure) have largely resolved this problem by preventing premature gas collapse and redistributing angular momentum.

4. Gas Cooling and the Star Formation Threshold

For gas to form a galaxy, it must be able to cool radiatively within a dynamical time. The cooling function $\Lambda(T)$ determines the rate of energy loss.

4.1 The Cooling Time

The cooling time for gas at temperature $T$ and number density $n$ is:

$$t_{\text{cool}} = \frac{3nk_BT/2}{n^2\Lambda(T)} = \frac{3k_BT}{2n\Lambda(T)}$$

For cooling to be efficient, $t_{\text{cool}} < t_{\text{dyn}} = R_{\text{vir}}/v_{\text{vir}}$. This condition defines a critical halo mass: below $\sim 10^{12}\,M_\odot$, gas cools efficiently and forms stars; above this mass, cooling is inefficient and gas remains hot (forming the hot atmospheres of galaxy groups and clusters).

4.2 The Kennicutt-Schmidt Law

Once gas has cooled and settled into a disk, star formation follows an empirical relation between the surface density of star formation and gas:

$$\boxed{\Sigma_{\text{SFR}} = A\,\Sigma_{\text{gas}}^N, \qquad N \approx 1.4}$$

where $A \approx 2.5 \times 10^{-4}\,M_\odot$ yr$^{-1}$ kpc$^{-2}$and $\Sigma_{\text{gas}}$ is measured in $M_\odot$ pc$^{-2}$. This relation, established by Robert Kennicutt (1998), reflects the gravitational collapse of gas on a dynamical time: $\Sigma_{\text{SFR}} \approx \epsilon_{\text{ff}}\Sigma_{\text{gas}}/t_{\text{ff}}$with a low efficiency $\epsilon_{\text{ff}} \sim 1\%$.

5. The Hubble Sequence and Galaxy Morphology

Edwin Hubble (1926) classified galaxies into a "tuning fork" diagram based on their visual appearance: ellipticals (E0–E7), lenticulars (S0), spirals (Sa–Sd), barred spirals (SBa–SBd), and irregulars (Irr).

5.1 Elliptical Galaxy Profiles

Elliptical galaxies follow the de Vaucouleurs (1948) $r^{1/4}$ surface brightness profile:

$$I(r) = I_e\,\exp\left\{-7.67\left[\left(\frac{r}{R_e}\right)^{1/4} - 1\right]\right\}$$

where $R_e$ is the effective (half-light) radius. More generally, the Sersic profile $I \propto \exp[-(r/R_e)^{1/n}]$ with Sersic index $n$ unifies the description: $n = 4$ for classical ellipticals, $n = 1$ (exponential) for disk galaxies.

5.2 The Fundamental Plane

Elliptical galaxies obey a tight scaling relation (the fundamental plane) connecting effective radius, surface brightness, and velocity dispersion:

$$R_e \propto \sigma^{1.24}\,I_e^{-0.82}$$

This is a tilted version of the virial theorem prediction $R_e \propto \sigma^2 I_e^{-1}$; the tilt arises from systematic variations in the mass-to-light ratio with mass.

5.3 Physical Origin of the Hubble Sequence

The morphological type is determined primarily by the merger history and the ratio of ordered (rotational) to random (dispersion-supported) kinetic energy. Gas-rich ("wet") major mergers destroy disks and produce ellipticals. Minor mergers and smooth accretion build and maintain disk structure. The fraction of elliptical galaxies increases in denser environments (the morphology-density relation), reflecting the higher merger rate in galaxy clusters.

Applications

The Milky Way's Assembly History

The Gaia space mission and ground-based spectroscopic surveys have revealed that the Milky Way assembled through a series of mergers. The largest was the Gaia-Sausage-Enceladus merger at $z \sim 2$ ($\sim 10$ Gyr ago), which contributed a substantial fraction of the inner stellar halo. The ongoing accretion of the Sagittarius dwarf spheroidal galaxy demonstrates that hierarchical assembly continues to the present day.

Galaxy Formation Simulations

State-of-the-art cosmological hydrodynamical simulations (IllustrisTNG, EAGLE, FIRE, SIMBA) self-consistently model gas dynamics, star formation, stellar feedback, and black hole growth in cosmological volumes. These simulations reproduce many observed galaxy properties including the galaxy stellar mass function, size-mass relation, morphological mix, and quenching of star formation in massive galaxies.

Galaxy Quenching Mechanisms

The cessation of star formation ("quenching") is one of the central puzzles in galaxy evolution. Massive galaxies ($M_* > 10^{11}\,M_\odot$) are predominantly quenched (red and dead), while lower-mass galaxies are star-forming (blue). Two main quenching channels are identified: (1) "mass quenching" at high stellar mass, driven by AGN feedback that heats or expels the gas supply; and (2) "environmental quenching" in dense environments (clusters), caused by ram pressure stripping of the gas disk as galaxies move through the hot intracluster medium, and strangulation (removal of the hot gas halo that feeds future star formation). The transition zone between the blue cloud and red sequence is the "green valley," where galaxies are actively quenching.

High-Redshift Galaxy Discovery with JWST

The James Webb Space Telescope has revolutionized our view of the earliest galaxies. Within its first year, JWST discovered candidate galaxies at redshifts $z \sim 12\text{--}16$, just 250–350 Myr after the Big Bang. These early galaxies are more luminous and numerous than predicted by pre-JWST models, suggesting that star formation was more efficient in the early Universe. Some massive quenched galaxies have been found at$z \sim 4\text{--}5$, implying rapid early assembly and early quenching mechanisms. These discoveries are reshaping our understanding of the timeline and physics of galaxy formation.

Historical Notes

The "Great Debate" of 1920 between Harlow Shapley and Heber Curtis centered on whether "spiral nebulae" were within the Milky Way or separate "island universes." Edwin Hubble resolved the debate in 1924 by identifying Cepheid variables in the Andromeda Nebula (M31), establishing its distance as far beyond the Milky Way. The modern theory of galaxy formation within the CDM framework was developed in the 1970s–1980s by White and Rees (1978), Fall and Efstathiou (1980), and Blumenthal et al. (1984). The Press-Schechter formalism (1974) provided the first analytic model for the halo mass function. The discovery that galaxies at high redshift are smaller, more irregular, and more actively star-forming than local galaxies confirmed the hierarchical assembly picture.

The Tully-Fisher relation (1977), connecting the luminosity of spiral galaxies to their rotation velocity ($L \propto V_{\text{rot}}^4$), became a key distance indicator and a fundamental scaling law reflecting the connection between baryonic and dark matter. The discovery of the bimodal color distribution of galaxies (the red sequence and blue cloud) by Strateva et al. (2001) using SDSS data revealed that galaxy evolution proceeds through a fundamental transformation from star-forming to quiescent states.

The era of precision galaxy formation simulations began with the Millennium Simulation (Springel et al., 2005), which followed $10^{10}$ dark matter particles in a (500 Mpc)$^3$ volume. Modern cosmological hydrodynamical simulations (IllustrisTNG, EAGLE, FIRE, SIMBA) self-consistently model gas dynamics, star formation, stellar and AGN feedback, and chemical enrichment, successfully reproducing the galaxy stellar mass function, size-mass relation, star formation main sequence, and morphological mix. The Gaia space mission (launched 2013) has provided proper motions and parallaxes for $\sim 2$ billion stars in the Milky Way, revealing the detailed kinematic structure and assembly history of our own galaxy with unprecedented precision. The combination of large spectroscopic surveys (SDSS, DESI, 4MOST, WEAVE) with photometric surveys (LSST, Euclid) will map the galaxy population across cosmic time from $z = 0$ to $z \sim 10$.

The Galaxy Stellar Mass Function

The galaxy stellar mass function (GSMF) describes the number density of galaxies as a function of their stellar mass. It is well described by a Schechter function:

$$\phi(M_*)\,dM_* = \phi_*\left(\frac{M_*}{M_*^c}\right)^\alpha\exp\left(-\frac{M_*}{M_*^c}\right)\frac{dM_*}{M_*^c}$$

with characteristic mass $M_*^c \approx 5 \times 10^{10}\,M_\odot$, faint-end slope $\alpha \approx -1.3$, and normalization $\phi_* \approx 10^{-2.8}$ Mpc$^{-3}$ dex$^{-1}$. The exponential cutoff at high masses reflects the efficiency of AGN feedback in suppressing star formation in the most massive halos. The power-law rise at low masses is shallower than the dark matter halo mass function, indicating that star formation efficiency decreases in low-mass halos due to supernova feedback and UV background heating.

The stellar-to-halo mass ratio $M_*/M_{\text{halo}}$ peaks at$\sim 20\%$ of the cosmic baryon fraction for halos of mass $\sim 10^{12}\,M_\odot$ (Milky Way-mass). This means galaxy formation is maximally efficient (but still only $\sim 3\%$ of baryons converted to stars) at precisely the halo mass where both supernova feedback (effective at lower masses) and AGN feedback (effective at higher masses) are least effective. This "abundance matching" technique powerfully constrains galaxy formation models.

Galaxy Scaling Relations

Galaxies obey a set of tight empirical scaling relations that constrain formation models:

The Tully-Fisher relation (spirals):$L \propto V_{\text{rot}}^4$ or equivalently $M_* \propto V_{\text{rot}}^{3.5}$. This reflects the virial theorem and the regularity of disk galaxy structure.

The Faber-Jackson relation (ellipticals):$L \propto \sigma^4$, connecting luminosity to stellar velocity dispersion. Together with the size-luminosity relation, this defines the fundamental plane.

The size-mass relation: Disk galaxies follow $R_e \propto M_*^{0.22}$ and ellipticals $R_e \propto M_*^{0.56}$, with ellipticals being more compact at fixed mass. At higher redshifts, galaxies of the same mass are systematically smaller, with $R_e \propto (1+z)^{-1}$ for late-type galaxies and even steeper evolution for early types.

The star formation main sequence: Star-forming galaxies follow a tight relation $\text{SFR} \propto M_*^{0.7}$ with scatter of only $\sim 0.3$ dex, implying that star formation is a smooth, quasi-steady process regulated by the gas supply rate. Galaxies above this main sequence are "starbursts" (often triggered by mergers), while those below are quenching.

The mass-metallicity relation: Galaxy stellar metallicity increases with stellar mass as $12 + \log(\text{O/H}) \propto 0.3\log M_*$, reflecting the deeper gravitational potential wells of massive galaxies that retain metals against supernova- driven outflows. The relation evolves with redshift, with galaxies at fixed mass being more metal-poor at higher redshift, tracing the progressive chemical enrichment of the Universe. The scatter in this relation correlates with star formation rate (the "fundamental metallicity relation"), suggesting that gas inflow dilutes the metal content.

These scaling relations, combined with the galaxy stellar mass function and the morphological distribution, provide the key observational benchmarks that any successful theory of galaxy formation must reproduce. The success of modern cosmological simulations in matching these relations simultaneously represents one of the major triumphs of the $\Lambda\text{CDM}$ framework for structure formation.

Computational Exploration

The following simulation computes the Jeans mass as a function of cosmic epoch, the Press-Schechter halo mass function, the spin parameter distribution, and galaxy surface brightness profiles.

Jeans Instability, Halo Mass Function, and Galaxy Profiles

Python

script.py234 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt

fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a0a')
for ax in axes.flat:
    ax.set_facecolor('#0f0f0f')
    ax.tick_params(colors='white')
    ax.xaxis.label.set_color('white')
    ax.yaxis.label.set_color('white')
    ax.title.set_color('white')
    for spine in ax.spines.values():
        spine.set_color('#333')

rng = np.random.default_rng(42)

# Constants
G = 6.674e-11
k_B = 1.381e-23
m_p = 1.673e-27
M_sun = 1.989e30
Mpc = 3.086e22
H_0 = 70e3 / Mpc  # s^-1
rho_crit = 3 * H_0**2 / (8 * np.pi * G)

# --- Plot 1: Jeans mass vs redshift ---
ax1 = axes[0, 0]

z_range = np.logspace(-1, 4, 500)
T_CMB_0 = 2.725

# Before recombination (z > 1100): radiation-coupled
# Sound speed ~ c/sqrt(3), but need baryon density
# After recombination: T ~ T_CMB(z), c_s = sqrt(5kT/3mp)

M_jeans = np.zeros_like(z_range)
c_light = 3e8

for i, z in enumerate(z_range):
    T = T_CMB_0 * (1 + z)

Omega_b = 0.045
    Omega_m = 0.3
    rho_b = Omega_b * rho_crit * (1 + z)**3

if z > 1100:
        # Radiation-coupled: c_s ~ c/sqrt(3)
        # But effective Jeans for baryons in radiation field
        rho_r = 4 * 5.67e-8 * T**4 / c_light**3
        c_s = c_light / np.sqrt(3 * (1 + 3*rho_b / (4*rho_r)))
    else:
        # Post-recombination: neutral gas
        T_gas = min(T, 3000 * ((1+z)/1101)**2)  # Adiabatic cooling
        c_s = np.sqrt(5 * k_B * T_gas / (3 * m_p))

rho_total = Omega_m * rho_crit * (1 + z)**3
    lambda_J = c_s * np.sqrt(np.pi / (G * rho_total))
    M_jeans[i] = (np.pi / 6) * rho_total * lambda_J**3 / M_sun

ax1.loglog(1 + z_range, M_jeans, color='#ffa500', linewidth=3)

# Mark key epochs
ax1.axvline(x=1101, color='#ff4444', linestyle='--', linewidth=2, alpha=0.5, label='Recombination (z=1100)')
ax1.axvline(x=3401, color='#44bbff', linestyle='--', linewidth=2, alpha=0.5, label='Matter-rad equality (z=3400)')

# Mark galaxy mass scales
ax1.axhspan(1e10, 1e13, alpha=0.1, color='#ffa500', label='Galaxy masses')
ax1.axhspan(1e4, 1e6, alpha=0.1, color='#44bbff', label='Globular cluster masses')

ax1.set_xlabel('1 + z', fontsize=12)
ax1.set_ylabel('Jeans Mass (M_sun)', fontsize=12)
ax1.set_title('Jeans Mass vs Cosmic Epoch', fontsize=14, fontweight='bold')
ax1.legend(fontsize=8, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white', loc='upper right')
ax1.set_xlim(1, 2e4)
ax1.set_ylim(1e3, 1e18)
ax1.grid(True, alpha=0.15)

# --- Plot 2: Press-Schechter mass function ---
ax2 = axes[0, 1]

def sigma_M(M, sigma_8=0.8):
    """Approximate sigma(M) using power-law CDM spectrum"""
    M_8 = 4.0/3.0 * np.pi * (8.0)**3 * Omega_m * rho_crit / M_sun * (Mpc / 1e6)**3
    # sigma(M) ~ sigma_8 * (M/M_8)^{-(n+3)/6} with effective n ~ -1.5
    n_eff = -1.5
    return sigma_8 * (M / 6e14)**((n_eff + 3) / (-6))

def press_schechter(M, z, sigma_8=0.8):
    """Press-Schechter halo mass function dn/dlog10M"""
    delta_c = 1.686
    # Growth factor (approximate)
    Omega_z = 0.3 * (1+z)**3 / (0.3*(1+z)**3 + 0.7)
    D_z = (1+z)**(-1) * (Omega_z)**0.6 / (0.3**0.6)
    D_z = D_z / D_z  # Normalize... simplified
    D_z = 1.0 / (1 + z)  # Very approximate

sig = sigma_M(M, sigma_8) * D_z
    nu = delta_c / sig

# dn/dlnM
    rho_mean = 0.3 * rho_crit / M_sun * Mpc**3  # M_sun / Mpc^3
    dlnsig_dlnM = -0.4  # approximate power-law slope

dn_dlnM = np.sqrt(2/np.pi) * rho_mean / M * nu * np.abs(dlnsig_dlnM) * np.exp(-nu**2 / 2)
    return dn_dlnM * np.log(10)  # convert to dlog10M

M_range = np.logspace(8, 16, 200)  # M_sun

redshifts = [0, 1, 3, 6]
colors_z = ['#ffa500', '#ffdd00', '#44bbff', '#ff4444']

for z_val, col in zip(redshifts, colors_z):
    dn = press_schechter(M_range, z_val)
    valid = dn > 0
    ax2.loglog(M_range[valid], dn[valid], color=col, linewidth=2.5, label=f'z = {z_val}')

ax2.set_xlabel('Halo Mass (M_sun)', fontsize=12)
ax2.set_ylabel('dn/dlog10(M) (Mpc^-3)', fontsize=12)
ax2.set_title('Halo Mass Function (Press-Schechter)', fontsize=14, fontweight='bold')
ax2.legend(fontsize=10, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax2.set_xlim(1e8, 1e16)
ax2.set_ylim(1e-8, 1e2)
ax2.grid(True, alpha=0.15)

# Mark characteristic masses
ax2.axvline(x=1e12, color='#888', linestyle=':', linewidth=1)
ax2.text(1.3e12, 1e-6, 'MW-like', color='#aaa', fontsize=9, rotation=90)

# --- Plot 3: Spin parameter distribution ---
ax3 = axes[1, 0]

# Log-normal distribution with median 0.035
lambda_median = 0.035
sigma_ln = 0.5
N_halos = 10000
lambda_sample = np.exp(rng.normal(np.log(lambda_median), sigma_ln, N_halos))

# Histogram
bins = np.linspace(0, 0.15, 60)
ax3.hist(lambda_sample, bins=bins, density=True, color='#ffa500', alpha=0.7, edgecolor='#333')

# Analytic distribution
lambda_x = np.linspace(0.001, 0.15, 200)
p_lambda = 1 / (lambda_x * sigma_ln * np.sqrt(2*np.pi)) * np.exp(-(np.log(lambda_x) - np.log(lambda_median))**2 / (2*sigma_ln**2))
ax3.plot(lambda_x, p_lambda, color='#ff4444', linewidth=3, label='Log-normal fit')

# Mark median
ax3.axvline(x=lambda_median, color='#44bbff', linestyle='--', linewidth=2, label=f'Median = {lambda_median}')

# Predict disk sizes for MW halo
R_vir_MW = 200  # kpc
for lam in [0.02, 0.035, 0.06]:
    R_d = lam * R_vir_MW / np.sqrt(2)
    ax3.text(lam, max(p_lambda)*0.3, f'R_d={R_d:.1f} kpc', color='white', fontsize=8,
             ha='center', rotation=90)

ax3.set_xlabel('Spin Parameter (lambda)', fontsize=12)
ax3.set_ylabel('Probability Density', fontsize=12)
ax3.set_title('Halo Spin Parameter Distribution', fontsize=14, fontweight='bold')
ax3.legend(fontsize=10, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax3.set_xlim(0, 0.15)
ax3.grid(True, alpha=0.15)

# --- Plot 4: Galaxy surface brightness profiles ---
ax4 = axes[1, 1]

r_range = np.linspace(0.01, 5, 200)  # r/R_e

# Sersic profile
def sersic(r, n, R_e=1.0, I_e=1.0):
    b_n = 2*n - 1.0/3.0 + 4.0/(405*n)  # approximation
    return I_e * np.exp(-b_n * ((r/R_e)**(1.0/n) - 1))

profiles = [
    ('n=1 (Exponential disk)', 1, '#44bbff'),
    ('n=2 (Pseudo-bulge)', 2, '#ffa500'),
    ('n=4 (de Vaucouleurs)', 4, '#ff4444'),
    ('n=8 (cD galaxy)', 8, '#ff69b4'),
]

for label, n, col in profiles:
    I = sersic(r_range, n)
    mu = -2.5 * np.log10(I / 1.0) + 20  # surface brightness in mag/arcsec^2
    ax4.plot(r_range, mu, color=col, linewidth=2.5, label=label)

ax4.set_xlabel('r / R_e', fontsize=12)
ax4.set_ylabel('Surface Brightness (mag/arcsec^2)', fontsize=12)
ax4.set_title('Sersic Surface Brightness Profiles', fontsize=14, fontweight='bold')
ax4.legend(fontsize=9, facecolor='#1a1a1a', edgecolor='#444', labelcolor='white')
ax4.invert_yaxis()
ax4.set_xlim(0, 5)
ax4.grid(True, alpha=0.15)

# Print summary
print("=" * 65)
print("GALAXY FORMATION: KEY RESULTS")
print("=" * 65)

Omega_m = 0.3
print("\nJeans mass at key epochs:")
# Post-recombination
T_rec = 3000
c_s_rec = np.sqrt(5*k_B*T_rec/(3*m_p))
rho_rec = Omega_m * rho_crit * 1101**3
lam_J = c_s_rec * np.sqrt(np.pi/(G*rho_rec))
M_J_rec = np.pi/6 * rho_rec * lam_J**3 / M_sun
print(f"  Post-recombination (z=1100): M_J = {M_J_rec:.2e} M_sun")
print(f"  Sound speed: c_s = {c_s_rec/1e3:.1f} km/s")

print("\nPress-Schechter characteristic mass:")
print(f"  z=0: M* ~ 10^13 M_sun (galaxy clusters)")
print(f"  z=2: M* ~ 10^12 M_sun (massive galaxies)")

print("\nSpin parameter statistics:")
print(f"  Median: {lambda_median}")
print(f"  Dispersion: sigma_ln(lambda) = {sigma_ln}")

print("\nDisk sizes (R_vir = 200 kpc):")
for lam in [0.02, 0.035, 0.06]:
    R_d = lam * 200 / np.sqrt(2)
    print(f"  lambda = {lam}: R_d = {R_d:.1f} kpc")

print("\nHubble types and Sersic indices:")
types = [("Irregular", "0.5-1"), ("Spiral disk", "1"), ("S0/Pseudo-bulge", "2-3"),
         ("Elliptical", "4"), ("cD galaxy", "6-10")]
for name, n in types:
    print(f"  {name:20s}: n ~ {n}")

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight', facecolor='#0a0a0a')
print("\nPlot saved to output.png")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Chapter Summary

The Jeans instability sets the minimum mass scale for gravitational collapse. In the expanding Universe, the Jeans mass drops dramatically at recombination from$\sim 10^{16}\,M_\odot$ to $\sim 10^5\,M_\odot$, but dark matter perturbations begin growing earlier, providing the gravitational potential wells for baryonic collapse.

Structure forms hierarchically in the CDM paradigm, with the Press-Schechter formalism predicting the halo mass function. Galaxy disks form through angular momentum conservation, with the spin parameter $\lambda \approx 0.035$ predicting disk sizes of $\sim 3\text{--}5$ kpc for Milky Way-mass halos.

Galaxy morphology reflects the formation history: disk galaxies form through quiescent gas accretion, while ellipticals form through major mergers. The Sersic profile unifies the surface brightness description across the Hubble sequence.

← Gamma-Ray Bursts Active Galactic Nuclei →