Structure Formation

Look at the night sky and you see stars organized into galaxies, galaxies gathered into clusters, and clusters connected by vast filaments spanning hundreds of millions of light-years. Between these structures lie enormous voids, nearly empty of matter. This intricate pattern, the cosmic web, grew from the tiny density fluctuations imprinted by inflation, amplified over billions of years by the relentless pull of gravity. The story of structure formation is the story of how quantum fluctuations became galaxies.

Jeans Instability

The fundamental mechanism of gravitational collapse was first understood by James Jeans in 1902. Consider a uniform medium with small density perturbations $\delta = \delta\rho/\bar{\rho}$. Gravity pulls material toward overdensities, while pressure pushes it away. The competition between these forces defines a critical scale, the Jeans length:

$$\lambda_J = c_s \sqrt{\frac{\pi}{G\bar{\rho}}}$$

where $c_s$ is the sound speed and $\bar{\rho}$ is the mean density. Perturbations with wavelength $\lambda > \lambda_J$ are gravitationally unstable and grow. Those with $\lambda < \lambda_J$ oscillate as sound waves, these are the acoustic oscillations we saw in the CMB.

The corresponding Jeans mass is the mass contained within a sphere of diameter $\lambda_J$:

$$M_J = \frac{4}{3}\pi \bar{\rho} \left(\frac{\lambda_J}{2}\right)^3$$

For dark matter, which is pressureless ($c_s = 0$), the Jeans length is effectively zero: perturbations at all scales are unstable. This is why dark matter is essential for structure formation, it begins to collapse long before baryons are released from the photon-baryon plasma at recombination.

Growth of Density Perturbations

In an expanding universe, density perturbations grow, but the expansion slows the growth. In the linear regime ($\delta \ll 1$), perturbations satisfy:

$$\ddot{\delta} + 2H\dot{\delta} - 4\pi G\bar{\rho}\delta = 0$$

The $2H\dot{\delta}$ term is the "Hubble friction" from the expansion. The solutions depend on the cosmological era:

Since the CMB perturbations are $\delta \sim 10^{-5}$ at $z \sim 1100$, and the scale factor has grown by a factor of ~1100 since then, the linear growth alone gives $\delta \sim 10^{-2}$ today. Nonlinear collapse (where $\delta \gg 1$) is needed to form galaxies and clusters with densities hundreds of times the cosmic mean. This is where dark matter's early start is crucial: it had been growing since matter-radiation equality at $z \sim 3400$.

The Transfer Function and Power Spectrum

Inflation predicts a nearly scale-invariant primordial power spectrum: $P_{\text{prim}}(k) \propto k^{n_s}$ with $n_s \approx 0.965$. But different scales enter the horizon at different times and experience different growth histories. The transfer function $T(k)$ encodes these modifications:

$$P(k, z) = P_{\text{prim}}(k) \cdot T^2(k) \cdot D^2(z)$$

where $D(z)$ is the linear growth factor. Scales that enter the horizon during the radiation era ($k > k_{\text{eq}}$) have their growth suppressed relative to scales that enter during matter domination. This produces a characteristic turnover in the matter power spectrum at the scale corresponding to matter-radiation equality, $k_{\text{eq}} \sim 0.01$ Mpc$^{-1}$.

Dark Matter Halos

When a dark matter perturbation reaches $\delta \sim 1$, it decouples from the expansion and collapses. The standard analytical model is spherical collapse: an overdense sphere expands more slowly than the background, reaches a maximum radius ("turnaround"), and then collapses. In reality, the collapsing dark matter virializes (reaches dynamical equilibrium) to form a dark matter halo with density about 200 times the critical density.

The abundance of halos as a function of mass is described by the halo mass function, first derived by Press and Schechter (1974) and refined with excursion set theory:

$$\frac{dn}{dM} \propto \frac{\bar{\rho}}{M^2} \frac{\delta_c}{\sigma(M)} \exp\left(-\frac{\delta_c^2}{2\sigma^2(M)}\right) \left|\frac{d\ln\sigma}{d\ln M}\right|$$

where $\delta_c \approx 1.686$ is the linear overdensity at collapse and $\sigma(M)$ is the variance of the density field smoothed on mass scale $M$. This predicts that small halos form first and merge to form larger ones, the hierarchical "bottom-up" picture of structure formation.

Baryonic Physics

Dark matter provides the gravitational skeleton, but the visible universe is built from baryons. After recombination, baryons fall into dark matter potential wells. The subsequent evolution involves rich physics:

Gas cooling: Baryonic gas must radiate away its kinetic energy to condense. Cooling mechanisms include atomic line emission, bremsstrahlung, and molecular hydrogen cooling. The cooling rate determines which halos can form galaxies: too massive and the gas becomes too hot to cool efficiently; too small and cooling is inefficient due to low density.

Star formation: When gas reaches high enough density, it fragments and forms stars. The complex physics of turbulence, magnetic fields, and feedback makes this one of the most challenging aspects of galaxy formation to model.

Feedback: Stars return energy to the interstellar medium through supernovae, stellar winds, and radiation pressure. Active galactic nuclei (powered by supermassive black holes) inject enormous energy into their host galaxies and surrounding cluster gas. These feedback processes regulate galaxy growth and are essential for reproducing observed galaxy properties.

The Cosmic Web

N-body simulations, tracking the gravitational evolution of billions of dark matter particles, reveal that large-scale structure forms a characteristic pattern: the cosmic web.

The cosmic web has three characteristic structures:

• Clusters: The most massive gravitationally bound objects ($10^{14}$-$10^{15}$ solar masses), sitting at the intersection of filaments. They contain hundreds to thousands of galaxies bathed in hot X-ray-emitting gas.
• Filaments: Elongated structures connecting clusters, containing most of the galaxies in the universe. They span tens to hundreds of megaparsecs.
• Voids: Vast underdense regions, typically 30-100 Mpc across, containing very few galaxies. They occupy most of the volume of the universe.

Baryon Acoustic Oscillations

The same acoustic oscillations that produced the peaks in the CMB power spectrum also left an imprint in the distribution of matter. Before recombination, sound waves propagated through the photon-baryon plasma. At recombination, these waves froze, leaving a characteristic scale, the sound horizon $r_s \approx 150$ Mpc, imprinted in the matter distribution.

This appears as a slight excess in the correlation function of galaxies at a separation of $\sim 150$ Mpc, or equivalently as oscillatory features in the matter power spectrum. Baryon acoustic oscillations (BAO) provide a "standard ruler" for cosmology:

$$d_A(z) = \frac{r_s}{\theta_{\text{BAO}}(z)}$$

By measuring the BAO scale at different redshifts, we map the expansion history of the universe. BAO measurements from galaxy surveys (SDSS, BOSS, DESI, Euclid) provide some of the tightest constraints on dark energy.

N-body Simulations and Galaxy Surveys

Modern cosmological N-body simulations, such as the Millennium Simulation, IllustrisTNG, and FLAMINGO, follow the evolution of billions of dark matter particles (and in some cases, baryonic gas) from high redshift to the present. These simulations:

• Reproduce the observed cosmic web with remarkable fidelity
• Predict the abundance and clustering of dark matter halos
• When coupled with baryonic physics, reproduce galaxy luminosity functions, color distributions, and morphologies
• Serve as essential tools for interpreting large-scale structure surveys

On the observational side, galaxy redshift surveys: SDSS, 2dFGRS, BOSS, DESI, and the upcoming Euclid and LSST/Rubin surveys, map the three-dimensional distribution of millions of galaxies. By comparing the observed galaxy power spectrum, BAO signal, and cluster counts with theoretical predictions, we constrain cosmological parameters with percent-level precision.

Key Insights

• The Jeans instability governs gravitational collapse: perturbations above the Jeans length grow under gravity
• Dark matter, being pressureless, collapses at all scales, forming the gravitational scaffolding for structure
• Perturbation growth is $\delta \propto a$ in the matter era but is suppressed during radiation and dark energy domination
• The matter power spectrum $P(k)$ has a characteristic turnover at the matter-radiation equality scale
• Dark matter halos form hierarchically: small structures merge to form larger ones (bottom-up)
• The cosmic web, clusters, filaments, and voids, emerges naturally from gravitational evolution
• BAO provide a standard ruler for measuring the expansion history and constraining dark energy
• N-body simulations and galaxy surveys have made structure formation a precision science

Looking Ahead

With this lesson, we complete our survey of cosmology, from the Big Bang through inflation, the CMB, dark matter, dark energy, and the formation of cosmic structure. The standard $\Lambda$CDM model is a remarkable achievement, but many deep questions remain. In Phase 10, we turn to the frontiers of physics, beginning with the quest for quantum gravity: the ultimate challenge of unifying general relativity with quantum mechanics.