14.1 Evidence for dark matter in galaxies and clusters

Galactic Evidence for Dark Matter

Dark matter can't be seen directly, but its gravitational effects show up everywhere we look. From the way stars orbit within galaxies to the bending of light around massive clusters, multiple independent lines of evidence point to a substance that outweighs visible matter roughly five to one.

Galaxy Rotation Curve Anomalies

A rotation curve plots the orbital velocity of stars and gas as a function of distance from a galaxy's center. These curves are one of the strongest pieces of evidence for dark matter.

For a galaxy where most mass is concentrated in the center (like most of the visible light suggests), Newtonian gravity predicts a Keplerian decline: orbital velocity should drop off as $v \propto r^{-1/2}$ beyond the bulk of the visible mass, just as planets farther from the Sun orbit more slowly.

That's not what we observe. Instead, rotation curves stay remarkably flat at large radii. Stars far from the galactic center orbit just as fast as those closer in. The galaxy NGC 3198 is a classic example: its rotation curve remains flat well beyond the edge of the visible disk.

What does a flat rotation curve imply? Since $v = \sqrt{\frac{GM(r)}{r}}$, a constant $v$ at increasing $r$ means $M(r)$ must keep growing linearly with radius. In other words, there's mass out there that we can't see. The ratio of total mass to visible mass grows larger and larger at greater radii, pointing to an extended dark matter halo that surrounds the visible galaxy and stretches far beyond it.

Gravitational Lensing in Clusters

General relativity predicts that mass curves spacetime, bending the path of light that passes near it. This effect, called gravitational lensing, gives us a way to "weigh" galaxy clusters independent of their visible matter.

• Strong lensing occurs near the cores of massive clusters, producing dramatic distortions: multiple images of the same background galaxy, stretched arcs, or complete Einstein rings. The cluster Abell 2218 is a textbook example, with dozens of arcs visible in Hubble images.
• Weak lensing is more subtle. Background galaxies are slightly stretched and aligned by the foreground cluster's gravity. By statistically analyzing the shapes of thousands of background galaxies, astronomers can reconstruct the cluster's total mass distribution.

Both methods consistently show that clusters contain far more mass than their visible galaxies and hot gas can account for.

The Bullet Cluster (1E 0657-56) provides some of the most direct evidence for dark matter. Two galaxy clusters collided, and the hot gas (visible in X-rays) was slowed by the collision and lagged behind. But weak lensing maps show that most of the mass passed right through, sitting ahead of the gas. This spatial separation between the visible matter (gas) and the lensing mass is very difficult to explain without dark matter. It also argues against modified gravity theories, which would predict the lensing signal to follow the visible mass.

Cosmic Structure and Dark Matter

Dark Matter in Large-Scale Structure

Dark matter doesn't just affect individual galaxies. It shaped the entire large-scale structure of the universe.

In the early universe, tiny density fluctuations existed in the matter distribution. Dark matter, because it doesn't interact with radiation, could begin gravitationally collapsing into denser regions earlier than ordinary matter could. These growing dark matter concentrations then pulled in normal matter, seeding the formation of galaxies and clusters.

The cold dark matter (CDM) model assumes dark matter particles are non-relativistic (slow-moving). This leads to hierarchical, bottom-up structure formation: small structures collapse first, then merge into progressively larger ones. This matches what we observe in the universe.

On the largest scales, dark matter forms the cosmic web, a network of dense filaments and sheets separated by vast voids. Simulations like the Millennium Simulation reproduce this web structure with striking accuracy when dark matter is included. Galaxies form along the filaments, and galaxy clusters sit at the nodes where filaments intersect.

Key features of dark matter's structural role:

• Dark matter halos surround galaxies and act as gravitational scaffolding, driving galaxy evolution through mergers and accretion of smaller halos
• Galaxy clusters are the largest gravitationally bound structures in the universe, with dark matter making up roughly 85% of their total mass (the Coma Cluster was one of the first where this mass discrepancy was noticed, by Fritz Zwicky in the 1930s)

Dark Matter Evidence from the CMB

The Cosmic Microwave Background (CMB) is relic radiation from when the universe was about 380,000 years old, at the epoch of recombination. Its tiny temperature fluctuations (on the order of $\Delta T / T \sim 10^{-5}$) encode a wealth of information about the universe's composition, including its dark matter content.

Before recombination, ordinary matter and radiation were coupled together, and competing forces created acoustic oscillations: gravity pulled matter inward toward denser regions, while radiation pressure pushed it back out. Dark matter, which doesn't interact with photons, only contributed gravitational pull without the radiation pressure counterforce. This asymmetry left a specific signature in the oscillation pattern.

The angular power spectrum of the CMB plots the amplitude of temperature fluctuations at different angular scales. The positions and relative heights of its peaks encode cosmological parameters:

• The odd-numbered peaks (1st, 3rd, etc.) correspond to compression phases of the oscillations, enhanced by gravity. More dark matter means stronger gravitational driving, which boosts these odd peaks relative to the even ones.
• The even-numbered peaks (2nd, 4th, etc.) correspond to rarefaction phases. The ratio of odd to even peak heights directly constrains the dark matter density.

Results from the Planck satellite (2018) provide the most precise CMB measurements to date. They confirm the $\Lambda$CDM model and pin down dark matter at approximately 26.8% of the universe's total energy density, compared to about 4.9% for ordinary (baryonic) matter. The remaining ~68.3% is dark energy.