Dark Matter & Dark Energy

For a star of mass m orbiting at radius r from a galactic centre containing enclosed mass M(r), gravitational centripetal balance gives v = √(GM(r)/r). If all mass were concentrated in the luminous disc, M(r) would be nearly constant beyond the visible edge, predicting v ∝ 1/√r (a Keplerian decline). Observations by Vera Rubin and Kent Ford showed v ≈ constant out to very large radii, implying M(r) ∝ r — mass keeps growing even where there is no visible matter. The simplest fit requires a spherical dark matter halo extending far beyond the stellar disc.

General relativity predicts that mass curves spacetime, bending light paths. The Bullet Cluster (1E 0657-558) is two galaxy clusters that passed through each other. The hot gas (detected in X-rays) slowed due to electromagnetic drag and piled up in the centre. But gravitational lensing maps show most mass concentrated around the two clusters of galaxies, not the gas. Since the galaxies themselves are mostly dark matter halos, the lensing mass 'passed through' the collision without interacting — exactly what collisionless, weakly-interacting dark matter predicts. This is one of the most direct lines of evidence.

The early universe was a hot plasma in which photons and baryons formed sound waves. When the universe cooled enough for atoms to form (recombination, ~380,000 years after the Big Bang), these waves froze. The CMB temperature map shows a characteristic angular power spectrum with peaks at specific multipoles. The relative heights of these peaks are exquisitely sensitive to the ratio of dark matter density to baryon density. Planck satellite data yield Ω_dm h² ≈ 0.120, giving dark matter density about 5× the baryon density. No modification of gravity has successfully reproduced all three observational pillars simultaneously without dark matter.

A Type Ia supernova results from a white dwarf accreting mass until it reaches the Chandrasekhar limit (~1.4 M☉) and undergoes thermonuclear runaway. The peak absolute magnitude is approximately M_B ≈ −19.3, making them excellent standard candles. The distance modulus μ = m − M gives the luminosity distance d_L. By plotting d_L against redshift z and comparing to theoretical Friedmann cosmology predictions, astronomers found supernovae at z ≈ 0.5 were ~0.25 magnitudes fainter than expected for a decelerating or coasting universe — they are farther away than expected, meaning the expansion accelerated. The simplest fit introduces a cosmological constant Λ, equivalent to a constant energy density of empty space, called dark energy.

ΛCDM (Lambda Cold Dark Matter) is the standard cosmological model. Λ is the cosmological constant representing dark energy with equation of state w = p/(ρc²) = −1 (negative pressure). CDM denotes cold dark matter — particles that were non-relativistic when they decoupled from other matter, enabling them to clump gravitationally and seed the large-scale structure we observe (galaxy clusters and cosmic web filaments). Candidates for CDM particles include WIMPs (Weakly Interacting Massive Particles) and axions, but none has been confirmed experimentally. The equation of state of dark energy is constrained by current data to w = −1.03 ± 0.03, tantalizingly close to Λ but not yet ruling out dynamical alternatives like quintessence fields.

• v_{rot} — Circular orbital speed (m s⁻¹): Tangential speed of a star or gas cloud on a circular orbit.
• G — Gravitational constant (m³ kg⁻¹ s⁻²): Universal constant, G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻².
• M_{enc}(r) — Enclosed mass (kg): Total mass contained within radius r, including dark matter halo.
• r — Orbital radius (m): Distance from the galactic centre to the orbiting object.

Problem. A galaxy's rotation curve is observed to be flat at v_rot = 220 km s⁻¹ out to a radius of r = 50 kpc (1 kpc = 3.086 × 10¹⁹ m). Estimate the enclosed dark matter mass within this radius, and compare it to the Milky Way's total stellar mass of approximately 5 × 10¹⁰ solar masses (M☉ = 1.989 × 10³⁰ kg).

Answer. Approximately 5.6 × 10¹¹ solar masses — about 11 times the visible stellar mass.

• Hubble Expansion and Redshift — The accelerating expansion evidence for dark energy comes directly from redshift–distance measurements of supernovae within the Hubble flow.
• White Dwarfs and the Chandrasekhar Limit — Type Ia supernovae used as standard candles to detect dark energy arise when white dwarfs reach the Chandrasekhar mass limit.
• Cosmic Microwave Background — CMB acoustic peak analysis provides the most precise measurements of dark matter and dark energy densities in the universe.
• General Relativity and Spacetime — Both gravitational lensing evidence for dark matter and the cosmological constant interpretation of dark energy are rooted in Einstein's field equations.