Dark Matter and Dark Energy

Picture the planets orbiting the Sun. Mercury, close in, whips around fast; far-out Neptune crawls. That is gravity's rule: the farther you sit from the central mass, the slower you orbit. Now look at a spiral galaxy. The stars near the bright center and the stars way out at the faint edge orbit at almost the same speed. Those outer stars are moving far too fast for the amount of glowing matter we can actually see. Either gravity behaves differently out there, or there is a huge amount of mass that we simply cannot see.

Astronomers name three things: the orbital speed $v$, the distance $r$ from the center, and the mass $M(<r)$ enclosed inside that distance. Setting gravity equal to the inward pull needed to travel in a circle gives one tidy rule.

If all the mass were the visible stars (packed near the middle), then past the glowing edge $M(<r)$ stops growing and $v$ should fade like $1/\sqrt{r}$. Our own Sun, about $8$ kpc from the galactic center, orbits at roughly $220$ km/s — and that speed barely drops even out past $30$ kpc. To hold $v$ flat, $M(<r)$ has to keep rising, $M(<r)\propto r$. That hidden, ever-growing mass is the dark matter halo.

Going deeper: a flat curve needs the halo density to fall as $\rho\propto 1/r^2$, and cold-dark-matter simulations sharpen this into the NFW profile $\rho(r)=\rho_0/[(r/r_s)(1+r/r_s)^2]$. Gravitational lensing and the Bullet Cluster confirm the extra mass is non-baryonic — not ordinary protons and neutrons. Dark energy is a separate puzzle. The Friedmann acceleration equation $\frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+3p/c^2\right)+\frac{\Lambda c^2}{3}$ shows that pressure gravitates too. When the equation of state $w=p/\rho c^2$ drops below $-1/3$, the $(\rho+3p)$ term flips negative and the expansion accelerates. The value $w=-1$ is Einstein's cosmological constant $\Lambda$. The sliders map straight onto this story: Galaxy Mass and DM Halo Radius set the rotation curve, DM Fraction is how much of the mass is dark, and Eq. of state w drives the expansion.

Try this in the sim above

Drag DM Fraction down to $0$ and watch the rotation curve nose-dive (pure Keplerian fall-off); push it back to $0.85$ and it flattens into the observed shape. Switch to Accelerating Expansion and slide $w$: at $w=-1$ the universe coasts outward forever, below $-1$ it races toward a "Big Rip," and above $-1/3$ it slows down instead. Finally open Cosmic Pie to see that everything we can see — every star, planet, and person — is only about $5\%$ of the whole universe.

Astronomical observations consistently show that ~95% of the universe's energy budget is in forms that don't emit light: 27% dark matter and 68% dark energy. They are inferred from gravitational effects: rotation curves, lensing, large-scale structure, CMB anisotropies, and accelerating expansion.

Step 1 — Galaxy rotation curves

For a visible disk, $M(<r)$ saturates beyond the optical radius, predicting $v \propto r^{-1/2}$ (Keplerian fall-off). Observed: $v$ stays FLAT — implying additional mass at large radii. Vera Rubin (1970s) showed this universally.

Step 2 — Dark matter halo profile (NFW)

Navarro-Frenk-White profile from N-body simulations of cold dark matter (CDM).

Step 3 — Gravitational lensing

Lensing of distant galaxies by clusters reveals total mass — including dark matter. Bullet Cluster (2006) showed DM offset from baryonic gas — definitive evidence DM is not baryonic.

Step 4 — Dark energy and Friedmann acceleration

For $w < -1/3$, the second term dominates and $\ddot{a} > 0$ — acceleration. Type Ia supernovae (1998) measured this directly.

Step 5 — Cosmological constant interpretation

Equivalently, $\rho_\Lambda = \Lambda c^2/(8\pi G)$. Observed value: $\sim 6 \times 10^{-27}$ kg/m³.

Mapping to the simulation

Rotation mode plots $v(r)$ for visible-only and visible+DM scenarios. Lensing mode shows light deflection by mass. Expansion mode demonstrates how $w$ controls $a(t)$.