The Dark Universe: Dark Matter and Dark Energy

Weighing a galaxy

The cleanest evidence for dark matter comes from a simple balance. A star orbiting the centre of a galaxy is held in its circular path by gravity supplying the centripetal force. Set gravity equal to the centripetal requirement and solve for the orbital speed at radius r; by the shell theorem, only the mass enclosed within that radius, M(r), matters.

\frac{G\,M(r)\,m}{r^2} = \frac{m\,v^2}{r} \;\;\Longrightarrow\;\; v(r) = \sqrt{\frac{G\,M(r)}{r}}

Orbital speed at radius r depends only on the mass enclosed within r.

Predict from what we see. Beyond the bright disk, almost all the visible mass is already enclosed, so M(r) is roughly constant. Then v(r)\propto 1/\sqrt{r}: orbital speeds should fall off with radius, just as the outer planets orbit the Sun more slowly than the inner ones (a Keplerian falloff).
Compare with observation. Measured rotation curves do the opposite: the speed stays roughly flat — around 200 to 250 km/s — out to radii far beyond the last visible star. The prediction fails badly.
Invert the logic. If v is constant then M(r)=v^2 r/G \propto r: the enclosed mass must keep growing linearly with radius, even where there is no visible matter to supply it. Something invisible and extended is there.
Put in numbers. Take v\approx220 km/s out to r\approx30 kpc \approx 9\times10^{20} m. Then M=v^2 r/G \approx (2.2\times10^5)^2(9\times10^{20})/(6.7\times10^{-11})\approx 6\times10^{41} kg, a few \times10^{11} solar masses — several times more than all the visible stars and gas combined.

The conclusion is forced: galaxies are embedded in vast halos of dark matter that outweigh their stars several times over, betrayed only by the gravity they exert.

Not one clue but many

Rotation curves would be easy to dismiss if they stood alone. They do not. Gravitational lensing — the bending of background light by a foreground mass — independently weighs galaxy clusters and finds the same missing matter. The colliding Bullet Cluster shows the gravitating mass (mapped by lensing) sitting apart from the visible hot gas, exactly as collisionless dark matter should. The CMB power spectrum and the growth of cosmic structure both require dark matter to fit at all.

The accelerating universe

In 1998 two teams tracking distant Type Ia supernovae — stellar explosions bright and uniform enough to serve as standard candles — expected to measure the expansion slowing down under gravity. Instead the far supernovae were dimmer, hence more distant, than a decelerating universe allowed: the expansion is speeding up. Something with repulsive gravity is pushing space apart. We call it dark energy.

Recall the acceleration equation from Guide 2: it is pressure that decides. A component with sufficiently negative pressure produces repulsive gravity. Physicists summarise a component by its equation of state w, the ratio of its pressure to its energy density.

w = \frac{p}{\rho c^{2}}, \qquad \text{acceleration} \iff w < -\tfrac{1}{3}, \qquad \Lambda:\; w = -1

Ordinary matter has w≈0, radiation w=1/3; a cosmological constant has w=-1, comfortably driving acceleration.

The cosmological constant and the deepest puzzle

The simplest form of dark energy is a cosmological constant \Lambda — an energy inherent to the vacuum itself, with w=-1, that does not dilute as space expands (recall \rho_\Lambda=\text{const}). Einstein first introduced \Lambda for the wrong reason (to hold up a static universe) and later regretted it; the accelerating universe brought it back as the leading description of the data, and it is the '\Lambda' in \LambdaCDM.