Cosmology — The Expanding Universe

In 1929, Edwin Hubble announced a discovery that shattered the notion of a static cosmos: distant galaxies are receding, and their velocity is proportional to their distance. Written as \( v = H_0 d \), this relationship, known as Hubble’s Law, is the first quantitative evidence that the universe is expanding. The constant \( H_0 \), the Hubble constant, measures today’s expansion rate.

Hubble’s observation relied on redshift: light from distant galaxies is stretched, shifting spectral lines toward the red. This redshift is not due to galaxies moving through space like bullets, but to the expansion of space itself. Each photon’s wavelength is stretched as the fabric of the universe grows.

Redshift, denoted \( z \), is a key observable in cosmology. It is defined by \[ 1 + z = \frac{\lambda_{\text{observed}}}{\lambda_{\text{emitted}}} \]. For nearby galaxies, redshift can be approximated by the Doppler effect. But at cosmological scales, redshift reflects the stretching of space itself. The larger the redshift, the farther back in time we are looking.

Distances in an expanding universe are subtle. Astronomers distinguish between proper distance (the instantaneous separation now), comoving distance (a measure that expands with the universe), and luminosity distance (inferred from brightness). Each distance measure depends on the expansion history encoded in cosmological parameters.

General relativity describes expansion not as galaxies flying outward into pre-existing space, but as the metric itself changing. The distance between galaxies grows because the scale factor \( a(t) \) in the metric increases with time. The line element for a homogeneous and isotropic universe is written as \[ ds^2 = -c^2 dt^2 + a^2(t) \left( \frac{dr^2}{1 - k r^2} + r^2 d\Omega^2 \right), \] known as the Friedmann–Lemaître–Robertson–Walker (FLRW) metric.

Here, \( a(t) \) encodes the size of the universe as a function of time, and \( k \) determines the geometry: \( k=0 \) flat, \( k=+1 \) closed (spherical), \( k=-1 \) open (hyperbolic). The dynamics of \( a(t) \) follow from Einstein’s field equations, leading directly to the Friedmann equations. This framework connects geometry, energy density, and cosmic fate in a single formula.

The heartbeat of cosmology lies in the Friedmann equations, derived from Einstein’s field equations under the cosmological principle. They describe how the scale factor \( a(t) \) evolves with time: \[ \left( \frac{\dot{a}}{a} \right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} + \frac{\Lambda}{3}, \] \[ \frac{\ddot{a}}{a} = -\frac{4 \pi G}{3} \left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda}{3}. \] These equations weave together matter density \( \rho \), pressure \( p \), spatial curvature \( k \), and the cosmological constant \( \Lambda \).

The first equation describes the expansion rate, often written with the Hubble parameter \( H(t) = \dot{a}/a \). The second governs acceleration or deceleration. Matter slows expansion, pressure can hasten collapse, while \( \Lambda \) — associated with dark energy — drives accelerated expansion. Together, they form the dynamical DNA of the universe.

Curvature, encoded in \( k \), determines the universe’s geometry. A flat universe (\( k=0 \)) follows Euclidean rules: parallel lines never meet, and the angles of a triangle sum to 180°. A closed universe (\( k=+1 \)) resembles the surface of a sphere: parallel lines converge, triangles exceed 180°, and the cosmos is finite but unbounded. An open universe (\( k=-1 \)) is hyperbolic: parallel lines diverge, triangles sum to less than 180°, and space is infinite.

Observations of the cosmic microwave background, particularly by the Planck satellite, show the universe is flat to within a fraction of a percent. This flatness is puzzling, as even tiny deviations in the early universe would have magnified dramatically over time — a riddle that inflationary cosmology was invented to solve.

The Friedmann equations allow speculation about cosmic destiny. If matter dominated, gravity might eventually halt expansion, leading to a Big Crunch. If dark energy dominates, expansion accelerates indefinitely, resulting in a Big Freeze: galaxies drift apart, stars burn out, and entropy approaches maximum. Some models even predict a Big Rip, where dark energy grows so strong that galaxies, stars, and atoms themselves are torn apart.

Cosmology thus becomes not only a science of origins but also of destiny. By measuring present parameters — matter density, dark energy fraction, and curvature — scientists attempt to forecast the ultimate end. These scenarios reveal the universe as a dynamic story with an uncertain final chapter.