Image by NASA Hubble Space Telescope on Unsplash
In the early 1920s, Russian Physicist Alexander Friedmann derived an equation that describe the rate of expansion of the universe. The equation came from Einstein and De Sitter’s assumption of an isotropic and homogeneous universe, together with Einstein’s Theory of General Relativity.
In the early 1910s, American Astronomer Vesto Slipher observed that distant galaxies emit redshifted light. This suggested that objects were receeding from the earth. In 1927, the cosmologist and Belgian priest Georges Lemaître derived a relation between the recessional velocity of galaxies and their proper distance, which was later derived by Edwin Hubble and became known as the Hubble-Lemaître Law. In Hubble’s paper, he also computed an important parameter, known as the Hubble constant, denoted by $H_0$. The Hubble constant does change over time with respect to the rate at which the universe expands, $R(t)$. The Hubble’s parameter is defined as $H = \frac{\dot{R}}{R}$. The Friedmann equation is
\[H^2 = \frac{8\pi G}{3}\rho - \frac{kc^2}{R^2} + \frac{\Lambda c^2}{3},\]where $G$ is the Gravitational constant originally introduced in Newton’s theory of gravity, and $c$ is the speed of light. The quantity $\rho$ represents the energy density of the universe. As the universe expands, the energy density dilutes. For a radiation-dominated universe the energy density dilutes much more quickly (proportional to $\frac{1}{R^4}$) than a matter-dominated universe (proportional to $\frac{1}{R^3}$).
The parameter $k$ is the curvature of the universe and describes it’s geometry. For example, $k=0$ corresponds to a flat universe, $k=1$ to a positively curved and $k=-1$ to a negatively curved universe. The curvature also determines how the universe expands. For instance, a universe with positve curvature is considered an closed universe. This universe would expand and then after a point would collapse back in on itself. If the universe was flat it would expand forever, but slow down over time. An open universe is a universe with negative curvature. This universe would expand linearly.
The constant $\Lambda$, known as the cosmological constant, represents the energy density of empty space itself. In 1917, Einstein included the cosmoligical constant in his field equation to counteract gravity and ensure a static universe. However, he later disregarded it, calling it his “biggest blunder”, when it was discovered the universe was in fact expanding. The constant has since gained renewed significance. As the universe expands, the vacuum energy density does not dilute, unlike the matter or radiation energy density, which in turn causes the expansion of the universe to accelerate. We refer to this vacuum energy as dark energy, simply because we cannot directy observe it.
From early observations of redshifted galaxies to the derivation of the Friedmann equation, our understanding of the universe has revealed it to be dynamic and ever-expanding. Observations tell us that the universe is remarkably close to flat. This apparent flatness may be the result of a process in the early universe called Inflation, a period of rapid expansion that drove the geometry of spacetime towards flatness. Today, the is dominated by the cosmological constant, placing us in the $\Lambda$-era of its expansion, where dark energy governs the large-scale behaviour of cosmic expansion.