The Friedmann Equations: The Math That Predicts How the Universe Expands

In 1922, a Russian mathematician and meteorologist named Alexander Friedmann did something almost nobody else dared to do: he took Einstein's own equations of general relativity and used them to show that the universe could not be standing still. It had to be expanding or contracting. Einstein initially called Friedmann's math 'suspicious,' and even added a fudge factor of his own (the cosmological constant) specifically to force his equations to describe a static universe. Seven years later, Edwin Hubble pointed a telescope at distant galaxies and found Friedmann had been right all along: the universe really is expanding.

- What the scale factor a(t) represents and why it is the main variable in cosmology - The first Friedmann equation and what each term (density, curvature, cosmological constant) contributes - How the equations predict three possible fates for the universe based on its density - Why dark energy forced a rewrite of what scientists thought the equations would show

The first Friedmann equation tracks a single quantity called the scale factor, written a(t), which represents how much the universe has stretched compared to some reference time. If a(t) doubles, every distance between galaxies that aren't gravitationally bound together has also doubled. The equation looks like this: (da/dt / a)^2 = (8*pi*G/3)*rho - kc^2/a^2 + (Lambda*c^2)/3 The left side, (da/dt / a), is called the Hubble parameter, and it describes the expansion rate at any given moment. On the right side, rho is the density of everything in the universe (matter, radiation, and dark energy combined), k describes the curvature of space (flat, open, or closed), and Lambda is the cosmological constant, the term Einstein originally added to cancel out expansion, which we now understand as dark energy.

Before dark energy was discovered in 1998, cosmologists used a simplified version of the Friedmann equation (ignoring Lambda) to predict three possible endings for the universe based on its average density compared to a 'critical density.' If the universe's density was above critical density, gravity would eventually win, expansion would reverse, and everything would collapse in a 'Big Crunch.' If density equaled exactly the critical density, the universe would expand forever, but the expansion rate would slow toward zero. If density was below critical density, the universe would expand forever and the expansion rate would keep slowing but never stop. Measurements from galaxy surveys in the 1990s kept finding that visible and dark matter together added up to only about 30% of the critical density, which was confusing: the equations seemed to need more.

In 1998, two independent teams studying distant exploding stars called Type Ia supernovae discovered something that shocked the entire field: the universe's expansion is not slowing down, it is speeding up. Plugging this observation back into the Friedmann equation meant the cosmological constant Lambda could not be zero. It had to be a real, positive term contributing roughly 68 to 70 percent of the universe's total energy density, a mysterious component now called dark energy. Combined with about 27% dark matter and less than 5% ordinary atoms, this matches the total needed to make the universe geometrically flat (k=0), which is what precise measurements of the cosmic microwave background later confirmed.