FRIEDMANN'S EQUATIONS

In 1922 a 34-year-old Russian named published four pages in the Zeitschrift für Physik that initially believed were wrong. Friedmann was not an astronomer. He had spent the First World War flying reconnaissance and dropping bombs over Przemyśl, and afterwards built the Soviet Union's theoretical meteorology from almost nothing. He approached Einstein's 1915 field equations the way a forecaster approaches the atmosphere: as a dynamical system to be integrated forward in time.

Einstein had assumed, in his 1917 cosmological paper, that the universe was static — eternal and unchanging on the largest scales. To make a static universe hold still against its own gravity he had inserted the Cosmological Constant $\Lambda$ by hand, tuned to exactly cancel the pull of matter. Friedmann refused the assumption. He asked the field equations a simpler question: if the universe is the same everywhere and in every direction, how does it evolve? The answer was not a number. It was an equation of motion for the size of space itself.

Einstein read the paper and fired off a note to the journal claiming Friedmann had made an error. He was wrong; he had made the error. A year later, prompted by Friedmann's colleague Yuri Krutkov, Einstein published a one-sentence retraction. Friedmann died of typhoid in 1925 at age 37, never knowing that the expanding universe he had derived on paper would be confirmed by telescopes within the decade. In 1927, independently, the Belgian priest rederived the same equations, connected them to the redshifts of galaxies, and took the further step Friedmann had not: he proposed that the expansion implied a beginning — a "primeval atom" from which everything unfolded.

We will not derive these from the tensor machinery — that path runs through the FLRW Metric and the FLRW geometry. Instead, read them as accounting. The only unknown is the Scale Factor $a(t)$: a single number, normalised to $a = 1$ today, that measures how stretched space is. Galaxies sit at fixed comoving coordinates; the physical distance between any two of them is proportional to $a(t)$. The Hubble rate is the fractional growth rate $H \equiv \dot a / a$.

The first Friedmann equation is an energy balance:

$H^2 = \left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k c^2}{a^2} + \frac{\Lambda c^2}{3}$

Read it left to right: the square of the expansion rate equals the gravitating energy density $\rho$ (matter and radiation, pulling expansion down), minus a curvature term set by the spatial geometry $k \in \{-1, 0, +1\}$, plus a constant vacuum term from $\Lambda$. It is exactly the statement that the kinetic energy of expansion plus the potential energy of gravity is a conserved constant — Newtonian energy conservation, dressed in relativistic clothing, applied to a shell of the universe.

The second Friedmann equation governs the acceleration:

$\frac{\ddot a}{a} = -\frac{4\pi G}{3}\left(\rho + \frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}$

This one says whether the expansion is speeding up or slowing down. Ordinary matter and radiation ($\rho > 0$, $p \geq 0$) make $\ddot a$ negative — gravity decelerates the expansion, exactly as you would expect. The surprise is the factor of $3p$: in general relativity, pressure gravitates too. A gas of photons, with pressure $p = \rho c^2/3$, decelerates the universe harder than the same energy density of dust. And anything with sufficiently negative pressure — like the vacuum, with $p = -\rho c^2$ — can flip the sign and make the expansion accelerate.

To run the equations forward you need to know how $\rho$ changes as space stretches. Each component of the cosmic energy budget dilutes at its own rate, and that single fact controls the entire history.

Matter — galaxies, gas, dark matter, anything moving slowly compared to light — is just stuff in a box. Double the size of the box in each direction and the number density falls by $a^3$. So $\rho_m \propto a^{-3}$.

Radiation — photons and, while they are relativistic, neutrinos — dilutes the same volumetric $a^{-3}$ way, but suffers an extra factor of $a$: every wavelength is stretched by the expansion, so each photon's energy $E = hc/\lambda$ drops as $a^{-1}$. Hence $\rho_r \propto a^{-4}$.

The cosmological constant is the strangest. Its energy density does not dilute at all: $\rho_\Lambda = \text{const}$. Stretch space and you get more vacuum energy, in exact proportion to the new volume. This is what it means for $\Lambda$ to be a property of empty space itself rather than of anything in it.

$\frac{H^2}{H_0^2} = E(a)^2 = \Omega_r\,a^{-4} + \Omega_m\,a^{-3} + \Omega_k\,a^{-2} + \Omega_\Lambda$

Dividing the first Friedmann equation by today's value $H_0^2$ collapses everything into this one dimensionless function. Each $\Omega_i = \rho_i / \rho_{\text{crit}}$ is a present-day density measured against the Critical Density $\rho_{\text{crit}} = 3H_0^2 / 8\pi G$ — the density that makes space exactly flat. The curvature term carries $\Omega_k = -kc^2/(a_0^2 H_0^2)$, and the flatness identity $\Omega_m + \Omega_r + \Omega_k + \Omega_\Lambda = 1$ is just the first equation evaluated today. The function $E(a)$ is the whole of cosmic dynamics in one line.

Because each ingredient dilutes differently, the universe passes through eras in which one component dominates the others, and the long-term fate is decided by whichever wins in the end.

Run the clock backward toward $a \to 0$ and the steepest term takes over: $\Omega_r a^{-4}$ blows up fastest, so the early universe was radiation-dominated. As space expanded, radiation thinned faster than matter, and at a scale factor $a_{\text{eq}} = \Omega_r/\Omega_m \approx 3 \times 10^{-4}$ — redshift $z \approx 3400$, about 50,000 years after the Big Bang — matter took over. The matter-dominated era is when galaxies formed and the cosmic web grew. Finally, because matter keeps diluting while $\Lambda$ does not, the constant vacuum term inevitably wins: at $a \approx 0.77$ ($z \approx 0.3$, roughly four billion years ago) the universe became Λ-dominated, and it will stay that way forever.

The geometry term $\Omega_k a^{-2}$ matters most at intermediate times and, crucially, decides the fate of a universe without $\Lambda$. In the matter-only models studied for most of the twentieth century, the question "will the universe expand forever or recollapse?" was identical to "is space open, flat, or closed?" — a closed ($k = +1$, $\Omega_m > 1$) universe has enough self-gravity to halt and reverse the expansion, ending in a Big Crunch; a flat or open universe expands forever. This is why measuring the mean density of the universe, and comparing it to $\rho_{\text{crit}}$, was the central project of observational cosmology for sixty years.

The clean dichotomy — open expands, closed recollapses — was shattered in 1998. Two teams measuring distant Type Ia supernovae found the expansion was not decelerating as a matter universe must. It was accelerating. The second Friedmann equation demanded a component with $\rho + 3p/c^2 < 0$: negative pressure, vacuum energy, $\Lambda$ resurrected.

In the concordance model the universe therefore did something counterintuitive. For its first nine billion years, matter dominated and the expansion decelerated — gravity was winning, slowing everything down. Then, as matter thinned past the point where the constant $\Lambda$ term could overpower it, the deceleration smoothly turned into acceleration. The handoff — sometimes called the "cosmic jerk" — happened at $a \approx 0.6$, redshift $z \approx 0.6$, when the universe was roughly seven billion years old.

Friedmann's two equations are the backbone of modern cosmology. Every quantity that observational cosmology measures — the age of the universe, the distance to a galaxy at a given redshift, the angular size of structures on the microwave sky — is an integral of $E(a)$. Fix the density parameters $\Omega_m$, $\Omega_r$, $\Omega_\Lambda$ and you have fixed the entire expansion history; the rest is computing integrals.

The same equations that gave Friedmann an expanding universe on paper in 1922 now serve as a precision instrument. Plug in the matter density and you predict the redshift at which expansion stopped decelerating; the supernova data confirm it. Run them back through the radiation era and they set the temperature history that produced the helium of big bang nucleosynthesis and the microwave background. The Hubble rate they define is what connects redshift to recession in Hubble's law and cosmological redshift, where the modern tension between two ways of measuring $H_0$ is currently the sharpest open problem in the field.

What the equations cannot tell you is what the ingredients are. They take $\Omega_m$, $\Omega_r$, and $\Omega_\Lambda$ as inputs, and roughly 95% of those inputs — the dark matter that dominates $\Omega_m$ and the dark energy that is $\Omega_\Lambda$ — remain unidentified. Friedmann's bookkeeping is exact; the entries in the ledger are still being read. That is the subject of dark matter and dark energy, and it is where a century of cosmic accounting hands the problem back to physics.