FLRW symmetry and Friedmann dynamics

Choose cosmic time $t$ as proper time along comoving worldlines and a dimensionless scale factor $a(t)$. For comoving radial coordinate $\chi$ with dimensions of length, the spatial-curvature parameter $K$ has dimensions of inverse length squared. A constant rescaling of the comoving coordinate changes the normalization of the scale factor. Observable redshifts depend on ratios of scale factors, so choosing unity at a reference epoch does not select a physical size for the universe.

Spatial symmetry makes the background stress tensor take perfect-fluid form with energy density $\rho$ and isotropic pressure $P$. Cold matter has negligible background pressure; freely propagating ultrarelativistic particles have $P=\rho/3$. Neither statement requires collisions at every instant. An isotropic collisionless radiation distribution can obey the radiation equation of state after decoupling. Local departures from homogeneity belong to perturbation theory, rather than an additional term inserted arbitrarily into the background metric.

For a component that exchanges no energy with other components, covariant stress-energy conservation reduces to a continuity equation. With constant equation-of-state parameter $w=P/\rho$, divide by the density and integrate against logarithmic scale factor. Matter then dilutes as inverse volume. Radiation dilutes faster because individual particle energies redshift as well. If components exchange energy, their separate continuity equations require equal-and-opposite source terms; only the total equation retains the source-free form.

Einstein gravity supplies the Friedmann constraint and acceleration equation. Positive density sets the expansion-rate magnitude, while density plus three times pressure controls acceleration. A cosmological constant can be represented as a fluid with negative pressure equal in magnitude to its density. Include it in the total density or as an explicit cosmological-constant term, never both. Expansion does not imply acceleration: a matter universe expands while its scale-factor growth slows.

Define each density fraction by dividing its density by the instantaneous critical density $3H^2/(8\pi G)$. In a flat universe these fractions sum to unity. They are not permanent mixture weights: radiation loses importance relative to matter as expansion proceeds. A convenient calculation first evolves each density relative to one reference critical density, adds those contributions to obtain the new squared expansion rate, and only then renormalizes the fractions at the new epoch.

For a single constant-pressure-ratio fluid in a flat expanding universe, integration gives a power-law scale factor when $w>-1$. This solution chooses an origin of cosmic time and assumes that fluid dominates throughout the interval. Applying it across matter-radiation equality without matching the changing mixture produces a false time-temperature history. Similarly, a successful homogeneous solution does not prove the initial state was thermal, identify dark matter, or explain why the large-scale background was smooth.