Matter-Radiation Equality

What you are seeing: an expanding comoving patch whose contents change as the scale factor $a$ grows. Early on a hot photon gas dominates ( $\rho_r \propto a^{-4}$ : photons redshift and dilute fastest); past equality matter takes over ( $\rho_m \propto a^{-3}$ ) and gravity switches on, so the matter clumps into structure; finally $\rho_\Lambda$ (constant) dominates. The lower strip plots the three components on a log-log scale with $a_{eq}$ and a marker at the current $a$ .

Two crossovers are marked: matter-radiation equality at $a_{eq} = \Omega_r / \Omega_m \approx 2.9 \times 10^{-4}$ ( $z_{eq} \approx 3410$ ), and matter-Lambda equality at $a_{m\Lambda} = (\Omega_m / \Omega_\Lambda)^{1/3} \approx 0.77$ . Below $a_{eq}$ the universe is radiation-dominated; between $a_{eq}$ and $a_{m\Lambda}$ it is matter-dominated; today (just past $a_{m\Lambda}$ ) it is Lambda-dominated and accelerating.

Figure 1. Density evolution and matter-radiation equality. Method: closed-form

\rho_m \propto a^{-3}

\rho_r \propto a^{-4}

Omega_m0.315

Omega_r9.24e-5

WHAT TO TRY

Raise Omega_r: equality moves to later times (lower z_eq), since more radiation keeps the photon gas dominant longer. The crossover marker slides along the scale-factor axis.
Watch the three eras hand off: radiation (rho ~ a^-4) dilutes fastest, then matter (rho ~ a^-3) takes over and gravity switches structure on, then the constant Lambda term wins.
Track z_eq and a_eq: equality sets the scale that entered the horizon then, imprinted as the turnover in the matter power spectrum.