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The Big Bang is the hot, dense early state from which the observable universe expanded. In classical Friedmann solutions its idealized initial boundary is often a singularity where the scale factor vanishes.

A Big Rip is a finite-future-time cosmological singularity at which the scale factor, Hubble parameter, and phantom energy density diverge.

Neglecting metric perturbations, a free massive scalar field fluctuation obeys $\delta {\chi }^{\prime \prime }+2\mathcal{H}\delta {\chi }^{\prime }-{\nabla }^2\delta \chi +a^2{m}^2\delta \chi =0$. Rescaling it by the scale factor removes the conformal-time friction and gives $v^{\prime \prime }-{\nabla }^{2}v+(a^2{m}^2-a^{\prime \prime }/a)v=0$. Growth of $v$ need not mean growth of the physical fluctuation $v/a$.

A positively curved pressureless Friedmann universe has the cycloidal parametrization $a=A(1-\cos \theta )$ and $t=B(\theta -\sin \theta )$, with $B=A/\sqrt{k}$. It expands from a Big Bang, reaches maximum size at $\theta =\pi$, and recollapses at $\theta =2\pi$.

With positive curvature, zero cosmological constant and radiation density ${\rho }_0{a}^{-4}$ in units $8\pi G=1$, $A=\sqrt{{\rho }_0/3}$. The scale factor is positive for $0<\tau <\pi$, reaches its maximum at $\pi /2$, and has a Big Crunch at conformal time $\pi$ or proper time $2A$. This complements the closed matter-dominated Friedmann solution.

A coasting fluid has pressure $p=-\rho /3$. The cosmological perfect-fluid continuity equation makes its density scale as $a^{-2}$. If it alone fills a flat expanding universe, the scale factor grows linearly in physical time. Its background density has the same scale-factor dependence as a spatial-curvature term, although the fluid does not itself create spatial curvature.

A comoving volume has fixed boundaries in comoving coordinates. Its physical volume grows as the cube of the scale factor. In a homogeneous adiabatic gas, cosmological entropy conservation makes $sV$ constant.

For conformal time $d\tau =dt/a$, the conformal Hubble parameter is $\mathcal{H}=a^{\prime }/a=aH=\dot{a}$, where primes and dots denote conformal and cosmic time derivatives respectively. Its inverse, in units $c=1$, is the comoving Hubble radius. It differs from the ordinary Hubble parameter by one factor of the scale factor.

At fixed scale factor and unchanged relativistic counts, instantaneous conversion of a relic into thermal radiation gives the displayed entropy-density enhancement. It exceeds one because the gas receives energy from outside its prior equilibrium sector. This does not contradict cosmological entropy conservation, whose source-free reversible assumptions fail during decay and thermalization.

If a nonrelativistic relic decays entirely into promptly thermalized photons, energy conservation at fixed scale factor gives ${\rho }_{\gamma }^{+}={\rho }_{\gamma }^{-}+{\rho }_{\chi }^{-}$. Since thermal photons have ${\rho }_{\gamma }\propto T_{\gamma }^4$, their temperature rises by the displayed factor while an already decoupled Cosmic neutrino background is not heated. Before decay the relic-to-photon density ratio grows as $a$.

Kinetic decoupling occurs when momentum-exchange collisions cease to maintain the temperature and velocity dispersion of a particle species in equilibrium with a surrounding bath. Afterwards, a freely moving nonrelativistic particle has momentum proportional to $a^{-1}$ and effective kinetic temperature proportional to $a^{-2}$, where $a$ is the scale factor.

In a flat matter-dominated era with $a\propto {\tau }^2$, one has $\mathcal{H}=2/\tau$ andThe growing mode is proportional to the scale factor.

The number of e-folds during an interval of expansion is the logarithmic growth of the scale factor:

A negatively curved universe with pressureless matter and no dark energy expands forever. With $a_0=1$ and $0<{\Omega }_{m,0}<1$, its scale factor is $a={\Omega }_{m,0}(\cosh \eta -1)/[2(1-{\Omega }_{m,0})]$ and its cosmic time is $t={\Omega }_{m,0}(\sinh \eta -\eta )/[2H_0(1-{\Omega }_{m,0}{)}^{3/2}]$. The parameter is conformal time scaled by $\sqrt{-K}$ and measured from the Big Bang.