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For radiation in cosmology with zero cosmological constant, $p=\rho /3$. The Friedmann acceleration equation gives $\ddot{a}/a=-8\pi G\rho /3$, so the deceleration parameter equals the cosmological density parameter, regardless of spatial curvature. This is a relation for a pure radiation fluid, not a general identity for a mixture.

A closed universe containing only a cosmic string network can remain static. Its Friedmann acceleration equation vanishes identically because $P_s=-{\rho }_s/3$, and its Friedmann equation requires ${\rho }_s{a}^2=3K/(8\pi G)$. Once this conserved normalization is tuned, every positive radius gives a static solution. The usual Hubble-normalized cosmological density parameter is undefined at a static epoch.

A homogeneous canonical scalar field with positive scalar potential has energy density ${\rho }_{\varphi }={\dot{\varphi }}^2/2+V$ and pressure $P_{\varphi }={\dot{\varphi }}^2/2-V$. When its potential energy dominates its kinetic energy, the equation-of-state parameter approaches $-1$. If other components and spatial curvature are negligible, the Friedmann acceleration equation gives accelerating expansion. Small initial kinetic energy is not enough to ensure long cosmic inflation; the field must also remain on a sufficiently flat part of the scalar potential, as measured by the potential slow-roll parameters.

The equation of state $p=-\varrho$ makes the cosmological continuity equation imply constant energy density ${\varrho }_0$. For ${\varrho }_0>0$, put $H_{\Lambda }^2=8\pi G{\varrho }_0/3$. Solving both the Friedmann equation and Friedmann acceleration equation givesUp to shifts of cosmic time and time reversal, the positive scale factors are $A{e}^{H_{\Lambda }t}$ for $k=0$, $H_{\Lambda }^{-1}\cosh (H_{\Lambda }t)$ for $k=1$, and $H_{\Lambda }^{-1}\sinh (H_{\Lambda }t)$ on $t>0$ for $k=-1$. They are the flat, closed and open slicings of de Sitter spacetime. The open zero is a coordinate boundary with finite Ricci scalar $12H_{\Lambda }^2$ and finite Kretschmann scalar $24H_{\Lambda }^4$.

For the expanding flat branch, the cosmological event horizon has comoving radial distance ${\int }_t^{\infty }ds/a(s)=1/[H_{\Lambda }a(t)]$ and proper distance in cosmology $H_{\Lambda }^{-1}$. An event on the horizon reaches the central comoving observer only at infinite reception time; events beyond it never do.