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An isotropic, noninteracting network of domain walls with energy density proportional to $a^{-1}$ has effective equation-of-state parameter $w=-2/3$. This follows from constant-equation-of-state density scaling $\rho \propto a^{-3(1+w)}$.

Choose a reference epoch with $H_0=\dot{a}(t_0)/a_0\ne 0$ and let $x=a/a_0$. Define the critical density and each cosmological density parameter byFor a closed universe ${\Omega }_{K,0}<0$. Take the dark-energy component to be a cosmological constant, as indicated by its $\Lambda$ label. The constant-equation-of-state density scaling givesSubstitution into the Friedmann equation givesAt the reference epoch, ${\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{s,0}+{\Omega }_{w,0}+{\Omega }_{\Lambda ,0}+{\Omega }_{K,0}=1$. The Friedmann acceleration equation isStrings have ${\rho }_s+3P_s=0$ and make no direct acceleration contribution, although they affect the Friedmann constraint. The normalized density parameters are undefined at a static epoch, where $H_0=0$; the static solution must therefore be described directly with dimensional densities.

For a constant equation-of-state parameter $w=P/\rho$, the cosmological perfect-fluid continuity equation integrates to $\rho \propto a^{-3(1+w)}$. Comparing the given powers yieldsThis constant-equation-of-state density scaling describes a cosmic string network and a cosmological domain wall network. An intuitive check is that the energy in a fixed comoving cell grows in proportion to the total string length, $a$, or total wall area, $a^2$, while the physical cell volume grows as $a^3$. Thus the densities scale as $a^{-2}$ and $a^{-1}$ respectively. These effective equations of state use the noninteracting, statistically isotropic network assumptions of the problem.

The cosmological perfect-fluid continuity equation and the constant barotropic equation of state $P=w\rho$ giveOn every interval where $a$ is monotone, separation of variables yieldsHence the constant-equation-of-state density scaling iswhere $\mu$ is a constant. The same expression extends through a regular turning point by continuity.

Take a fixed comoving region, whose physical volume is $V=V_0{a}^3$. Its energy is $E=\rho V$, so the first-law relation $dE=-PdV$ givesDifferentiating with respect to cosmic time and using the Hubble parameter $H=\dot{a}/a$ yields the cosmological perfect-fluid continuity equation

For the barotropic equation of state $P=w\rho$, separation gives the constant-equation-of-state density scalingSubstituting this into the flat Friedmann equation and integrating the expanding branch gives, for $w\ne -1$,For $w=-1$, the density and Hubble parameter are constant and instead $a(t)\propto e^{Ht}$.

Conformal time is defined byAt radiation–string equality, let each component have density ${\rho }_{\mathrm{eq}}$ at $a=a_{\mathrm{eq}}$. Since radiation in cosmology has $w=1/3$ and a cosmic string network has $w=-1/3$,ConsequentlyThusOn the expanding branch, integration givesChoosing the Big Bang to occur at $\tau =0$ gives the radiation--cosmic-string Friedmann solution

Separate stress-energy conservation and the constant equation-of-state parameter $w$ give the constant-equation-of-state density scalingBecause $w<-1$, the phantom energy density grows with the scale factor, whereas the pressureless matter density decays as $a^{-3}$. Phantom energy therefore dominates at late times. With $\Lambda =0$, the Friedmann equation becomesIntegrating and absorbing the integration constant into the finite time $t^{\ast }$ givesThe exponent is negative, so $a(t)\to \infty$ as $t\to t^{\ast -}$. This is a Big Rip.

For a component with the barotropic equation of state $P=w\rho$, the cosmological perfect-fluid continuity equation isUsing $H=\dot{a}/a$ makes this a separable differential equation:Taking $a_0=1$ gives the constant-equation-of-state density scalingThe spatially flat Friedmann equation and the cosmological density parameters therefore giveAlong a radial light ray, $d\chi =dt/a$. Since the redshift-time relation is $dt=-dz/[(1+z)H]$ and $a=(1+z{)}^{-1}$,Hence the comoving radial distance is

For the barotropic equation of state $P=w\rho$, the continuity equation separates asSince $a(t_0)=1$,which is the constant-equation-of-state density scaling.

When $Q=0$, the expanding branch satisfieswhere $H_0=\dot{a}(t_0)>0$. Because $w>-1$, integration givesand hence the flat constant-equation-of-state scale factorIt vanishes atas $|w|<1$ ensures $1+w>0$.

For a component with $P=w\rho$ and constant $w$, the cosmological perfect-fluid continuity equation becomesIntegration gives the constant-equation-of-state density scalingDefine the present cosmological density parameter bySubstitution into the spatially flat Friedmann equation gives the Hubble parameter for constant-equation-of-state componentsSpatial flatness is what allows the expression to contain only the listed density components, with ${\sum }_i{\Omega }_{i,0}=1$.

The constant-equation-of-state density scaling givesFor a spatially flat universe dominated by this component, the Friedmann equation gives $H^2\propto a^{-1}$. Integrating $\dot{a}/a\propto a^{-1/2}$ yieldsThis is accelerated expansion because $w=-2/3<-1/3$.