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For $0<\Omega <1$, choose the Big Bang to occur at $\tau =t=0$ and retain the expanding solution. The first integral found above isUse $a=(\Omega /b){\sinh }^{2}u$. Substitution gives $u^{\prime }=H_0\sqrt{b}/2=\alpha /2$, so $u=\alpha \tau /2$. This gives the flat matter-coasting-fluid Friedmann solution:Integrate $dt/d\tau =a$, with the same zero of time:As a check, $a^{\prime }=(\Omega \alpha /2b)\sinh (\alpha \tau )$, and the identity ${\sinh }^{2}v=(\cosh v-1)(\cosh v+1)$ verifies $a^{\prime 2}=H_0^2(\Omega a+b{a}^2)$. Also $t^{\prime }=a$ exactly. At early times $a\simeq \Omega H_0^2{\tau }^2/4$ and $t\simeq \Omega H_0^2{\tau }^3/12$, reproducing the matter-dominated relation $a\propto t^{2/3}$.

The endpoint $\Omega =1$ is obtained by taking the smooth limit: $a=H_0^2{\tau }^2/4$ and $t=H_0^2{\tau }^3/12$. At $\Omega =0$ the coasting cosmic-string universe instead has $a=H_{0}t$ and $\mathcal{H}=H_0$. Its Big Bang is at $\tau =-\infty$, so a finite conformal-time origin at the bang is no longer available; choosing $a=1$ at $\tau =0$ gives $a=e^{H_0\tau }$ and $t=H_0^{-1}{e}^{H_0\tau }$. The physical-age limit remains regular.

The prescribed cosmic string network scaling gives $\dot{\rho }=-2H\rho$. Comparing with the cosmological perfect-fluid continuity equation givesThe averaged pressure is therefore $P=-\rho /3$, provided the network is treated as the stipulated isotropic homogeneous component. The Friedmann acceleration equation gives $\ddot{a}=0$. On an expanding branch let $\rho =C/a^2$ and $B^2=8\pi GC/3-K>0$; thenThis coasting cosmic-string universe has no growing flatness instability, but it does not drive an initially nonflat universe toward flatness. The comoving Hubble radius remains constant, so there is also no shrinking-radius mechanism of the usual cosmic inflation type.

The global causal issue is more precise than saying that a constant comoving Hubble radius solves every Horizon problem. If this ideal coasting solution extends to its zero-size time $t_B$, then ${\int }_{t_B}^{t}d{t}^{\prime }/[B(t^{\prime }-t_B)]$ diverges logarithmically, so it has no finite particle horizon. If string domination starts at a finite earlier time after another cosmological phase, its causal history must instead be evaluated from that earlier phase. The string era alone does not establish that previously disconnected regions had time to equilibrate. A static branch with $B=0$ is a separate possibility; its Hubble-normalized $\Omega$ is undefined and is not covered by the expanding formulas.