Given the component parameters, solve the first-order equation
$$
\dot a=aH_0E(a^{-1}-1),
\qquad
t-t_i=\int_{a_i}^a\frac{d\widetilde a}{\widetilde aH(\widetilde a)}.
$$
A <cosmic string network> has $w=-1/3$, so $\rho\propto a^{-2}$ and $H\propto a^{-1}$. Therefore $\dot a$ is constant and a string-dominated expanding universe has
$$
\boxed{a(t)\propto t}
$$
after choosing the Big Bang as $t=0$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 310 / 1 / b / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-310/1/b/solution