I am reading this old paper: https://arxiv.org/pdf/1309.6637.pdf and trying to work out the units in equation 63. It gives the number density of cosmic strings in the radiation era as
$$ \frac{n(\ell,t)}{a^3(t)}\approx\frac{0.18}{t^{3/2}(\ell+\Gamma G\mu t)^{5/2}} $$
with $n$ the number density, $a$ the scale factor, $t$ coordinate time, $\ell$ the loop size, $\Gamma$ the ratio of power radiated between GWs and EM and $G\mu$ the characteristic string tension.
I expect the LHS to have units of $1/m^3$, as the scale factor is unitless. Making the substitution $t\rightarrow ct$ and $G\mu\rightarrow\frac{G\mu}{c^2}$ in order to convert $s$ to $m$ and $\frac{m^2}{s^2}$ to unitless turns the units of the RHS into a quantity with units $1/m^4$, which doesn't reconcile with the LHS.
What mistake am I making converting back to SI?
