Zeeman energy $q$. The temperature is set to $1\mu$K, density to $10^{14}$ cm$^{-3}$, and we use the scattering lengths for $^{87}$Rb, $a_{0}(a_{2}) = 5.39(5.31)$nm. For the homogenous gas in consideration here, this is well above the temperature for Bose-Einstein condensation. As shown in Fig. \[fig:-1\](b), at $q=0$, $\omega^{2}(k)$ is negative for small $k$, indicative of an instability.

The timescale for the onset of this instability ($t_{ins}$) is determined by the most unstable mode, which from (\[eq:8\]) occurs at $\omega^{2} = - (2 |c_{2}|n_{0})^{2}$, giving $t_{ins} \sim \frac{1}{2|c_{2}|n_{0}}$.The corresponding wave-vector is $k_{ins}\Lambda_{T} =  \frac{\sqrt{2 |c_{2}|c_{0}}n_{0}}{k_{B} T}$. Solving for where the mode frequency vanishes, we find that all wave-vectors $k$ smaller than a critical wave-vector $k_{c} =  \sqrt{2}k_{ins}$ lead to unstable modes. For achievable experimental densities ($\sim 10^{14}$cm$^{-3}$) and temperatures ($\sim 1\mu$K), this wave-length of the instability $\lambda_{c} \sim 100\Lambda_{T} = 10\mu$m, which is comparable to typical cloud sizes. The time for the onset of this instability $\sim 0.25$s. Although this timescale is easily achievable in experiment, small collisional timescales ($\sim 50$ms) at these temperatures may make the instability undetectable in experiment.

*Microwave induced instability in a ferromagnetic polar gas*:$-$ [^1] Gerbier *et al.* [@gerbier] have shown that a weak, microwave driving 