This instability is associated with splay deformations of the director and with spatial gradients normal to the di- rection of mean order, suggesting that it may bear no relevance to the stripe formation observed in the numer- ics due to the fact that the particles in the stripes are always aligned along the short direction thereby retain- ing the long-wavelength nature of the splay mode. How- ever, as shown in Fig. (1), the splay instability line agrees remarkably well with the numerically observed onset of stripe formation at high density. This result, discussed further below, suggests that nonlinear pattern selection mechanisms may play an important role in stripe forma- tion. Finally, Eq. (17) shows that v S c ∼ 1 ρ 0 for ρ$_{0}$ >> ρ$_{c}$ and appears to diverge as we approach the phase tran- sition. This apparent divergence is regularized when the effect of overdamped fluctuations of the magnitude of the polar order is incorporated in the mode analysis. This is done in Appendix A, where rather than just neglecting δ ˜ P entirely, we approximate its behavior by assuming that on the time scales of interest | ∂$_{t}$δ ˜ P | ≪ | 2 a$_{20}$δ ˜ P | in Eqs. (6) and (7). We then neglect the time derivative of δ ˜ P , solve for the overdamped δ ˜ P , and use this result to eliminate it from the equations for density and director fluctuations. We find that the resulting modes still ex- hibit a splay instability as described above, but the crit- ical speed monotonically approaches the constant value v S c ( ρ$_{c}$ ) = [ π $_{2}$ρ$_{c}$ ] − 1 as ρ$_{0}$ → ρ$_{c}$ . The renormalized bound- ary v S c ( ρ$_{0}$ ) of this splay instability given in Eq.(A11) is plotted in Fig. 1 as a red dashed line. In addition, fluctu- ations in the magnitude of the polarization renormalize of the diffusion constant associated with the decay of density fluctuations. the hydrodynamic mode s L − takes the form with where we have used the microscopic parameters given in Table 1. When D$_{eff}$ < 0, density fluctuations grow in time and the ordered state is unstable. At the phase transition, i.e., for ρ$_{0}$ = ρ$_{c}$ , The condition D$_{eff}$ < 0 is satisfied for all values of v$_{0}$ and the ordered state is always unstable. Away from the transition, by considering the exact modes in (18) we find that for densities in a range ρ$_{c}$ ≤ ρ$_{0}$ ≤ ρ L c there exists a range of self propulsion speeds v L c 1 ≤ v$_{0}$ ≤ v L c 2 where the propagating density fluctuations are unstable. The lower instability boundary v L c $_{1}$( ρ$_{0}$ ) is shown in Fig. (1) as a purple dashed-dotted line. The results of the linear stability analysis are sum- marized in Fig. 2. The linear stability analysis predicts that near the mean field order-disorder transition the ho- mogeneous ordered state is destabilized at small v$_{0}$ by the growth of coupled density and polarization fluctua- tions. The instability occurs for spatial gradients along the direction of mean order, signaling the onset of spa- tial structures that are inhomogeneous in this direction, like the stripes found numerically. The wavevector k$_{c}$ of the fastest growing mode for this instability scales as k$_{c}$ ∼ ( v$_{0}$ − v$_{c}$ ) − 1 / 2 at fixed density ρ$_{0}$ and as ( ρ$_{0}$ − ρ$_{c}$ ) 1 / 2 at fixed self propulsion speed. The boundary of stabil- ity v L c $_{1}$( ρ$_{0}$ ) obtained from the linear theory vanishes at ρ$_{c}$ , in agreement with the onset of the striped phase obtained by numerical solution of the nonlinear equa- tions, as shown in Fig. (1), but grows faster with v$_{0}$ than obtained numerically. This discrepancy is likely to stem from the fact that the full nonlinear dynamics of amplitude fluctuations must be incorporated to account for the behavior in these regions. In addition, the lin- ear stability analysis predicts that the homegeneous or- dered state is again stable at large self-propulsion speed for v$_{0}$ > v L c $_{2}$( ρ$_{0}$ ). This second line is shown in Fig. 2. The numerics, however, yield a striped phase in this re- gion. Finally, the longitudinal instability only exists for ρ$_{0}$ ≤ ρ L c ≃ 1 . 1, while numerically stripes are observed at all densities above a critical velocity. Deep in the or- dered state, the linear stability analysis predicts that the homogeneous flocking state is destabilized by splay fluc- tuations of the order parameter. In this case the insta- bility is associated with spatial gradients in the direction normal to that the mean order. The wavevector of the fastest growing mode also scales as k$_{c}$ ∼ ( v$_{0}$ − v$_{c}$ ) 1 / 2 at fixed density ρ$_{0}$ and as ( ρ$_{0}$ − ρ$_{c}$ ) 1 / 2 at fixed self propul- sion speed, but it is clear that nonlinear pattern selection mechanisms must be involved to yield the formation of the observed transverse stripes (associated with spatial gradients in the longitudinal direction) in this region. On As the continuous order-disorder transition is ap- proached from above, a$_{20}$ → 0 and the separation of time scales between the decay of speed/magnitude fluc- tuations δP and the true hydrodynamic variables δρ and δp$_{⊥}$ no longer holds. To capture the physics of the system in the vicinity of the order disorder transition, we need to retain the dynamics of the ”non-hydrodynamic” variable δP and examine the three coupled equations (6). One can show that the splay instability described above for k normal to the direction of mean order survives and is qualitatively unchanged. On the other hand, for k along the direction of broken symmetry, director fluctuations δp$_{⊥}$ decouple from density and speed fluctuations and decay at the rate (13b). The coupled modes for the dy- namics of density and speed fluctuations are then given by where A$_{ij}$ are the elements of the matrix A ( k , t ) given in Eq. (7) for k = k$_{‖}$ ˆ p$_{0}$ . It is easy to see that one of the two dispersion relations describes a non-hydrodynamic mode, s L $_{+}$(0) = 2 a$_{20}$ , but with a decay rate that becomes vanishingly small for ρ$_{0}$ → ρ$_{c}$ . The other mode vanishes at k = 0. At small wavevectors the dispersion relation of s L ± = 1 2 ( A$_{11}$ + A$_{22}$ ) ± 1 2 √ ( A$_{11}$ − A$_{22}$ ) 2 + 4 A$_{21}$A$_{12}$ (18) D$_{eff}$ = D + v 2 0 4 | a$_{20}$ | − v 2 0 ( α + 1) 2 2 a$_{40}$ + πv 3 $_{0}$ρ$_{0}$ ( α + 1) 2 a$_{40}$ (20) s L $_{−}$( k ) = ikv$_{0}$P$_{0}$ ( α + 1) − D$_{eff}$ k 2 + O ( k $^{3}$) , (19) 6