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for ρ$_{0}$ ≫ ρ$_{c}$ , i.e., 

With this choice α = ρ 0 ρ c ρ 2 $_{0}$− ρ 2 c is always positive. 

We look for solutions of the form $^{δ}$˜ y ( k , t ) ∼ e s α ( k ) $^{t}$, where the rates s$_{α}$ ( k ) are the hydrodynamic modes of the system. These are defined as those with decay rates (here proportional to Re [ s$_{α}$ ( k )]) that vanishes in the long wavelength limit k → 0. Modes with Re [ s$_{α}$ ( k )] < 0 decay at long times, while modes with Re [ s$_{α}$ ( k )] > 0 grow, rendering the homogeneous state linearly unstable. 

The general form of the dispersion relation of the hydro- dynamic modes is readily obtained by solving a quadratic equation and is given in Appendix A. Here we discuss some limiting cases. For wavevectors k along the direc- tion ˆ p$_{0}$ of broken symmetry, i.e., k = k$_{‖}$ and k$_{⊥}$ = 0, density and orientation fluctuations decouple and decay with rates 

Both modes are stable and propagating, albeit with dif- ferent speeds. Deep in the ordered region, P$_{0}$ ≃ 1. The 

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We discuss the hydrodynamic modes by considering some simplified cases. Further details are given in Appendix A. 

First, we consider the behavior of the system for ρ$_{0}$ ≫ ρ$_{c}$ , i.e., deep in the ordered state. The rate of decay of long wavelength fluctuations of the magnitude δP of the order parameter is controlled by A$_{22}$ ∼ 2 a$_{20}$ , which is always finite for ρ ≫ ρ$_{c}$ , away from the mean field continuous transition. In other words δP is a non- hydrodynamic variable that decays on microscopic time scales. In this regime we can then neglect fluctuations δP and simply consider the dynamics of density and director fluctuations governed by the two coupled equations 

propagation speed of density fluctuations is then simply v$_{0}$ , while the propagation speed of director fluctuations is λ$_{1}$ρ$_{0}$ ∼ v 2 $_{0}$ρ$_{0}$ . A Galilean invariant system would have λ$_{1}$ = v$_{0}$/ρ$_{0}$ and the two modes would have the same propagation speed, v$_{0}$ . The difference in the propaga- tion speed of the two modes can then be considered a signature of the violation of Galilean invariance. 

Next, we consider wavevectors k transverse to the di- rection ˆ p$_{0}$ of broken symmetry, i.e., k = k$_{⊥}$ and k$_{‖}$ = 0. In this case the equations for density and director fluctu- ations are coupled and the two hydrodynamic modes are given by 

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The mode s T + can become positive, yielding an instability, for k < k$_{c}$ , with 

provided 

The parameter λ$_{3}$ has been estimated for a few mi- croscopic models and found to be of order v 2 0 [21, 23]. Eq. (16) then identifies a value v S c ( ρ$_{0}$ ) of the self propul- sion speed above which the homogeneous polarized state 

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becomes unstable, with k$_{c}$ ∼ ( v$_{0}$ − v S c ) 1 / $^{2}$. The instabil- ity boundary v S c ( ρ$_{0}$ ) depends on microscopic parameters and is therefore model-dependent. Using the parameter values obtained for the model of self-propelled hard rods discussed in [23], where the nonlinear terms in the polar- ization equation arise from momentum-conserving colli- sions between the self-propelled rods, and summarized in Table I, we obtain 

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