Description of the evolution of inhomogeneities on a dark matter halo with the Vlasov equation

Abstract

We use a direct numerical integration of the Vlasov equation in spherical symmetry with a background gravitational potential to determine the evolution of a collection of particles in different models of a galactic halo in order to test its stability against perturbations. Such collection is assumed to represent a dark matter inhomogeneity which is represented by a distribution function defined in phase-space. Non-trivial stationary states are obtained and determined by the virialization of the system. We describe some features of these stationary states by means of the properties of the final distribution function and final density profile. We compare our results using the different halo models and find that the NFW halo model is the most stable of them, in the sense that an inhomogeneity in this halo model requires a shorter time to virialize.

Kinetic theory

1.1 Zero angular momentum

When the motion is purely radial and the angular momentum is zero, $L_0=0$, the derivation of the particle density $\rho _f$ in physical space, the momentum density $j_f$, the mean value over momentum space, the average value of an arbitrary function g, and the total number of particles has to begin with a Dirac delta dependence of the distribution function in the angular momenta

$$\begin{aligned} f(t,r,p_r,p_\theta ,p_\varphi )= F(t,r,p_r) \; \delta (p_\theta ) \; \delta (p_\varphi /\sin \theta ) \end{aligned}$$

(A.1)

Transforming the Dirac deltas from coordinates ($p_\theta$,$p_\varphi$) to the new coordinates (L,$\psi$) yields

$$\begin{aligned} \delta (p_\theta ) \; \delta (p_\varphi / \sin \theta ) = \frac{1}{L} \; \delta (L) \; \delta (\psi ). \end{aligned}$$

(A.2)

So that the expressions for the density, current, mean value and average in the case of zero angular momentum become

$$\begin{aligned} \rho _f(t,r)= & {} \frac{1}{r^2} \int F(t,r,p_r) \; dp_r , \end{aligned}$$

(A.3)

$$\begin{aligned} j_f(t,r)= & {} \frac{1}{r^2} \int p_r \; F(t,r,p_r) \; dp_r, \end{aligned}$$

(A.4)

$$\begin{aligned} \bar{g}(t,r)= & {} \frac{1}{r^2} \int g \; F(t,r,p_r) \; dp_r, \end{aligned}$$

(A.5)

$$\begin{aligned} \langle g\rangle (t)= & {} 4 \pi \int g \; F(t,r,p_r) \; dp_r \; dr, \end{aligned}$$

(A.6)

$$\begin{aligned} N= & {} 4 \pi \int F(t,r,p_r) \; dp_r \; dr. \end{aligned}$$

(A.7)

1.2 Continuity equation

We start from the Vlasov equation written as

$$\begin{aligned} \frac{\partial f}{\partial t} + \frac{p_r}{m}\frac{\partial f}{\partial r} - \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \; \frac{\partial f}{\partial p_r} = 0, \end{aligned}$$

(A.8)

where $\varPhi _{\mathrm{eff}}(r) \equiv \varPhi (r)+ L^2/2 m r^2$ is the effective potential. Integrating over momentum space we find

$$\begin{aligned} \int \frac{\partial f}{\partial t} \; d \bar{\omega } + \int \frac{p_r}{m} \; \frac{\partial f}{\partial r} \; d \bar{\omega } - \int \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \; \frac{\partial f}{\partial p_r} \; d \bar{\omega } = 0, \end{aligned}$$

(A.9)

with $d \bar{\omega }$ the volume element given in Eq. (2.7).

For now on we will assume that the distribution function is of compact support in phase space (or decays very rapidly). The first term of the above equation is easy to simplify:

$$\begin{aligned} \int \frac{\partial f}{\partial t} \; d \bar{\omega } = \frac{\partial }{\partial t} \int f \; d \bar{\omega } = \frac{\partial \rho _f}{\partial t}, \end{aligned}$$

(A.10)

where have used the definition of the particle density $\rho _f$, Eq. (2.9). In a similar way we find for the second term:

$$\begin{aligned} \int \frac{p_r}{m} \; \frac{\partial f}{\partial r} \; d \bar{\omega }= & {} \frac{2 \pi }{m r^2} \; \int p_r \; \frac{\partial f}{\partial r} \; L \; dp_r dL \nonumber \\= & {} \frac{2 \pi }{m r^2} \; \frac{\partial }{\partial r} \int p_r f L \; dp_r dL \nonumber \\= & {} \frac{1}{m r^2} \; \frac{\partial }{\partial r} \left( r^2 j_f \right) , \end{aligned}$$

(A.11)

where we have now used the definition of the momentum density $j_f$, Eq. (2.10). Finally, the last term can be easily shown to vanish for a distribution function of compact support:

$$\begin{aligned} \int \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \; \frac{\partial f}{\partial p_r} \; d \bar{\omega }= & {} \frac{2 \pi }{r^2} \; \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \int \frac{\partial f}{\partial p_r} \; L \; dp_r dL \nonumber \\= & {} \frac{2 \pi }{r^2} \; \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \int \left( \int \frac{\partial f}{\partial p_r} \; d p_r \right) L \; dL \nonumber \\= & {} 0, \end{aligned}$$

(A.12)

where we assumed that f is zero for large values of $|p_r|$. If we now define the mass density as $\rho _m := m \rho _f$, the Vlasov equation integrated over momentum space reduces to

$$\begin{aligned} \frac{\partial \rho _m}{\partial t} + \frac{1}{r^2} \; \frac{\partial }{\partial r} \left( r^2 j_f \right) = 0, \end{aligned}$$

(A.13)

which is nothing more than the standard continuity equation in spherical symmetry. By integrating the continuity equation in physical space it can now be easily shown, by using the divergence theorem, that the number of particles N defined above in Eq. (2.12) is conserved in the sense that $\partial N / \partial t = 0$.

1.3 Virial theorem

To obtain the virial theorem, the Vlasov equation in spherical symmetry is multiplied by $r p_r$, and integrated over phase space:

$$\begin{aligned} \int r p_r \; \frac{\partial f}{\partial t} \; dV d\bar{\omega }+ & {} \frac{1}{m} \int r p_r^ 2 \; \frac{\partial f}{\partial r} \; dV d\bar{\omega } \nonumber \\- & {} \int r p_r \; \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \; \frac{\partial f}{\partial p_r} \; dV d\bar{\omega } = 0. \end{aligned}$$

(A.14)

The first term above can be rewritten as:

$$\begin{aligned} \int r p_r \; \frac{\partial f}{\partial t} \; dV d\bar{\omega } = \frac{\partial }{\partial t} \int r p_r f \; dV d \bar{\omega } = \frac{\partial }{\partial t} \langle r p_r \rangle . \end{aligned}$$

(A.15)

For the second term in Eq. (A.14) we find:

$$\begin{aligned} \frac{1}{m} \int r p_r^ 2 \; \frac{\partial f}{\partial r} \; dV d\bar{\omega }= & {} \frac{8 \pi ^2}{m} \int p_r^2 \left( r \frac{\partial f}{\partial r} \right) L \; dr dL dp_r \nonumber \\= & {} - \frac{8 \pi ^2}{m} \int p_r^2 f L \; dr dL dp_r \nonumber \\= & {} - \frac{1}{m} \; \langle p_r^2 \rangle \nonumber \\= & {} - 2 \langle K_r \rangle , \end{aligned}$$

(A.16)

where in the second line above we integrated by parts over r, using the fact that f has compact support, and where $K_r=p_r^2/2m$ is the radial kinetic energy.

For the third term of (A.14) we find:

$$\begin{aligned} \int r p_r \; \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \; \frac{\partial f}{\partial p_r} \; dV d\bar{\omega }= & {} 8 \pi ^2 \int r \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} \left( p_r \; \frac{\partial f}{\partial p_r} \right) L \; dr dL dp_r \nonumber \\= & {} - 8 \pi ^2 \int r \frac{\partial \varPhi _{\mathrm{eff}}(r)}{\partial r} f L \; dr dL dp_r \nonumber \\= & {} \langle r F_{\mathrm{eff}}(r) \rangle , \end{aligned}$$

(A.17)

where again we have integrated by parts, but now over $p_r$, and where we defined the effective force as $F_{\mathrm{eff}}(r) = - \partial \varPhi _{{\mathrm{eff}}}(r) / \partial r$.

Collecting our results we find that the integrated Vlasov equation becomes

$$\begin{aligned} \frac{\partial }{\partial t} \langle r p_r \rangle = 2\langle K_r \rangle + \langle r F_{\mathrm{eff}}(r) \rangle . \end{aligned}$$

(A.18)

The last result is known as the virial theorem. In a steady state, the virial theorem implies

$$\begin{aligned} 2 \langle K_r \rangle + \langle r F_{\mathrm{eff}}(r) \rangle = 0. \end{aligned}$$

(A.19)