Modifying PyUltraLight to model scalar dark matter with self-interactions

We introduce a modification of the pysiultralight code that models the dynamical evolution of ultralight axionlike scalar dark matter fields. Our modified code, pysiultralight, adds a quartic, self-interaction term to reflect the one which arises naturally in axionlike particle models. Using a particle mass of ${10}^{-22}\mathrm{eV}/{\mathrm{c}}^2$, we show that pysiultralight produces spatially oscillating solitons, exploding solitons, and collapsing solitons which prior analytic work shows will occur with attractive self-interactions. Using our code we calculate the oscillation frequency as a function of soliton mass and equilibrium radius in the presence of attractive self-interactions. We show that when the soliton mass is below the critical mass ($M_c=\frac{\sqrt{3}}{2}{M}_{\max }$) described by Chavanis [Phys. Rev. D 94, 083007 (2016)] and the initial radius is within a specific range, solitons are unstable and explode. We test the maximum mass criteria described by Chavanis [Phys. Rev. D 94, 083007 (2016)] and Chavanis and Delfini [Phys. Rev. D 84, 043532 (2011)] for a soliton to collapse when attractive self-interactions are included. We also analyze both binary soliton collisions and a soliton rotating around a central mass with attractive and repulsive self-interactions. We find that when attractive self-interactions are included, the density profiles get distorted after a binary collision. We also find that a soliton is less susceptible to tidal stripping when attractive self-interactions are included. We find that the opposite is true for repulsive self-interactions in that solitons would be more easily tidally stripped. Including self-interactions might therefore influence the survival timescales of infalling solitons.

Figure 1

This plot shows how changing the equilibrium radius changes the frequency of oscillation. The green line shows the fit of Eq. (30) scaled by the best fit dynamical time squared.

Figure 2

This plot shows changing the equilibrium radius changes the frequency of oscillation. The green line shows the fit of Eq. (33) scaled by the best fit dynamical time squared.

Figure 3

The energy components (the total energy, the energy associated with a central potential, the gravitational potential energy, and the kinetic and quantum energies) of an oscillating soliton system. Note that the total energy is conserved (to within 1 part in ${10}^4$).

Figure 4

A single soliton with a mass of 1.0 code units ($2.3\times {10}^6{M}_{\odot }$) explodes. These plots show contours of constant density. Time progresses from top to bottom, and the time under each frame is indicated in code units. In the Gaussian ansatz, $R(0)=1$ code unit. This is well below the critical mass of about 2.3 ($5.3\times {10}^6{M}_{\odot }$) code units. Here, $\kappa =-2.0$. The box length is 45 code units and the time step is 0.3.

Figure 5

Similar to Fig. 3: the energy components of a collapsing soliton system. The total energy (in blue) is under the orange line.

Figure 6

Two colliding solitons with no phase shift and no self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The total duration of the simulation is 7.6 Gyr. The two solitons have the same shape and size from when they leave the collision area and when they enter. Each soliton passes though the other unaffected.

Figure 7

Two colliding solitons with no phase shift with an attractive self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The duration for the simulation is 7.6 Gyr. $\kappa =-0.02$ or $\lambda =-9.7\times {10}^{-91}$. The inclusion of an attractive self-interaction causes the solitons to become distorted. With relatively small masses, the solitons do not collapse into a black hole, and instead merge and oscillate in size.

Figure 8

Two colliding solitons with no phase shift with a repulsive self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The duration is 7.6 Gyr. $\kappa =0.02$ or $\lambda =9.7\times {10}^{-91}$. The results are similar to that in Fig. 6 but with the resultant solitons slightly enlarged. There are also minor differences in the intermediate time steps.

Figure 9

Two colliding solitons with a phase shift of pi and no self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The total duration of the simulation is 7.6 Gyr. The effective repulsive force from the phase difference causes the solitons to decelerate and then move in opposite directions. The initial and final soliton profiles are the same.

Figure 10

Two colliding solitons with a phase shift of pi with an attractive self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The total duration is 7.6 Gyr. $\kappa =-0.02$ or $\lambda =-9.7\times {10}^{-91}$. The repulsive force caused by the phase shift is strong enough to repel the solitons, however, the attractive self-interaction still causes the solitons to oscillate in size.

Figure 11

Two colliding solitons with a phase shift of pi with a repulsive self-interaction. These plots show contours of constant density. Time progresses across each row left to right, and the time under each frame is indicated in code units. The duration is 7.6 Gyr. $\kappa =0.02$ or $\lambda =9.7\times {10}^{-91}$. The results are similar to that in Fig. 9.

Figure 12

Plot (a) shows how a soliton behaves when rotating around a central potential at different times when there are no self-interactions. Plot (b) has the same initial conditions except it has a coupling of $\kappa =-0.05$. The most noticeable difference between the two plots is that the soliton with an attractive self-interaction is less spread out by the central potential. Plot (c) has a repulsive coupling of $\kappa =0.05$. Here, we see that the soliton is more spread out than the simulations with no coupling or with an attractive coupling.