Self-Similar Cosmic String Wake Solution Below are two movies of the phase space of particles falling into the wake of a cosmic string. Each frame shows the phase space at that time of all the particles in the simulation. The initial redshift of simulation 1 is Z = 100; the initial redshift of simulation two is Z = 10000; and the final is Z ~ 6 for the first and Z ~ 2 for the second simulation. The vertical axis is velocity, the horizontal axis is position, units are CGS. (Note: 1 Mpc = 3 x 10^24 cm) The speed of the cosmic string which made the velocity perturbation is 0.5c. There are 1024 particles in the first simulation and 2048 in the second.
The upper graph in this picture shows how the particle turnaround radius evolves in time. The cyan crosses indicate the numerical data, and the yellow line is a plot of time to the two-thirds power. The lower graph shows how the velocity of particles outside the wake evolves in time. Again the cyan crosses indicate the numerical data, the yellow line is a plot of time to the negative one-third power.
Self-Similar Cosmic String Wake Solution Here is the phase space where velocity is scaled as time to the minus one-third power and the turnaround position is scaled as time to the two-thirds power. Notice that now the trajectory in phase space is fixed. This is known as a self-similar solution. Also notice that just before the end of the simulation, as particle infall ceases, the wake begins to expand.
The self-similar solution shown above predicts that the current radius of the overdensity is 0.34 Mpc for a wake formed at Z = 10000, string velocity of 0.5c and h = 0.5. That makes the width of the overdensity w = 0.68 Mpc. The streaming velocity of the matter outside the same wake is predicted to be around 34 km/s.
Hot dark matter in cosmic string wakes eventually also forms a self-similar solution. Self-similarity isn't reached for a longer time after the wake is formed than for cold dark matter due to the velocity dispersion of the dark matter, giving it an effective pressure and preventing condensation before the mean free path of the dark matter particles is below the Jean's length. Below is a movie of the phase space of hot dark matter in a planar symmetric cosmic string wake. Velocity is scaled to the Zel'dovich solution velocity, and positions are scaled to time to the two-thirds power. Disregard the values on the axes, they are random.
Note the initial velocity dispersion damp out due to the Hubble expansion as time goes on. Also, note the relatively long time it takes for the solution to go self-similar, relative to cold dark matter.
The width of the overdensity in the solution above is 0.29 Mpc for a wake formed at Z = 10000, string velocity of 0.5c and h = 0.5. This is smaller than that of the cold dark matter.
The next two selections are movies of the mass density of the wake as it evolves in time. The horizontal axis is bin number and the vertical axis is density. Units are CGS. This is the same simulation as in the above movie.
Below is a simulation of the matter density scaled to the turnaround radius. Notice that, as in the self-similar phase space movie above, the density profile remains constant (up to numerical resolution) in time. Also, at the end of the simulation, notice the change away from self-similarity due to depletion of matter (and therefore lack of infalling matter) outside the wake overdensity.
In the following plot, the streaming velocity of particles outside the overdense region is compared to the Zel'dovich approximation (which is exact in one dimension). This shows how closely the code tracks the correct solution. The ratio of the two solutions is plotted. Time is in seconds; a ratio of 1.0 means the solutions are exactly the same. Notice that the numerical solution is within 0.004 of the Zel'dovich solution.
Comparison of Streaming Velocity with Zel'dovich Solution A lower resolution (64 gridzones) simulation shows the temperature of the baryonic matter in the overdense region of the wake.
Below is a movie of results from a simulation of a string with wiggles on it. We don't resolve the wiggles, but they give an effective mass per length on the string, thus, there is an associated Newtonian gravitational attraction, which dominates the velocity kick due to the deficit angle. Notice that the string is also slow moving and thus creates a filamentary structure. Since the velocity imparted to the fluid by the gravitational force is not greater than the sound speed, an outward moving shock is created, depleting the core of the wake of baryonic matter and creating high a temperature and pressure at the core of the wake.
Note that the color axis plotted for the temperature is wrong and should be multiplied by a factor of 10^6, NOT 10^7 as is indicated in the graph. There was some bug that I couldn't figure out that caused the graphing routine to put the wrong number. The scale is correct, however.