In this section we show some images of simulation and artificial halos and discuss the stability of artificial halos. We construct three artificial halos using potential ob-tained by (a)simulation, (b)soliton plus NFW profile, and (c) self-consistent solution with parameters β = 3.4 and µ = −2.5. Figure 2.1 shows the z = 0 slices of density, real part and imaginary part of wave function of simulation halo Run03 Halo02 03.
And figure 3.19 shows corresponding slices of artificial halo construct by simulation potential. This potential is calculated form density profile of Run03 Halo02 03, which is the most massive halo we have. Finally, figure 3.20 shows artificial halo using potential of self-consistent solution.
From these figures, we can see that simulation halo is anisotropic in outer region compare with artificial halo, and both of them are isotropic in inner region. This re-sult shows evidence that distribution function should depend on angular momentum
3.3. Artificial halos 27
(a) (b)
(c) (d)
Figure 3.6: OM King model Run03 Halo01 06 time072
or it is not virialized in the outer region, while direct fitting face the challenge that is hard to determine the dependence of angular momentum, which was discussed in section 3.1.
We also shows the evolution of density profile by running simulation for about one free-fall time at virial radius. Figure 3.21 shows the evolution of simulation halo Run03 Halo02 03 and artificial halo made by simulation potential. They evolve roughly the same way, a stable outer halo with an oscillating soliton, while the density of inner part of artificial halo decrease to form a more stable configuration in very short time. This may be caused by the fact that simulation potential and parameters of artificial are not self-consistent. On the other hand, the evolution of artificial halo made by self-consistent solution is shown in figure 3.22(black curve).
The whole halo is stable and soliton also oscillate, but with a smaller amplitude compare with simulation halo. We also show the evolution of artificial halo made by
28 3.4. Time correlation function
(a) (b)
(c) (d)
Figure 3.7: OM King model Run03 Halo05 time072
soliton plus NFW potential. It relax into more stable configuration within a free-fall time at virial radius.
3.4 Time correlation function
The result of time correlation function defined in section 2.7 are shown in figure 3.23. We use self-consistent solution with β = 3.4 and µ = −2.5. r1 to r9 denote radius of shells in which we calculate correlation function. We take log spacing in this case and the largest radius r9 is about half of the virial radius. The width of the shells is 3001 virial radius, around 3 grids of our simulation box. The unit of time is 1/6 ground state period. Note that we exclude ground state when we calculate correlation function. We can see from this figure that the inner most radius r1 have a strong correlation after one hundred steps. As radius increase, the correlation after
3.4. Time correlation function 29
Figure 3.8
first drop gradually decay. This feature is caused by the fact that lower excited states are dominant at small radius, and there are no enough states to make correlation function drop to zero.
30 3.4. Time correlation function
(a)
(b)
Figure 3.9: (a)density profile of self-consistent solution(green) and simula-tion halo(blue). (b)input(red) and output(green) potential of fifth iterasimula-tion
3.4. Time correlation function 31
(a)
(b)
Figure 3.10: (a)density profile of self-consistent solution(green) and sim-ulation halo(blue). (b)input(red) and output(green) potential of fifth
iter-32 3.4. Time correlation function
(a)
(b)
Figure 3.11: (a)density profile of self-consistent solution(green) and sim-ulation halo(blue). (b)input(red) and output(green) potential of fifth
iter-3.4. Time correlation function 33
(a)
(b)
Figure 3.12: (a)density profile of self-consistent solution(green) and sim-ulation halo(blue). (b)input(red) and output(green) potential of fifth
iter-34 3.4. Time correlation function
(a)
(b)
Figure 3.13: (a)density profile of self-consistent solution(green) and sim-ulation halo(blue). (b)input(red) and output(green) potential of fifth
iter-3.4. Time correlation function 35
(a)
(b)
Figure 3.14: (a)density profile of self-consistent solution(green) and sim-ulation halo(blue). (b)input(red) and output(green) potential of fifth
iter-36 3.4. Time correlation function
(a)
(b)
Figure 3.15: (a)density profile of self-consistent solution with different β.
(b)output potential of self-consistent solution with different β.
3.4. Time correlation function 37
(a)
(b)
Figure 3.16: (a)density profile of self-consistent solution with different µ.
(b)output potential of self-consistent solution with different µ
38 3.4. Time correlation function
Figure 3.17: Distribution function fitted by β = 3.4, µ = −2.5, and Ec= 0(red) compare with direct fitting(green).
3.4. Time correlation function 39
(a)
(b)
Figure 3.18: Self-consistent solution with parameters obtained from fitting(a)density profile of self-consistent solution(green) and simulation
40 3.4. Time correlation function
(a)
(b)
(c)
3.4. Time correlation function 41
(a)
(b)
(c)
Figure 3.20: Density slice of ψDM halo
42 3.4. Time correlation function
Figure 3.21: Density profile evolves about one free-fall time at virial radius.
Here the red curve denote artificial halo construct by simulation potential, and blue curves denote simulation halo. Soliton oscillate in both cases and the outer part of halos are stable as well.
3.4. Time correlation function 43
Figure 3.22: Density profile evolves about one free-fall time at virial radius.
The Arti-Old denote the artificial halo constructed by soliton plus NFW potential. Arti-New denote artificial halo constructed by self-consistent so-lution potential with β = 3.4 and µ = −2.5. The self-consistent soso-lution halo is stable while with soliton oscillate slightly compare to simulation halo.
While soliton plus NFW halo need time to relax to a steady state.
44 3.4. Time correlation function
Figure 3.23: Time correlation function of self-consistent solution with β = 3.4 and µ = −2.5
Chapter 4 Conclusion
To conclude, we have shown that the probability distribution function of a spher-ically symmetric ψ dark matter halo which obeys Schr¨odinger-Poisson equation at late time can be describe by several classical distribution function models. More-over, we show that we can obtain self-consistent solutions of density and potential of ψDM by utilizing some technics as well as fermionic King model. The artificial halos construct by self-consistent solution is stable and similar with simulation halo for the region whose radius is smaller than one third of virial radius, while fail to represent simulation halo at large radius. We also calculate time correlation function for this halo, and find that the correlation time increase as radius increase.
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