On Nonlinear Schrödinger Equation as a Model for Dark Matter

Comments on Galactic Collisions, Supermassive Black Holes and Analogue Laboratory Implementations
  • Angel ParedesEmail author
  • Humberto Michinel
Part of the Understanding Complex Systems book series (UCS)


In this chapter, we present an overview of the problem of dark matter and the scalar field dark matter model, which assumes the existence of a cosmological matter wave describing a condensate of ultralight axions. The mathematical description is in terms of a nonlinear Schrödinger-Poisson system of equations. We introduce the framework in a pedagogical way, for readers interested in nonlinear science assuming no prior knowledge of cosmology. We describe a split-step pseudospectral numerical method which is useful to compute the evolution in time of dark matter distributions. We then discuss two aspects of the model: an explanation of the so-called offsets between dark matter and stars in galactic clusters and the laws relating supermassive black holes and dark matter distributions. Finally, we emphasize the formal connections to particular situations of other physical systems, including cold atom Bose-Einstein condensates and laser beam propagation in thermo-optical media, which may lead to tabletop laboratory analogues of cosmological phenomena.


Dark matter Axion-like particle Solitons Nonlinear Schrödinger equation Schrödinger-Poisson equation Nonlocal nonlinearities Scalar field dark matter 



We acknowledge financial support from Ministerio de Economía y Competitividad (MINECO) through grants FIS2014-58117-P, FIS2014-61984-EXP, and from Xunta de Galicia through grant GPC2015/019.


  1. 1.
    Ade, P., Aghanim, N., Arnaud, M., Ashdown, M., Aumont, J., Baccigalupi, C., Banday, A., Barreiro, R., Bartlett, J., Bartolo, N., et al.: Planck 2015 results-XIII. Cosmological parameters. Astron. Astrophys. 594, A13 (2016)CrossRefGoogle Scholar
  2. 2.
    Agrawal, G.P.: Nonlinear Fiber Optics. Academic press (2007)Google Scholar
  3. 3.
    Alberucci, A., Jisha, C.P., Smyth, N.F., Assanto, G.: Spatial optical solitons in highly nonlocal media. Phys. Rev. A 91(1), 013841 (2015)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Alexandre, J.: Dynamical mechanism for ultralight scalar dark matter. Phys. Rev. D 92(12), 123524 (2015)ADSCrossRefGoogle Scholar
  5. 5.
    Amendola, L., Barbieri, R.: Dark matter from an ultra-light pseudo-Goldsone-boson. Phys. Lett. B 642(3), 192–196 (2006)ADSCrossRefGoogle Scholar
  6. 6.
    Armengaud, E., Avignone, F., Betz, M., Brax, P., Brun, P., Cantatore, G., Carmona, J., Carosi, G., Caspers, F., Caspi, S., et al.: Conceptual design of the International Axion Observatory (IAXO). J. Instrum. 9(05), T05002 (2014)CrossRefGoogle Scholar
  7. 7.
    Arvanitaki, A., Dimopoulos, S., Dubovsky, S., Kaloper, N., March-Russell, J.: String axiverse. Phys. Rev. D 81(12), 123530 (2010)ADSCrossRefGoogle Scholar
  8. 8.
    Barranco, J., Bernal, A., Degollado, J.C., Diez-Tejedor, A., Megevand, M., Alcubierre, M., Núñez, D., Sarbach, O.: Schwarzschild black holes can wear scalar wigs. Phys. Rev. Lett. 109(8), 081102 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Bauer, D., Buckley, J., Cahill-Rowley, M., Cotta, R., Drlica-Wagner, A., Feng, J.L., Funk, S., Hewett, J., Hooper, D., Ismail, A., Kaplinghat, M., Kusenko, A., Matchev, K., McKinsey, D., Rizzo, T., Shepherd, W., Tait, T.M., Wijangco, A.M., Wood, M.: Dark matter in the coming decade: complementary paths to discovery and beyond. Phys. Dark Universe 7–8, 16–23 (2015)ADSCrossRefGoogle Scholar
  10. 10.
    Behroozi, P.S., Wechsler, R.H., Conroy, C.: The average star formation histories of galaxies in dark matter halos from z= 0–8. Astrophys. J. 770(1), 57 (2013)ADSCrossRefGoogle Scholar
  11. 11.
    Bekenstein, R., Schley, R., Mutzafi, M., Rotschild, C., Segev, M.: Optical simulations of gravitational effects in the Newton-Schrodinger system. Nat. Phys. 11, 872–878 (2015)CrossRefGoogle Scholar
  12. 12.
    Bernal, A., Guzman, F.S.: Scalar field dark matter: head-on interaction between two structures. Phys. Rev. D 74(10), 103002 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    Bernal, A., Guzman, F.S.: Scalar field dark matter: nonspherical collapse and late-time behavior. Phys. Rev. D 74(6), 063504 (2006)ADSCrossRefGoogle Scholar
  14. 14.
    Bernal, T., Fernández-Hernández, L.M., Matos, T., Rodríguez-Meza, M.A.: Rotation curves of high-resolution LSB and SPARC galaxies in wave (fuzzy) and multistate (ultra-light boson) scalar field dark matter. Mon. Not. R. Astron. Soc. 475(2), 1447–1468 (2017)., arXiv:1701.00912. 1 April 2018ADSCrossRefGoogle Scholar
  15. 15.
    Biasi, A.F., Mas, J., Paredes, A.: Delayed collapses of Bose-Einstein condensates in relation to anti-de sitter gravity. Phys. Rev. E 95, 032216 (2017)ADSCrossRefGoogle Scholar
  16. 16.
    Bogdán, Á., Goulding, A.D.: Connecting dark matter halos with the galaxy center and the supermassive black hole. Astrophys. J. 800(2), 124 (2015)ADSCrossRefGoogle Scholar
  17. 17.
    Böhmer, C., Harko, T.: Can dark matter be a Bose-Einstein condensate? J. Cosmol. Astropart. Phys. 06, 025 (2007)ADSCrossRefGoogle Scholar
  18. 18.
    Bousso, R.: The cosmological constant. Gen. Relat. Gravit. 40(2–3), 607–637 (2008)CrossRefGoogle Scholar
  19. 19.
    Bower, R., Benson, A., Malbon, R., Helly, J., Frenk, C., Baugh, C., Cole, S., Lacey, C.G.: Breaking the hierarchy of galaxy formation. Mon. Not. R. Astron. Soc. 370(2), 645–655 (2006)ADSCrossRefGoogle Scholar
  20. 20.
    Bozek, B., Marsh, D.J., Silk, J., Wyse, R.F.: Galaxy UV-luminosity function and reionization constraints on axion dark matter. Mon. Not. R. Astron. Soc. 450(1), 209–222 (2015)ADSCrossRefGoogle Scholar
  21. 21.
    Briscese, F.: Theoretical foundations of the Schrödinger method for LSS formation. Eur. Phys. J. C. 2017(77), 623 (2016)., arXiv:1612.04572
  22. 22.
    Carrasco, E., Gomez, P., Verdugo, T., Lee, H., Diaz, R., Bergmann, M., Turner, J., Miller, B., West, M.: Strong gravitational lensing by the super-massive cD galaxy in Abell 3827. Astrophys. J. Lett. 715(2), L160 (2010)ADSCrossRefGoogle Scholar
  23. 23.
    Chavanis, P.H.: BEC dark matter, Zeldovich approximation, and generalized Burgers equation. Phys. Rev. D 84(6), 063518 (2011)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Chavanis, P.H., Delfini, L.: Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions. II. Numerical Results. Phys. Rev. D 84(4), 043532 (2011)ADSGoogle Scholar
  25. 25.
    Conti, C., Peccianti, M., Assanto, G.: Route to nonlocality and observation of accessible solitons. Phys. Rev. Lett. 91(7), 073901 (2003)ADSCrossRefGoogle Scholar
  26. 26.
    Cotner, E.: Collisional interactions between self-interacting nonrelativistic boson stars: effective potential analysis and numerical simulations. Phys. Rev. D 94(6), 063503 (2016)ADSCrossRefGoogle Scholar
  27. 27.
    Cyburt, R.H., Fields, B.D., Olive, K.A., Yeh, T.H.: Big bang nucleosynthesis: present status. Rev. Mod. Phys. 88(1), 015004 (2016)ADSCrossRefGoogle Scholar
  28. 28.
    Dalfovo, F., Giorgini, S., Pitaevskii, L.P., Stringari, S.: Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys. 71(3), 463 (1999)ADSCrossRefGoogle Scholar
  29. 29.
    Davies, G., Widrow, L.M.: Test-bed simulations of collisionless, self-gravitating systems using the Schrödinger method. Astrophys. J. 485(2), 484 (1997)ADSCrossRefGoogle Scholar
  30. 30.
    De Blok, W.: The core-cusp problem. Adv. Astron. 2010, 789293 (2010)ADSGoogle Scholar
  31. 31.
    Diósi, L.: Gravitation and quantum-mechanical localization of macro-objects. Phys. Lett. A 105, 199 (1984)ADSCrossRefGoogle Scholar
  32. 32.
    Feng, J.L.: Dark matter candidates from particle physics and methods of detection. Annu. Rev. Astron. Astr. 48, 495–545 (2010)ADSCrossRefGoogle Scholar
  33. 33.
    Garay, L., Anglin, J., Cirac, J., Zoller, P.: Sonic black holes in dilute Bose-Einstein condensates. Phys. Rev. A 63(2), 023,611 (2001)Google Scholar
  34. 34.
    Giulini, D., Großardt, A.: Gravitationally induced inhibitions of dispersion according to the Schrödinger-Newton equation. Class. Quant. Grav. 28(19), 195026 (2011)ADSCrossRefGoogle Scholar
  35. 35.
    Gonzáles-Morales, A.X., Marsh, D.J., Peñarrubia, J., Ureña-López, L.: Unbiased constraints on ultralight axion mass from dwarf spheroidal galaxies. Mon. Not. R. Astron. Soc. 472(2), 1346–1360 (2017)., arXiv:1609.05856. 1 December 2017ADSCrossRefGoogle Scholar
  36. 36.
    González, J., Guzmán, F.: Interference pattern in the collision of structures in the Bose-Einstein condensate dark matter model: comparison with fluids. Phys. Rev. D 83(10), 103513 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    Goodman, J.: Repulsive dark matter. New Astron. 5(2), 103–107 (2000)ADSCrossRefGoogle Scholar
  38. 38.
    Gupta, P.D., Thareja, E.: Supermassive black holes from collapsing dark matter Bose-Einstein condensates. Class. Quant. Grav. 34(3), 035006 (2017)ADSCrossRefGoogle Scholar
  39. 39.
    Guth, A.H., Hertzberg, M.P., Prescod-Weinstein, C.: Do dark matter axions form a condensate with long-range correlation? Phys. Rev. D 92(10), 103513 (2015)ADSCrossRefGoogle Scholar
  40. 40.
    Guzmán, F., González, J., Cruz-Pérez, J.: Behavior of luminous matter in the head-on encounter of two ultralight BEC dark matter halos. Phys. Rev. D 93(10), 103535 (2016)ADSCrossRefGoogle Scholar
  41. 41.
    Guzmán, F.S., Urena-López, L.A.: Evolution of the Schrödinger-Newton system for a self-gravitating scalar field. Phys. Rev. D 69(12), 124033 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    Guzman, F.S., Urena-Lopez, L.A.: Gravitational cooling of self-gravitating Bose condensates. Astrophys. J. 645(2), 814 (2006)ADSCrossRefGoogle Scholar
  43. 43.
    Harko, T.: Bose-Einstein condensation of dark matter solves the core/cusp problem. J. Cosmol. Astropart. P. 2011(05), 022 (2011)CrossRefGoogle Scholar
  44. 44.
    Harko, T.: Cosmological dynamics of dark matter Bose-Einstein condensation. Phys. Rev. D 83(12), 123515 (2011)ADSMathSciNetCrossRefGoogle Scholar
  45. 45.
    Harrison, R., Moroz, I., Tod, K.: A numerical study of the Schrödinger-Newton equations. Nonlinearity 16(1), 101 (2002)ADSCrossRefGoogle Scholar
  46. 46.
    Harvey, D., Massey, R., Kitching, T., Taylor, A., Tittley, E.: The nongravitational interactions of dark matter in colliding galaxy clusters. Science 347(6229), 1462–1465 (2015)ADSCrossRefGoogle Scholar
  47. 47.
    Helfer, T., Marsh, D.J., Clough, K., Fairbairn, M., Lim, E.A., Becerril, R.: Black hole formation from axion stars. J. Cosmol. Astropart. Phys. 03, 055 (2017)., arXiv:1609.04724CrossRefGoogle Scholar
  48. 48.
    Hlozek, R., Grin, D., Marsh, D.J., Ferreira, P.G.: A search for ultralight axions using precision cosmological data. Phys. Rev. D 91(10), 103512 (2015)ADSCrossRefGoogle Scholar
  49. 49.
    Hu, W., Barkana, R., Gruzinov, A.: Fuzzy cold dark matter: the wave properties of ultralight particles. Phys. Rev. Lett. 85(6), 1158 (2000)ADSCrossRefGoogle Scholar
  50. 50.
    Hui, L., Ostriker, J.P., Tremaine, S., Witten, E.: Ultralight scalars as cosmological dark matter. Phys. Rev. D 95, 043541 (2017)ADSCrossRefGoogle Scholar
  51. 51.
    Jee, M., Mahdavi, A., Hoekstra, H., Babul, A., Dalcanton, J., Carroll, P., Capak, P.: A study of the dark core in A520 with the Hubble Space Telescope: The mystery deepens. Astrophys. J. 747(2), 96 (2012)ADSCrossRefGoogle Scholar
  52. 52.
    Jiang, S., Greengard, L., Bao, W.: Fast and accurate evaluation of nonlocal Coulomb and dipole-dipole interactions via the nonuniform FFT. SIAM J. Sci. Comput. 36(5), B777–B794 (2014)MathSciNetCrossRefGoogle Scholar
  53. 53.
    Kahlhoefer, F., Schmidt-Hoberg, K., Kummer, J., Sarkar, S.: On the interpretation of dark matter self-interactions in Abell 3827. Mon. Not. R. Astron. Soc. Lett. 452(1), L54–L58 (2015)ADSCrossRefGoogle Scholar
  54. 54.
    Khlopov, M.Y., Malomed, B.A., Zeldovich, Y.B.: Gravitational instability of scalar fields and formation of primordial black holes. Mon. Not. R. Astron. Soc. 215(4), 575–589 (1985)ADSCrossRefGoogle Scholar
  55. 55.
    Khmelnitsky, A., Rubakov, V.: Pulsar timing signal from ultralight scalar dark matter. J. Cosmol. Astropart. P. 2014(02), 019 (2014)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Kiritsis, E.: Gravity and axions from a random UV QFT. In: EPJ Web of Conferences, vol. 71, p. 00068. EDP Sciences (2014)Google Scholar
  57. 57.
    Klypin, A., Kravtsov, A.V., Valenzuela, O., Prada, F.: Where are the missing galactic satellites? Astrophys. J. 522(1), 82 (1999)ADSCrossRefGoogle Scholar
  58. 58.
    Kumar, R.K., Young-S, L.E., Vudragović, D., Balaž, A., Muruganandam, P., Adhikari, S.: Fortran and C programs for the time-dependent dipolar Gross-Pitaevskii equation in an anisotropic trap. Comput. Phys. Commun. 195, 117–128 (2015)ADSCrossRefGoogle Scholar
  59. 59.
    Laux, S.E., Stern, F.: Electron states in narrow gate-induced channels in Si. Appl. Phys. Lett. 49(2), 91–93 (1986)ADSCrossRefGoogle Scholar
  60. 60.
    Lee, J.W., Koh, I.G.: Galactic halos as boson stars. Phys. Rev. D 53(4), 2236 (1996)ADSCrossRefGoogle Scholar
  61. 61.
    Lee, J.W., Lee, J., Kim, H.C.: The M-sigma relation of super massive black holes from the scalar field dark matter (2015). arXiv:1512.02351
  62. 62.
    Lee, J.W., Lim, S., Choi, D.: BEC dark matter can explain collisions of galaxy clusters (2008). arXiv:0805.3827
  63. 63.
    Liddle, A.: An Introduction to Modern Cosmology. Wiley (2015)Google Scholar
  64. 64.
    Lončar, V., Young-S, L.E., Škrbić, S., Muruganandam, P., Adhikari, S.K., Balaž, A.: OpenMP, openMP/MPI, and CUDA/MPI C programs for solving the time-dependent dipolar Gross-Pitaevskii equation. Comput. Phys. Commun. 209, 190–196 (2016)ADSCrossRefGoogle Scholar
  65. 65.
    Lovell, M.R., Frenk, C.S., Eke, V.R., Jenkins, A., Gao, L., Theuns, T.: The properties of warm dark matter haloes. Mon. Not. R. Astron. Soc. 439, 300–317 (2014)ADSCrossRefGoogle Scholar
  66. 66.
    Magana, J., Matos, T.: A brief review of the scalar field dark matter model. J. Phys. Conf. Ser. 378, 012012 (2012) (IOP Publishing)Google Scholar
  67. 67.
    Markevitch, M., Gonzalez, A., Clowe, D., Vikhlinin, A., Forman, W., Jones, C., Murray, S., Tucker, W.: Direct constraints on the dark matter self-interaction cross section from the merging galaxy cluster 1E 0657–56. Astrophys. J. 606(2), 819 (2004)ADSCrossRefGoogle Scholar
  68. 68.
    Marsh, D.J.: Axion cosmology. Phys. Rep. 643, 1–79 (2016)ADSMathSciNetCrossRefGoogle Scholar
  69. 69.
    Marsh, D.J.E., Pop, A.R.: Axion dark matter, solitons and the cusp-core problem. Mon. Not. R. Astron. Soc. 451, 2479 (2015)ADSCrossRefGoogle Scholar
  70. 70.
    Massey, R., Williams, L., Smit, R., Swinbank, M., Kitching, T.D., Harvey, D., Jauzac, M., Israel, H., Clowe, D., Edge, A., et al.: The behaviour of dark matter associated with four bright cluster galaxies in the 10 kpc core of Abell 3827. Mon. Not. R. Astron. Soc. 449(4), 3393–3406 (2015)ADSCrossRefGoogle Scholar
  71. 71.
    Mavromatos, N.E., Argüelles, C.R., Ruffini, R., Rueda, J.A.: Self-interacting dark matter. Int. J. Mod. Phys. D 26, 1730007 (2017)ADSCrossRefGoogle Scholar
  72. 72.
    McGaugh, S.S., Lelli, F., Schombert, J.M.: Radial acceleration relation in rotationally supported galaxies. Phys. Rev. Lett. 117(20), 201101 (2016)ADSCrossRefGoogle Scholar
  73. 73.
    van Meter, J.R.: Schrödinger-Newton "collapse" of the wavefunction. Class. Quantum Gravity 28(21), 215013 (2011)ADSCrossRefGoogle Scholar
  74. 74.
    Milgrom, M.: A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J. 270, 365–370 (1983)ADSCrossRefGoogle Scholar
  75. 75.
    Moore, B., Ghigna, S., Governato, F., Lake, G., Quinn, T., Stadel, J., Tozzi, P.: Dark matter substructure within galactic halos. Astrophys. J. Lett. 524(1), L19 (1999)ADSCrossRefGoogle Scholar
  76. 76.
    Moroz, I.M., Penrose, R., Tod, P.: Spherically-symmetric solutions of the Schrödinger-Newton equations. Class. Quantum Gravity 15(9), 2733 (1998)ADSCrossRefGoogle Scholar
  77. 77.
    Navarrete, A., Paredes, A., Salgueiro, J.R., Michinel, H.: Spatial solitons in thermo-optical media from the nonlinear Schrödinger-Poisson equation and dark matter analogs. Phys. Rev. A 95, 013844 (2017)ADSCrossRefGoogle Scholar
  78. 78.
    Nguyen, J.H., Dyke, P., Luo, D., Malomed, B.A., Hulet, R.G.: Collisions of matter-wave solitons. Nat. Phys. 10(12), 918–922 (2014)CrossRefGoogle Scholar
  79. 79.
    O’Dell, D., Giovanazzi, S., Kurizki, G., Akulin, V.: Bose-Einstein condensates with 1/r interatomic attraction: electromagnetically induced "gravity". Phys. Rev. Lett. 84(25), 5687 (2000)ADSCrossRefGoogle Scholar
  80. 80.
    Paredes, Á., Feijoo, D., Michinel, H.: Coherent cavitation in the liquid of light. Phys. Rev. Lett. 112(17), 173901 (2014)ADSCrossRefGoogle Scholar
  81. 81.
    Paredes, A., Michinel, H.: Interference of dark matter solitons and galactic offsets. Phys. Dark Universe 12, 50–55 (2016)., Scholar
  82. 82.
    Peccei, R.D.: The strong CP problem and axions. In: Axions, pp 3–17. Springer (2008)Google Scholar
  83. 83.
    Peebles, P.J.E.: Principles of Physical Cosmology. Princeton University Press (1993)Google Scholar
  84. 84.
    Penrose, R.: On gravity’s role in quantum state reduction. Gen. Relat. Gravit. 28(5), 581–600 (1996)ADSMathSciNetCrossRefGoogle Scholar
  85. 85.
    Penrose, R.: On the gravitization of quantum mechanics 1: Quantum state reduction. Found. Phys. 44(5), 557–575 (2014)ADSMathSciNetCrossRefGoogle Scholar
  86. 86.
    Pontzen, A., Governato, F.: Cold dark matter heats up. Nature 506(7487), 171–178 (2014)ADSCrossRefGoogle Scholar
  87. 87.
    Pontzen, A., Slosar, A., Roth, N., Peiris, H.V.: Inverted initial conditions: exploring the growth of cosmic structure and voids. Phys. Rev. D 93(10), 103519 (2016)ADSMathSciNetCrossRefGoogle Scholar
  88. 88.
    Primack, J.R.: Precision cosmology. New Astron. Rev. 49(2), 25–34 (2005)ADSCrossRefGoogle Scholar
  89. 89.
    Qin, J., Dong, G., Malomed, B.A.: Hybrid matter-wave-microwave solitons produced by the local-field effect. Phys. Rev. Lett. 115(2), 023901 (2015)ADSCrossRefGoogle Scholar
  90. 90.
    Qin, J., Dong, G., Malomed, B.A.: Stable giant vortex annuli in microwave-coupled atomic condensates. Phys. Rev. A 94(5), 053611 (2016)ADSCrossRefGoogle Scholar
  91. 91.
    Riess, A.G., Filippenko, A.V., Challis, P., Clocchiatti, A., Diercks, A., Garnavich, P.M., Gilliland, R.L., Hogan, C.J., Jha, S., Kirshner, R.P., et al.: Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron. J. 116(3), 1009 (1998)ADSCrossRefGoogle Scholar
  92. 92.
    Roger, T., Maitland, C., Wilson, K., Westerberg, N., Vocke, D., Wright, E.M., Faccio, D.: Optical analogues of the Newton-Schrödinger equation and boson star evolution. Nat. Commun. 7, 13492 (2016)ADSCrossRefGoogle Scholar
  93. 93.
    Roos, M.: Introduction to Cosmology. Wiley (2015)Google Scholar
  94. 94.
    Rotschild, C., Cohen, O., Manela, O., Segev, M., Carmon, T.: Solitons in nonlinear media with an infinite range of nonlocality: first observation of coherent elliptic solitons and of vortex-ring solitons. Phys. Rev. Lett. 95(21), 213904 (2005)ADSCrossRefGoogle Scholar
  95. 95.
    Ruffini, R., Bonazzola, S.: Systems of self-gravitating particles in general relativity and the concept of an equation of state. Phys. Rev. 187(5), 1767 (1969)ADSCrossRefGoogle Scholar
  96. 96.
    Schaller, M., Robertson, A., Massey, R., Bower, R.G., Eke, V.R.: The offsets between galaxies and their dark matter in \(\Lambda \) cold dark matter. Mon. Not. R. Astron. Soc. Lett. 453(1), L65–L69 (2015)ADSCrossRefGoogle Scholar
  97. 97.
    Schive, H.Y., Chiueh, T., Broadhurst, T.: Cosmic structure as the quantum interference of a coherent dark wave. Nat. Phys. 10(7), 496–499 (2014)CrossRefGoogle Scholar
  98. 98.
    Schive, H.Y., Chiueh, T., Broadhurst, T., Huang, K.W.: Contrasting galaxy formation from quantum wave dark matter, \(\psi \)DM, with \(\Lambda \)CDM, using Planck and Hubble data. Astrophys. J. 818(1), 89 (2016)ADSCrossRefGoogle Scholar
  99. 99.
    Schive, H.Y., Liao, M.H., Woo, T.P., Wong, S.K., Chiueh, T., Broadhurst, T., Hwang, W.P.: Understanding the core-halo relation of quantum wave dark matter from 3D simulations. Phys. Rev. Lett. 113(26), 261302 (2014)ADSCrossRefGoogle Scholar
  100. 100.
    Schwabe, B., Niemeyer, J.C., Engels, J.F.: Simulations of solitonic core mergers in ultralight axion dark matter cosmologies. Phys. Rev. D 94(4), 043513 (2016)ADSCrossRefGoogle Scholar
  101. 101.
    Sin, S.J.: Late-time phase transition and the galactic halo as a Bose liquid. Phys. Rev. D 50(6), 3650 (1994)ADSCrossRefGoogle Scholar
  102. 102.
    Suárez, A., Robles, V.H., Matos, T.: A review on the scalar field/Bose-Einstein condensate dark matter model. In: Accelerated Cosmic Expansion, pp. 107–142. Springer (2014)Google Scholar
  103. 103.
    Taha, T.R., Ablowitz, M.I.: Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear schrödinger equation. J. Comput. Phys. 55(2), 203–230 (1984)ADSMathSciNetCrossRefGoogle Scholar
  104. 104.
    Urena-Lopez, L.A., Liddle, A.R.: Supermassive black holes in scalar field galaxy halos. Phys. Rev. D 66(8), 083005 (2002)ADSCrossRefGoogle Scholar
  105. 105.
    Valle, D.C., Mielke, E.W.: Solitonic axion condensates modeling dark matter halos. Ann. Phys. 336, 245–260 (2013)ADSCrossRefGoogle Scholar
  106. 106.
    Veltmaat, J., Niemeyer, J.C.: Cosmological particle-in-cell simulations with ultralight axion dark matter. Phys. Rev. D 94(12), 123523 (2016)ADSCrossRefGoogle Scholar
  107. 107.
    Weinberg, D.H., Bullock, J.S., Governato, F., de Naray, R.K., Peter, A.H.: Cold dark matter: controversies on small scales. P. Natl. Acad. Sci. USA 112(40), 12249–12255 (2015)ADSCrossRefGoogle Scholar
  108. 108.
    Weinberg, S.: Gravitation and cosmology: principles and applications of the general theory of relativity, vol. 1. Wiley, New York (1972)Google Scholar
  109. 109.
    Widrow, L.M., Kaiser, N.: Using the Schrödinger equation to simulate collisionless matter. Astrophys. J. 416, L71 (1993)ADSCrossRefGoogle Scholar
  110. 110.
    Williams, L.L., Saha, P.: Light/mass offsets in the lensing cluster Abell 3827: evidence for collisional dark matter? Mon. Not. R. Astron. Soc. 415(1), 448–460 (2011)ADSCrossRefGoogle Scholar
  111. 111.
    Woo, T.P., Chiueh, T.: High-resolution simulation on structure formation with extremely light bosonic dark matter. Astrophys. J. 697(1), 850 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Área de óptica, Departamento de Física AplicadaOurenseSpain
  2. 2.Área de óptica, Escola de Enxeñaría AeroespacialOurenseSpain

Personalised recommendations