1. Bose-Einstein condensation of dark matter axions
Pierre Sikivie
Axion Dark Matter Conference
Nordita, Stockholm
December 5 - 9, 2016

2. Outline Cosmic axion Bose-Einstein condensation
Evidence for axion dark matter from the
study of caustics
(axions are different)
(axions are better)

3. number density velocity dispersion phase space density
Cold axion properties
number density
velocity dispersion
phase space density if
decoupled

4. QFT has two classical limits:
limit of point particles (WIMPs, …)
limit of classical fields (axions)

5. A classical scalar field behaves like CDM
in the non-relativistic limit
on distance scales much larger than the wavelength
with

6. "quantum pressure"

7. In the expanding universe, the Fourier components
of the relative overdensity grow according to:
M. Khlopov, B. Malomed, Y. Zel'dovich 1985
M. Bianchi, D. Grasso, R. Ruffini, 1990
There is a Jeans length

8. Ultra-light axion-like particles have been proposed
S.-J. Sin 1994
J. Goodman 2000
W. Hu, R. Barkana & A. Gruzinov 2000
.....
H.-Y. Schive et al., 2014
D. Grin, R. Hlozek, D. Marsh & P. Ferreira, 2015
L. Hui, J. Ostriker, S. Tremaine and E. Witten, 2106
see talk by J. Niemeyer

9. A quantum scalar field differs from CDM on time scales longer than its thermal relaxation time

10. Bose-Einstein Condensation
if identical bosonic particles
are highly condensed in phase space
and their total number is conserved
and they thermalize
then most of them go to the lowest energy
available state

11. by yielding their energy to the non-condensed particles, the
why do they do that?
by yielding their energy to the
non-condensed particles, the
total entropy is increased.
preBEC
BEC

12. the axions thermalize and form a BEC after a time
the axion fluid obeys
classical field equations,
behaves like CDM
the axion fluid does not obey
classical field equations,
does not behave like CDM

13. the axion BEC rethermalizes
the axion fluid obeys
classical field equations,
behaves like CDM
the axion fluid does not obey
classical field equations,
does not behave like CDM

14. from M.R. Andrews, C.G. Townsend, H.-J. Miesner, D.S. Durfee,
D.M. Kurn and W. Ketterle, Science 275 (1997) 637.

15. Axion field dynamics From self-interactions
From gravitational self-interactions

16. In the “particle kinetic” regime
implies
When

17. 4 1 2 3

18. D. Semikoz & I. Tkachev, PRD 55 (1997) 489D.

19. After , axions thermalize in the “condensed” regime
implies
for
and
for self-gravity

20. Toy model thermalizing in the condensed regime:
with
i.e.

21. 50 quanta among 5 states 316 251 system states Start with
Number of particles
Total energy
Thermal averages

22. Integrate
Calculate
Do the approach the on
the predicted time scale?

24. P.S. and Elisa Todarello, arXiv 1607.00949
Quantum
evolution
of the
initial state
|12,25,4,12,1>
Classical
evolution
of the
corresponding
initial state

25. The quantum system thermalizes on the expected time scale
The classical system does not thermalize even after a very long time
The classical system tracks the quantum system only briefly
Is the degeneracy large enough?
Fermi, Pasta &
Ulam, 1954

26. Evolution of the 4th oscillator when the initial state is |6r, 4r, 5r, 4r, 3r> with r = 1,2,3,4,5,6.

27. The classical system is invariant under rescalings
The quantum system is not scale invariant for small but does approach a scale invariant limit when is increased.
The classical and quantum systems behave differently in the limit of large
Would coherent states behave differently?

28. Evolution of the initial state |0,12,16,0,0> :
classical (dotted), quantum (solid) & coherent (dashed)

29. Thermalization occurs due to gravitational interactions
PS + Q. Yang, PRL 103 (2009)
at time

30. Gravitational interactions thermalize the axions and cause them to form a BEC when the photon temperature
After that

31. Galactic halos live in phase space
ordinary fluid
dark matter (collisionless) fluid

32. . DM forms caustics in the non-linear regime . x x x x x x
DM particles
in phase space x x x x

33. Phase space distribution of DM in a homogeneous universe. z z
for WIMPs
for axions (preBEC)
for sterile neutrinos

34. The dark matter particles lie on a 3-dimensional sheet in 6-dimensional phase space. z
the physical density is the projection of the phase space sheet onto position space z

35. The cold dark matter particles lie on a 3-dimensional sheet in 6-dimensional phase space. z
the physical density is the projection of the phase space sheet onto position space z

36. Phase space structure of spherically symmetric halos

37. Implications:
At every point in physical space, the distribution of velocities is discrete, each velocity corresponding to a particular flow at that location (J. Ipser and PS, 1992)
2. At some locations in physical space, where the number of flows changes, there is a caustic, i.e. the density of dark matter is very high there.

38. - the number of flows at our location in the Milky Way halo is of order 100
- small subhalos from hierarchical structure formation produce an effective velocity dispersion
but do not destroy the sheet structure in phase space
the known inhomogeneities in the distribution of matter are insufficient to diffuse the flows by gravitational scattering
present N-body simulations have barely enough particles to resolve the flows and caustics

39. Hierarchical clustering introduces effective velocity dispersion

40. (from Binney and Tremaine’s book)

42. Phase space structure of spherically symmetric halos

43. Galactic halos have inner caustics as
well as outer caustics.
If the initial velocity field is dominated by net
overall rotation, the inner caustic is a ‘tricusp ring’.
If the initial velocity field is irrotational, the inner
caustic has a ‘tent-like’ structure.
(Arvind Natarajan and PS, 2005).

44. A shell of particles, part of a continuous flow
A shell of particles, part of a continuous flow. The shell has net angular momentum. As the shell falls in and out of the galaxy, it turns itself inside out. . . . . . .

45. Sphere turning inside out

46. in case of net overall rotation
simulations by Arvind Natarajan
in case of net overall rotation

47. The caustic ring cross-sectionD -4
an elliptic umbilic catastrophe

49. in case of irrotational flow

50. On the basis of the self-similar infall model (Filmore and Goldreich, Bertschinger) with angular momentum (Tkachev, Wang + PS), the caustic rings were predicted to be
in the galactic plane
with radii
was expected for the Milky Way halo from the effect of angular momentum on the inner rotation curve.

51. Galactic rotation curves
mass
rotation speed

52. Effect of a caustic ring of dark matter upon the galactic rotation curve

53. Composite rotation curve (W. Kinney and PS, astro-ph/9906049)
combining data on
32 well measured
extended external
rotation curves
scaled to our own galaxy

54. Inner Galactic rotation curve
from Massachusetts-Stony Brook North Galactic Pane CO Survey (Clemens, 1985)

55. IRAS

56. IRAS

57. Inner Galactic rotation curve
from Massachusetts-Stony Brook North Galactic Pane CO Survey (Clemens, 1985)

58. Outer Galactic rotation curve
R.P. Olling and M.R. Merrifield, MNRAS 311 (2000) 361

59. Monoceros Ring of stars
H. Newberg et al. 2002; B. Yanny et al., 2003; R.A. Ibata et al., 2003;
H.J. Rocha-Pinto et al, 2003; J.D. Crane et al., 2003; N.F. Martin et al., 2005
in the Galactic plane
at galactocentric distance
appears circular, actually seen for
scale height of order 1 kpc
velocity dispersion of order 20 km/s
may be caused by the n = 2 caustic ring of
dark matter (A. Natarajan and P.S. ’07)

60. Rotation curve of M31 obtained by Rubin and Ford in 1970
Galaxies are surrounded by halos of dark matter.

61. Rotation curve of Andromeda Galaxy
from L. Chemin, C. Carignan & T. Foster, arXiv:

62. 10.3
15.4
29.2 kpc
10 arcmin = 2.2 kpc

63. The caustic ring halo model assumes
L. Duffy & PS
PRD78 (2008)
063508
net overall rotation
axial symmetry
self-similarity

64. The specific angular momentum distribution on the turnaround sphere
Is it plausible in the context of
tidal torque theory?

65. Tidal torque theory neighboring protogalaxy
Stromberg 1934; Hoyle 1947; Peebles 1969, 1971

66. Tidal torque theory with ordinary CDM
neighboring
protogalaxy
the velocity field remains irrotational

67. For collisionless particles
If initially,
then for ever after.

68. in case of irrotational flow

69. Tidal torque theory with axion BEC
net overall rotation is obtained because, in the lowest energy state,
all axions fall with the same angular momentum

70. in case of net overall rotation

71. The specific angular momentum distribution on the turnaround sphere
Is it plausible in the context of
tidal torque theory?

72. Tidal torque theory with axion BEC
net overall rotation is obtained because, in the lowest energy state,
all axions fall with the same angular momentum

73. Magnitude of angular momentum
G. Efstathiou et al. 1979, 1987
from caustic rings
fits perfectly ( )

74. The specific angular momentum distribution on the turnaround sphere
Is it plausible in the context of
tidal torque theory?

75. Self-Similarity
a comoving volume

76. Self-Similarity (yes!)
time-independent axis of rotation
provided

77. Conclusions
Axions differ from the other cold dark matter candidates because they thermalize by gravitational self-interactions and form a Bose-Einstein condensate
Axion BEC explains the evidence for caustic rings of dark matter in all its aspects