Brief discussion of some the problems of cold dark matter in cosmological structure formation, the idea of `fuzzy' dark matter and some applications of the Schrodinger-Poisson system for cosmic reconstruction. Talk given at `Post-Planck Cosmology', Inter-University Centre for Astronomy and Astrophysics, Pune, India (12th October 2017).

1. Alternative(s) to
Cold Dark Matter
Peter Coles
(@telescoper)

6. 12 October, 2017

8. “CONCORDANCE”

10. Ingredients of the Standard
Cosmology
•General Relativity
•Cold Dark Matter
•Cosmological Constant
•Cosmological Principle
•Primordial Gaussian fluctuations
•Inflation
•Baryons
•Neutrinos
•Radiation…

11. Questionable Aspects of the
Standard Cosmology
•General Relativity
•Cold Dark Matter
•Cosmological Constant
•Cosmological Principle
•Primordial Gaussian fluctuations
•Inflation
•Baryons
•Neutrinos
•Radiation…

12. Questionable Aspects of the
Standard Cosmology
•General Relativity
•Cold Dark Matter
•Cosmological Constant
•Cosmological Principle
•Primordial Gaussian fluctuations
•Inflation
•Baryons
•Neutrinos
•Radiation…

13. Why Should The Dark Sector
Be Simple?

14. Problems with Cold Dark Matter
Astrophysics:
• Cuspy Halo Problem
• Missing Satellite Problem
• Planar Structures
• Galaxy Morphology
• …
Particle Physics
• No Evidence for Supersymmetry
• No evidence from Direct Searches
• …

17. RA Ibata et al. Nature 493, 62-65 (2013) doi:10.1038/nature11717
Satellite galaxy positions as viewed from Andromeda.

20. Alternatives (not exhaustive)
•HDM (historical; see also CHDM)
•WDM
•SIDM
•SADM
•Axions
•Fuzzy DM
•……
•Modified Gravity (e.g. MOND)

21. arXiv:1509.07471

22. Alternatives (not exhaustive)
•HDM (historical; see also CHDM)
•WDM
•SIDM
•SADM
•Fuzzy DM
•……
•Modified Gravity (e.g. MOND)

23. • Simple idea): DM is a (very) light particle (m~ 10-22 eV) then
the Compton wavelength can be a galactic scale.
• In this case the `quantum pressure’ is a real physical effect.
• Something like `warm’ dark matter arises (actually `fuzzy’ dark
matter), but with quantum pressure.
• Sometimes called Fuzzy Dark Matter
Might dark matter be quantum-
mechanical?

24. The Schrodinger-Poisson
System


V
mt
i 

 2
2
2



  GGV 442

25. From Schive et al., arXiv: 1406.6586 (also published in Nature)

26. From Schive et al., arXiv: 1406.6586 (also published in Nature)

27. arXiv:1607.08208

28. arXiv:1710.03747

32. Structure formation basics
• Small primordial density perturbations grow via the
mechanism of gravitational instability.
• Large density fluctuations observable today thought to be
dominated by non-baryonic matter.
• Observations of clustering support some form of collisionless
CDM.
• Approximate model of structure formation:
Large-scale structure is the result of the gravitational
amplification of small inhomogeneities in the primordial
CDM distribution.

33. The fluid approach
• Treat collisionless CDM as a fluid.
• Linear perturbation theory gives an equation for the density
contrast
• In a spatially flat CDM-dominated universe
where:
• Comoving velocity associated with the growing
mode is irrotational:
1/  b
2/3
)(
)(





aaD
aaD Growing mode
Decaying mode
)X()(),X( iaDa  
dad /XU 
UXU 

34. • Linear theory only valid at early times when fluctuations in
physical fluid quantities are small.
• Perturbations grow and the system becomes non-linear in
nature.
• Linear theory predicts the existence of spatial regions with
negative density , which is unphysical.
Problems with the fluid approach
1

35. The Zel’dovich approximation
• Follows perturbations in particle trajectories:
• Mass conservation leads to:
• Zel’dovich approximation remains valid in the quasi-linear
regime, after the breakdown of the linearised fluid approach.
)Q(UQ),Q(X aa 
1
)1(
1
),X( 3
1

 

i
ia
a



36. Problems with the Zel’dovich
approximation
• The Zel’dovich approximation fails
when particle trajectories cross –
shell crossing.
• A region where shell-crossing
occurs is called a caustic.
• At caustics the mapping
is no longer unique and the density
becomes infinite.
• Particles pass through caustics
without responding to the large
gravitational force non-linear
regime described very poorly.
XQ 


37. From Zel’dovich to Burgers
Define
Then
b
x





  0



 

b
Modify RHS   
 

2



b
Burgers’ Equation
“viscosity”

38. The wave-mechanical approach
• Apply the Madelung transformation
to the fluid equations.
• Obtain the Schrodinger equation:
• is the quantum pressure term.
• DeBroglie wavelength
)/exp(  Ui


 








PV
a
i 2
X
2
2

 2
X
2
2

P
 dB

39. The Madelung Transformation










22
2
2
*
2
22
)/exp(
)(
2
1
0)(












V
t
i
i
V
t
t
Quantum Pressure
} v

40. The wave-mechanical approach
• Assume the comoving velocity is irrotational:
• The equations of motion for a fluid of gravitating CDM
particles in an expanding universe are then:
where and
UXU 
0)(
0)(
2
1
Xx
2
X






U
U
U
a
V
a




Bernoulli
Continuity
b /
U
p
a
a
aaa
V 






  222
2



‘Modified potential’

41. The Trouble with 
• The classical limit has 0…
• BUT the “weight” oscillates wildly as this
limit is approached.
• For a finite computation, need a finite value of 
• Also, system misbehaves
• Note  is dimensionally a viscosity; c.f. Burgers
equation
)/exp( iS


2
d
Rs

42. • Assume a sinusoidal initial density profile in 1D:
where is the comoving period of the perturbation.
• Free parameters are:
1. The amplitude of the initial density fluctuation.
2. The dimensionless number
• Quantum pressure
• DeBroglie wavelength
Gravitational collapse in one
dimension






D
X
Xi


2
cos)( 0
D
0
ie aDR /2

2
/1 eRP 
edB R/1

43. Gravitational collapse in one
dimension
Evolution of a periodic 1D self-gravitating system with )/2cos()( 0 DXXi  
3
0
100.1
001.0


eR


44. Gravitational collapse in one
dimension
Evolution of a periodic 1D self-gravitating system with )/2cos()( 0 DXXi  
7
0
102.1
001.0


eR


45. Cosmological Reconstruction
Problems
• We observe redshifts and (sometimes)
estimated distances in the evolved local
Universe for some galaxies
• Problem I. What is the real space
distribution of dark matter
• Problem II. What were the initial data that
evolved into the observed data?

47. Problems with the Zel’dovich
approximation
• The Zel’dovich approximation fails
when particle trajectories cross –
shell crossing.
• Regions where shell-crossing
occurs are associated with
caustics.
• At caustics the mapping
is no longer unique and the density
becomes infinite.
• Particles pass through caustics
non-linear regime described
very poorly.
XQ 


48. The ‘free-particle’ Schrodinger
equation
• In a spatially flat CDM-dominated universe, the ‘potential’
in the linear regime.
• Neglecting quantum pressure, the Schrodinger equation to be
solved is then the ‘free-particle’ equation:
• Can be solved exactly!
0V


 2
X
2
2



a
i

49. The wave-mechanical approach
• For a collisionless
medium, shell-crossing
leads to the generation
of vorticity velocity
flow no longer
irrotational
• Possible to construct
more sophisticated
representations of the
wavefunction that allow
for multi-streaming
(Widrow & Kaiser 1993).

Phase-space evolution of a 1D self-gravitating
system with ,
0)( Xvi
)/exp()( 22
0 LXXi  

50. Uhlemann&Kopp(2014)

51. • In Eulerian space the Zel’dovich approximation becomes:
• Reconstruction process:
1. Determine present comoving velocity potential
2. Smooth to remove non-linearities.
3. Integrate ZB equation backwards from to
4. Use linear theory to calculate initial density field
The Zel’dovich-Bernoulli method
)2(
0)(
2
1
22
2
X
aaaa
a
p
U
U
U
 








Zel’dovich-Bernoulli
0,U
10 a 001.0~ia
i

52. • The Zel’dovich-Bernoulli equation can be replaced by the
‘free-particle’ Schrodinger equation!
• Currently testing the `free-particle’ reconstruction method on a
2D N-body simulation.
• If successful, possible extensions are:
1. Model errors in galaxy position and velocity measurements by
exploiting the nature of quantum mechanics.
2. Work in redshift space coordinates
3. Generalise to 3D.
Wave-mechanics and the
Zel’dovich-Bernoulli method
)SˆU(SˆXS  a

53. Behaviour of the free-particle
approximation

54. The role of quantum pressure
• Recall:
•
• Define a ratio:
P
C
a



 
2
2
1


x
C
P

22
DP 

55. Point-by-point comparisons:
Mpc1
4 
 hrsm
2/122/12



nb
nb
r 

56. Point-by-point comparisons:
Mpc1
8 
 hrsm

57. One-point PDFs

58. Comparisons in Fourier space
 







 

 22
2
ˆˆ
ˆˆ
nb
nb
k




59. Summary
• The wave-mechanical approach can overcome some of the
main difficulties associated with the fluid approach and the
Zel’dovich approximation.
• More sophisticated representations of the wavefunction can
be used to allow for multi-streaming; the quantum pressure
term is crucial in determining how well the wave-mechanical
approach performs.
• The `free-particle’ Schrodinger equation can be applied to the
problem of reconstruction.
• Dark Matter may even be quantum-mechanical!