The classical field approximation for
Ultra Light Dark Matterarxiv.2310.07119

Andrew Eberhardt [Kavli IPMU]
andrew.eberhardt@ipmu.jp
Kashiwa DM 2023
Dec, 5th 2023

Ultra light dark matter

2

Ultra light dark matter

Lin arxiv 1904.07915

3

Ultra light dark matter●

At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales

4Ultra light dark matter●

At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales●
Interference patterns

Density fields for different
particle masses

5Ultra light dark matter●

At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales●
Interference patterns

Schive et al (Nature 2014)6

Ultra light dark matter●

At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales●
Interference patterns●“Quantum” pressure

Gravitational collapse in
1D

7

Ultra light dark matter●

At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales●
Interference patterns●“Quantum” pressure●
Granular density patterns

Typical halo density
Gosenca [,Eberhardt] et al., PRD (2023)

8

Ultra light dark matter
At the lowest masses dark matter manifests wave-likephenomena on astrophysical scales●
Gives a rich phenomenology of constraints●

Image credit: Kier Rogers

9Ultra light dark matter●

Most ULDM constraints rely on the predictions of classicalfield theory10Ultra light dark matter
Most ULDM constraints rely on the predictions of classicalfield theory●
Requires accuracy of classical approximation of a wide
range of scales●

11Ultra light dark matter
Most ULDM constraints rely on the predictions of classicalfield theory●
Requires accuracy of classical approximation of a wide
range of scales●

12Ultra light dark matter
Most ULDM constraints rely on the predictions of classicalfield theory●
Requires accuracy of classical approximation of a wide
range of scales●

Understanding these simulations is essential tounderstanding the model and its constraints

13

Simulations

14

Simulations●

Simulations rely on a hierarchy of approximations

15

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

16

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Ap
pr
ox

im
ati
on
s

17

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Non relativistic

Quantum Schrodinger equation

Ap
pr
ox

im
ati
on
s

18

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Non relativistic

Quantum Schrodinger equation

Ap
pr
ox

Classical

im
ati
on
s

Classical Schrodinger equation

19

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Non relativistic

Quantum Schrodinger equation

Ap
pr
ox

Classical

im
ati
on
s

Classical Schrodinger equation
Discretize

20

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Non relativistic

Quantum Schrodinger equation

Ap
pr
ox

Classical

im
ati
on
s

Classical Schrodinger equation
Discretize

Predictions are made here
21

Simulations●

Simulations rely on a hierarchy of approximations
Quantum Klein Gordon equation

Non relativistic

Quantum Schrodinger equation

Ap
pr
ox

Classical

im
ati
on
s

Classical Schrodinger equation
Discretize

Predictions are made here
22

Classical approximation

23

Classical approximation●

Classical field theory is an approximation ofquantum field theory replacing operators withnumbers

24Classical approximation●●

Classical field theory is an approximation ofquantum field theory replacing operators withnumbers
A classical field places a number at every pointin space25

Classical approximation●●

C

Classical field theory is an approximation ofquantum field theory replacing operators withnumbers
A classical field places a number at every pointin space26

Classical approximation●●●

Coherent stateClassical field theory is an approximation ofquantum field theory replacing operators withnumbers
A classical field places a number at every pointin space
The quantum field places a probability
distribution at each point

27

Classical approximation●●●

Number eigenstateClassical field theory is an approximation ofquantum field theory replacing operators withnumbers
A classical field places a number at every pointin space
The quantum field places a probability
distribution at each point

28

Classical approximation
●

●

●

●

Classical field theory is an approximation ofquantum field theory replacing operators withnumbers
A classical field places a number at every pointin space
The quantum field places a probability
distribution at each point
If the distribution is tightly peaked around theclassical value then we can approximate the
distribution using this number

29Classical approximation●

The classical field approximation is usually
motivated in two ways

30

Classical approximation●

Coherent state

The classical field approximation is usually
motivated in two ways–

The misalignment mechanism produces aquantum coherent state (specifies a
distribution shape)

Number eigenstate31Classical approximation●

The classical field approximation is usually
motivated in two ways–

The misalignment mechanism produces aquantum coherent state–

Occupation numbers are very large (gives
the fractional variance)

32

Classical approximation●●

The classical field approximation is usually
motivated in two ways–

The misalignment mechanism produces aquantum coherent state–

Occupation numbers are very large

Both conditions are necessary for the classicalfield equations to make accurate predictions[Eberhardt et al (PRD 2021)]

33

Classical approximation●

In the absence of nonlinearities wewould expect this description to survive

34

Classical approximation●●

In the absence of nonlinearities wewould expect this description to survive
Nonlinearities introduce quantum
corrections on some timescale

35

Central Questions●

Which case is relevant for the simulation of ultra light dark matter?

Weakly nonlinear

Strongly nonlinear
36

Let’s look at an analogous system that
contain all the important components ofthis problem

37

Let’s look at an analogous system that
contain all the important components ofthis problem

https://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat#/media/File:Schrodingers_cat.svg

38

Schrödinger’s Cat●

Let’s look at an analogous systemthat contain all the importantcomponents of this problem

Cs

39

Schrödinger’s Cat●

The system starts in a state welldescribed by classical mechanics

Cs

40

Schrödinger’s Cat●●

The system starts in a state welldescribed by classical mechanics

Cs

On some timescale nonlinear
interactions will create a system
poorly described by classicalmechanics

41

Schrödinger’s Cat
●

●

●

Cs

The system starts in a state welldescribed by classical mechanics
On some timescale nonlinear
interactions will create a system
poorly described by classicalmechanics

50%

50%

On some timescale environmental
interactions (“observers”) will send
this system to its pointer states

42

Schrödinger’s Cat●●●

The system starts in a state welldescribed by classical mechanics

50%

50%

On some timescale nonlinear
interactions will create a system
poorly described by classicalmechanics
On some timescale environmental
interactions (“observers”) will send
this system to its pointer states

Pointer states

43

Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence

44Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence45Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence46Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence47

Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence48

Schrödinger’s Cat
Initial conditions
Life time of Cs

Quantum corrections

Quantum state

Pointer states
Env interaction

Decoherence49Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions

Pointer states

Decoherence
Quantum state

Cat
Pointer states

Analytic estimates

Analytic estimates

Quantum corrections

Decoherence

Dark matter
50

Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions

Pointer states

Decoherence
Quantum state

Pointer states

Analytic estimates

Analytic estimates

Quantum corrections

Decoherence51Approaches to answers●●●

Previous approaches generally
separate into two groups–

Order of magnitude estimates–

Simulations of small quantum“number eigenstates”

We directly simulate the evolution ofquantum corrections for coherent
states on a variety of scales
Made difficult by the scaling of
Quantum Hilbert spaces

Eberhardt et al., PRD (Feb 2022)
52

Methods: truncated Wigner approximation●

For large systems we want to move to“quantum phase space”

Large Systems:

53

Methods: truncated Wigner approximation●

Large Systems:

For large systems we want to move to“quantum phase space”
This is done using the Weyl symbol of
the quantum state, the Wigner
function
Quantum state:

Wigner function:

7.5

f (ψ, ψ ∗)

5.0
2.5

I[ψ]

●

0.0

−2.5

Go from a vector in Hilbert space
to a probability functional on field
configurations

−5.0
−7.5

−0.4

−0.2

0.0
R[ψ]

0.2

0.4

54

Methods: truncated Wigner approximation●●●

For large systems we want to move to“quantum phase space”

Large Systems:

This is done using the Weyl symbol of
the quantum state, the Wigner
function
Rewrite the Von Neumann equationsin this space
Von Neumann eqn:

55

Methods: truncated Wigner approximation
●

●

●

●

For large systems we want to move to“quantum phase space”

Large Systems:

This is done using the Weyl symbol of
the quantum state, the Wigner function
Rewrite the Von Neumann equations inthis space
Approximate a Moyal bracket with aPoisson bracket
Von Neumann eqn:

56

Methods: truncated Wigner approximation
●

●

●

●

7.5

For large systems we want to move to“quantum phase space”
This is done using the Weyl symbol of
the quantum state, the Wigner function
Rewrite the Von Neumann equations inthis space
Approximate a Moyal bracket with aPoisson bracket
Approximate Wigner function as
ensemble of classical fields drawn from
Wigner distributionf (ψ, ψ ∗)

5.0
2.5

I[ψ]

●

0.0

−2.5
−5.0
−7.5

−0.4

−0.2

0.0
R[ψ]

0.2

0.4

57

Methods: truncated Wigner approximation
●

●

●

●

●

7.5

For large systems we want to move to“quantum phase space”
This is done using the Weyl symbol of thequantum state, the Wigner function
Rewrite the Von Neumann equations in thisspace
Approximate a Moyal bracket with a PoissonbracketApproximate Wigner function as ensemble ofclassical fields drawn from Wigner distribution
Resulting evolution is each stream evolves
classically and independent of the others,
resulting in a highly parallel algorithmf (ψ, ψ ∗)

5.0
2.5

I[ψ]

●

0.0

−2.5
−5.0
−7.5

−0.4

−0.2

0.0
R[ψ]

0.2

0.4

58

Methods: truncated Wigner approximation
●

●

●

●

●

●

For large systems we want to move to“quantum phase space”

Large Systems:

This is done using the Weyl symbol of thequantum state, the Wigner function
Rewrite the Von Neumann equations in thisspace
Approximate a Moyal bracket with a PoissonbracketCPU
CPU

Approximate Wigner function as ensemble ofclassical fields drawn from Wigner distribution
Resulting evolution is each stream evolves
classically and independent of the others,
resulting in a highly parallel algorithm
59Methods: truncated Wigner approximation●

The truncated Wigner approximation has a lot of good properties:

60

Methods: truncated Wigner approximation●

The truncated Wigner approximation has a lot of good properties:–

Good scaling with problem size

61

Methods: truncated Wigner approximation●

The truncated Wigner approximation has a lot of good properties:–

Good scaling with problem size–

Highly parallelizable62

Methods: truncated Wigner approximation●

The truncated Wigner approximation has a lot of good properties:–

Good scaling with problem size–

Highly parallelizable–

Accurate for a long time

63

Methods: truncated Wigner approximation●

We can also model decoherence64Methods: truncated Wigner approximation●●

We can also model decoherence
We start by defining a system which includes an environment and dark mattercomponent
dark matterallenvironment

65

Methods: truncated Wigner approximation●●●

We can also model decoherence
We start by defining a system which includes an environment and dark mattercomponent
We then model the dark matter-environment interaction

Gravitational
interaction

66

Methods: truncated Wigner approximation
●

●

●

●

We can also model decoherence
We start by defining a system which includes an environment and dark mattercomponent
We then model the dark matter-environment interaction
Use this Hamiltonian and a joint Wigner function to describe the evolution of thesystemTest particle
phase spaceposition initiallylocalized at
67

Methods: truncated Wigner approximation
●

●

●

●

●

We can also model decoherence
We start by defining a system which includes an environment and dark mattercomponent
We then model the dark matter-environment interaction
Use this Hamiltonian and a joint Wigner function to describe the evolution of thesystemBecause we know luminous matter has well defined phase space trajectories weknow that the decoherence rate must be at least as fast as the test particle
enters into a macroscopic super position in phase space

68

Results

69

Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions

Pointer states

Decoherence
Quantum state

Pointer states

Analytic estimates

Analytic estimates

Quantum corrections

Decoherence70Results●●

We quantify the size of corrections using a parameter Q which measures how
the average spread in the wavefunction compares to the mean value
Q goes from 0 to 1 in all systems

71

Results●●

We quantify the size of corrections using a parameter Q which measures how
the average spread in the wavefunction compares to the mean value
Not a unique choice (or only one we looked at) but reliable indicator ofdifferences between quantum and classical evolutions72

Results

5●First analysis we performed was totest how long it takes for Q to growto a certain size (this defined thequantum breaktime) as a functionof the total number of particles
keeping the mean field evolution
fixed

tbr

4

∝ 1.33 log(ntot)

Full quantum
simulations

3

2

1
101

ntot

73

Results

74

Results

5∝ 1.33 log(ntot)

Full quantum

First analysis we performed was totest how long it takes for Q to growto a certain size (this defined thequantum breaktime) as a functionof the total number of particles
keeping the mean field evolution
fixed

3

2

0.09 log(x) + −0.56

2.0

1
101

ntot

tbr [td]

●

tbr

4

1.5

approximation

1.0

0.5

104

106
ntot

108

1010
75

Results

5∝ 1.33 log(ntot)

Full quantum●

First analysis we performed was totest how long it takes for Q to growto a certain size (this defined thequantum breaktime) as a functionof the total number of particles
keeping the mean field evolution
fixed
See a logarithmic enhancement in
the breaktime with particle number

3

2

0.09 log(x) + −0.56

2.0

1
101

ntot

tbr [td]

●

tbr

4

1.5

approximation

1.0

0.5

104

106
ntot

108

1010
76

Results

5∝ 1.33 log(ntot)

Full quantum●●

First analysis we performed was totest how long it takes for Q to growto a certain size (this defined thequantum breaktime) as a functionof the total number of particles
keeping the mean field evolution
fixed
See a logarithmic enhancement in
the breaktime with particle number
Well known prediction for systems
that exhibit classical chaos

3

20.09 log(x) + −0.56

2.0

1
101

ntot

tbr [td]

●

tbr

4

1.5

approximation

1.0

0.5

104

106
ntot

108

1010
77

Results

5∝ 1.33 log(ntot)

Full quantum

●

●

●

First analysis we performed was totest how long it takes for Q to grow toa certain size (this defined thequantum breaktime) as a function of
the total number of particles keeping
the mean field evolution fixed
See a logarithmic enhancement in
the breaktime with particle number
Well known prediction for systems
that exhibit classical chaos
Straightforward to understand in thetruncated Wigner context

3

2

0.09 log(x) + −0.56

2.0

1
101

ntot

tbr [td]

●

tbr

4

1.5

approximation

1.0

0.5

104

106
ntot

108

1010
78

Results●

Small quantum perturbations in initial conditions spread exponentially79

Results
Second analysis is to look at how Q grows

10−1
10−2
Q(t)

●

10−3
10−4
10−5
100

200
300
t [Myr]

400

500
80

Results
Second analysis is to look at how Q grows

10−1
10−2
Q(t)

●

10−3
10−4
10−5
100

200
300
t [Myr]

400

500
81

Results●

Second analysis is to look at how Q grows●

Staged growth82

Results●

Second analysis is to look at how Q grows●

Staged growth–

Initial quadratic growth83Results●

Second analysis is to look at how Q grows●

Staged growth–

Initial quadratic growth–

Exponential growth during collapse

84

Results●

Second analysis is to look at how Q grows●

Staged growth–

Initial quadratic growth–

Exponential growth during collapse–

Powerlaw after collapse

85

Results●

Second analysis is to look at how Q grows●

Staged growth●–

Initial quadratic growth–

Exponential growth during collapse–

Powerlaw after collapse

Any powerlaw growth is too slow but
exponential growth may be a problem for theclassical theory

86Results●

How does behavior generalize to 3D systems?

87Results●

How does behavior generalize to 3D systems?

●

Used 3 test problems: collapse of a random field88

Results●

How does behavior generalize to 3D systems?

●

Used 3 test problems: collapse of a random field, stable collapsed object

89Results●●

How does behavior generalize to 3D systems?
Used 3 test problems: collapse of a random field, stable collapsed object,merging of two collapsed objects

90Results
Results corroborate 1D expectations
Collapsing
10−92

Already collapsed
10−92∝ e7.1 t/tc

∝ t2.210−90∝ e7.1 t/tc

10−93
Q(t)

10−93
Q(t)

Merging

10−94

Q(t)

●

10−94

10−92

10−95
10−94

10−95
10−96
500

1000
t [Myr]

1500

2000

10−96

200

400
600
t [Myr]

800

1000

500

1000
t [Myr]

1500

2000

91

Results–

Nonlinear collapse/merging is exponential–

Powerlaw very early and post collapse

10∝ e7.1 t/tc

10−2

Merging∝ t2.2

10−90

10−4
Q(t)

Q(t)

10−94

10−94

10−92

10−5

10−95

10

−95

10

−94

10−96
500

1000
t [Myr]

1500

2000

10−96

10−3

∝ e7.1 t/tc

10−93

10−93
Q(t)

−92

10−1

Q(t)

Already collapsed

Collapsing
10−92

1D

Results corroborate 1D expectations

●

100
200

400
600
t [Myr]

800

1000

500

1000
t [Myr]

1500

2000

200
300
t [Myr]

400

500

3D
92

Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions

Pointer states

Decoherence
Quantum state

Pointer states

Analytic estimates

Analytic estimates

Quantum corrections

Decoherence93Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions

Pointer states

Decoherence
Quantum state

Pointer states

Analytic estimates

Analytic estimates

Quantum corrections

Decoherence94Results●

What predictions do quantum corrections effect?

95

Results●

Leading order effect is to remove density fluctuations from interference

96Results●

Leading order effect is to remove density fluctuations from interference97Results●

Leading order effect is to remove density fluctuations from interference98Results●●

Gravitational lensing

Leading order effect is to remove
density fluctuations from interference
Effects constraints that rely on the
granularity of the density profile–

Heating of ultra faint dwarf stellar
dispersions–

Constraints from gravitational
lensing

Stellar dispersions

Power, et al. MNRAS 2023

99

Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions
Simulated results

Quantum corrections

Pointer states

Decoherence
Quantum stateLess granulardensity

Pointer states
Analytic estimates

Decoherence100Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions
Simulated results

Quantum corrections

Pointer states

Decoherence
Quantum stateLess granulardensity

Pointer states
Analytic estimates

Decoherence101Results●

Test decoherence by coupling our dark matter state to a test particle

102

Results●

Test decoherence by coupling our dark matter state to a test particle●

Over time the test particle will evolve in a super position on phase space

103

Results●

Test decoherence by coupling our dark matter state to a test particle●

Over time the test particle will evolve in a super position on phase space●

This occurs at the same rate as quantum corrections are introduced

104Results●

Test decoherence by coupling our dark matter state to a test particle●

Over time the test particle will evolve in a super position on phase space●

This occurs at the same rate as quantum corrections are introduced●

Unlike Schrodinger’s cat, both the quantum corrections and the decoherence
are caused by the same thing, gravity

105

Results●

Test decoherence by coupling our dark matter state to a test particle●

Over time the test particle will evolve in a super position on phase space●

This occurs at the same rate as quantum corrections are introduced

●

●

●

Unlike Schrodinger’s cat, both the quantum corrections and the decoherence
are caused by the same thing, gravity
Difficult to evolve into a state with large quantum corrections without also
putting test particles into macroscopic super positions which we do not
observe
The decoherence time scale must be at least as fast as the nonlinear
timescale106

Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions
Simulated results

Quantum corrections

Pointer states

Decoherence
Quantum stateLess granulardensity

Pointer states
Analytic estimates

Decoherence107Schrödinger’s Cat
Initial conditions

Quantum state

Life time of Cs

Env interaction

Quantum corrections
Initial conditions
Simulated results

Quantum corrections

Pointer states

Decoherence
Quantum stateLess granulardensity

Pointer states
Simulated results

Decoherence108Conclusions

109

Conclusions
–

–

Q(t)

Quantum corrections:

10−93
10−94
10−95

grow exponentially in systems that areexperience nonlinear growth (collapsing,merging, etc)

10−96
500

Grow slowly in systems alreadycollapsed systems

10−92

1000
t [Myr]

1500

2000

800

1000

∝ t2.2

10−93
Q(t)

●

∝ e7.1 t/tc

10−92

10−94
10−95
10−96

200

400
600
t [Myr]110

Conclusions●●

Quantum corrections:–grow exponentially in systems that are experience nonlinear growth(collapsing, merging, etc)–

Grow slowly in systems already collapsed systems

Corrections remove granular/interference structures from the density

111

Conclusions●●●

Quantum corrections:–grow exponentially in systems
that are experience nonlinear
growth (collapsing, merging, etc)–

Grow slowly in systems alreadycollapsed systems

Corrections remove
granular/interference structures fromthe density
Decoherence occurs at least as fast
as quantum corrections grow112

Conclusions●

Quantum corrections:–grow exponentially in systems that are experience nonlinear growth(collapsing, merging, etc)–

Grow slowly in systems already collapsed systems●

Corrections remove granular/interference structures from the density●

Decoherence occurs at least as fast as quantum corrections grow●●

Small systems will be most effected by corrections but have the longest
dynamical times
Decoherence means states with large corrections are unlikely
113

Conclusions●

Quantum corrections:–grow exponentially in systems that are experience nonlinear growth (collapsing,merging, etc)–

Grow slowly in systems already collapsed systems●

Corrections remove granular/interference structures from the density●

Decoherence occurs at least as fast as quantum corrections grow●

Small systems will be most effected by corrections but have the longest dynamical
times

●

Decoherence means states with large corrections are unlikely●

Strong support that the predictions of the classical theory are accurate
114

Questions

115

Extra slides: Truncated Wigner Approximation
Equations of motion for the truncated Wigner expansion [see Polkovnikov (Annals of Physics 2010)]
If I approximate my Wigner function as an ensemble of streams:

Is the independent classical evolution:

The same as approximating the evolution of the Wigner function to this order?:116Extra slides: Truncated Wigner Approximation
Yes

117

Extra slides: Other interactions/quantum states●

Looked at field number and number
eigenstates–

Coherent states are the states
associated with the misalignment
mechanism–

Number state is immediately
nonclassical–Field number state would be interesting
follow up

118

Extra slides: Other interactions/quantum states●

Looked at contact interaction–

Found spreads wavefunction too slowly (powerlaw, cf. Kerr oscillator)

119Extra slides: Other interactions/quantum states

Coherent state

Field number state120