Schrödinger method

Large scale structure formation
with the
Schrödinger method
Cora Uhlemann
Arnold Sommerfeld Center, LMU
& Excellence Cluster Universe
to be continued in
Utrecht University, Netherlands
visitor grant sponsored by
Balzan Centre for Cosmological Studies
Outline
1. Structure formation
2. Analytical description of
cold dark matter
a. dust model
b. Schrödinger method
c. coarse-grained dust model
3. Summary
Cosmological Structure Formation
-13.8 billion years: nearly uniform initial state
Inflation
• established `boring` initial conditions
• quantum fluctuations get amplified
• primordial plasma cools → recombination → CMB
Structure formation
• hierarchical
• tiny over-densities act as seeds
• congregation via gravitational instability
• collapse into bound structures
Large scale structure: Dark Matter
• linear regime
✓ analytically understood
• nonlinear stage
?! N-body simulations inevitable
WANTED
theoretical
N-body
double
today: rich structures in cosmic web
Kravtsov & Klypin (simulations @NCSA)
Describing Cold Dark Matter
with the
Schrödinger method
Describing Cold Dark Matter
phase space distribution function f(t,x,p)
• describes number density & distribution of momenta p
numerical realization
theoretical expectation
W
fcg
Schrödinger
fN N-body
p
p
x
x
p
x
Pueblas & Scoccimarro (2009, PRD 80)
Schrödinger method: Widrow & Kaiser (1993, ApJ 416)
Widrow (1997, PRD 55)
Describing Cold Dark Matter
N-body picture
• N non-relativistic particles
• only gravitational interaction
fN (x, p, ⇥ ) =
Vlasov - Poisson equation
) ⇥t fN (x, p, ) =
p
rx fN + amrx V rp fN
am
N
X
D (x
xi (⇥ ) )
D (p
pi (⇥ ) )
i=1
pi
⇤ xi =
,
am
⇤ pi = amrV (xi )
1
0
N
4⇥Gm @X
xj ) hnN iA
V (xi ) =
D (xi
a
j6=i
Describing Cold Dark Matter
phase space distribution function f(t,x,p)
• N-body: non-relativistic, only gravitationally
• continuous: ensemble average, no collisions
Vlasov - Poisson equation
fN =
f
n=
d3p f
i
D (x
xi )
D (p
pi )
p
⇤ gravitational
f (x, p, ⇥ ) =
rx f + amrx V rp f
potential
am
p
⇤4f (x,
Gmp, ⇥ ) = am rx f + am
V (x, ⇥ ) =
(n(x, ⇥ ) hni)
Z aV (x, ⇥ ) = 4 Gm (n(x, ⇥ )
Z a density
n = integro
d3p f number
p
⇤ f (x, p, ⇥ ) =
rx f + amrx V rp f
am
4 Gm
partial
V3+3+1
(x, ⇥ ) =
(n(x,
⇥ ) hni)
nonlinear
Z a
variables
X
Solving is hard!
have to choose a special ansatz for f(x,p)
reduce some information content
n=
d3p f
Describing Cold Dark Matter
continuous distribution function f(t,x,p)
• ensemble average
• dropping collision terms ~1/N
Vlasov - Poisson equation
fN =
f
X
density n(x): M (0) = n(x) , velocity v(x): M (1) = nv(x)
velocity dispersion 𝞼(x): M (2) = n (vv + ) (x) , …
@t M
(n)
=
1
(n+1)
r
·
M
a2 m
infinite coupled hierarchy
xi )
D (p
pi )
p
⇤ gravitational
f (x, p, ⇥ ) =
rx f + amrx V rp f
potential
am
4 Gm
V (x, ⇥ ) =
(n(x, ⇥ ) hni)
Z a
p
⇤ f (x, p, ⇥ ) =
rx f + amrx V rp f
am
4 Gm
V (x, ⇥ ) =
(n(x, ⇥ ) hni)Z
n=
a
Z
(n)
3
M
(x)
=
d
p p i 1 . . . pin f
Hierarchy of Moments
3
n= d p f
•
•
i
D (x
cumulant
mrV · M (n
1)
d3p f
Describing Cold Dark Matter
phase space distribution function f(t,x,p)
• N-body: non-relativistic, only gravitationally
• continuous: ensemble average, no collisions
Vlasov - Poisson equation
p
⇤ f (x, p, ⇥ ) =
rx f + amrx V rp f
am
4 Gm
V (x, ⇥ ) =
(n(x, ⇥ ) hni)
Z a
Hierarchy of Cumulants
3
n=
•
•
dp f
fN =
f
X
i
D (x
xi )
D (p
p
⇤ gravitational
f (x, p, ⇥ ) =
rx f + amrx V rp f
potential
am
4 Gm
V (x, ⇥ ) =
(n(x, ⇥ ) hni)
Z a
n=
d3p f
density n(x): C (0) = ln n(x) , velocity v(x): C (1) = v(x)
velocity dispersion 𝞼(x): C (2) = (x) , …
@t C (n) =
n
h
X
1
(n+1)
(n+1
r
·
C
+
C
a2 m
|S|=0
consistent truncation C
( n)
⌘0
pi )
|S|)
· rC (|S|)
9n : |S|  1 ^ |S|
i
n1
· mrV
!
n 8S ) n = 2
Dust model
dust model
• only consistent truncation of hierarchy
• pressureless fluid: density and velocity
fd (x, p, ⌧ ) = n(x, ⌧ )
Continuity ⇥⌧ n =
Euler
⇥⌧
=
(3)
D (p
r (x, ⌧ ))
1
r(nr )
am
1
(r )2 amV
2am
•
•
limited to single-stream
no velocity dispersion, …
•
•
shell-crossing singularities
no virialization
fd
impossible
with dust
fd
Schrödinger method
Schrödinger method
self-gravitating field
=
p
n exp
i
~
Schrödinger - Poisson equation

~2
i~@t =
+ mV
2
2a m
4⇡G⇢0
V =
(| |2 1)
a
=
Schrödinger method
Schrödinger method
Wigner function, constructed from self-gravitating field
Z
3 0 3 0
d xd p
exp
3
( ⇥x ⇥ p )

(x
0 2
x)
2⇥x2
Wigner-Vlasov equation
@ t fW

p2
=
+ mV
2
2a m
Vlasov equation

2
sin
~
0 2
(p p )
fW (x, p)
=
2
2⇥p
✓
!
~
(rx rp
2
⇣ !
p2
@t f =
+ mV
rx rp
2
2a m
Z
n exp
i
~

3
d x̃
i 0
0
¯ 0 + x̃)
exp
2
p
·
x̃
⇤(x
x̃)
⇤(x
(2 ~)3
~
Schrödinger - Poisson equation
◆
!
rp rx ) fW
!⌘
rp rx f
=
p
Problems
• Wigner distribution function not manifestly positive
• time evolution not in good correspondence to Vlasov

~2
i~@t =
+ mV
2
2a m
4⇡G⇢0
V =
(| |2 1)
a
Schrödinger method
Schrödinger method
• Coarse-grained Wigner function, constructed from self-gravitating field
W
fcg
(x, p)
=
Z
3 0 3 0
d xd p
exp
3
( ⇥x ⇥ p )

(x
0 2
x)
2⇥x2
(p
0 2
p)
2⇥p2
Z
3
=

p
n exp
d x̃
i 0
0
exp
2
p
·
x̃
⇤(x
(2 ~)3
~
i
~
¯ 0 + x̃)
x̃)⇤(x
Schrödinger - Poisson equation
Wigner-Vlasov equation
@ t fW

p2
=
+ mV
2
2a m
Vlasov equation

2
sin
~
✓
!
~
(rx rp
2
⇣ !
p2
@t f =
+ mV
rx rp
2
2a m
◆
!
rp rx ) fW
!⌘
rp rx f
Problems
• Wigner distribution function not manifestly positive
• time evolution not in good correspondence to Vlasov
• solution: add a coarse-graining x p & ~/2

~2
i~@t =
+ mV
2
2a m
4⇡G⇢0
V =
(| |2 1)
a
Schrödinger method
Schrödinger method
• Coarse-grained Wigner function, constructed from self-gravitating field
W
fcg
(x, p)
=
Z
3 0 3 0
d xd p
exp
3
( ⇥x ⇥ p )

(x
0 2
x)
2⇥x2
(p
0 2
p)
2⇥p2
Z
3
=

p
n exp
d x̃
i 0
0
exp
2
p
·
x̃
⇤(x
(2 ~)3
~
i
~
¯ 0 + x̃)
x̃)⇤(x
Schrödinger - Poisson equation
degrees of freedom
• 2: amplitude n & phase ϕ
parameters
• coarse-graining x , p
• fundamental resolution
•
Schrödinger scale ~
• degree of restriction
• dust as special case

~2
i~@t =
+ mV
2
2a m
4⇡G⇢0
V =
(| |2 1)
a
x p
& ~/2
Continuity ⇥⌧ n =
Euler
⇥⌧
=
1
r(nr )
quantum potential
am
✓ p ◆
2
1
~
n
2
(r )
amV +
p
2am
2am
n
Schrödinger method at a glance
time-evolution
f
Wigner
Vlasov
WANTED
shellcrossing
N-body
double
fN
x
p
f
W
~!0
fd
dust
⌧ xtyp
⌧ ptyp
~/2 .
coarse grained
Vlasov
fcg
x p
Takahashi
(1989, PTP 98)
coarse grained
Wigner
W
fcg
Features of Schrödinger Method
✅ prevention of shell-crossing singularities
⁉ occurrence of phase jumps
‼
=
p
nei
/~
free of pathologies
Features of Schrödinger Method
phase space density
Multi-streaming
❌ dust model: fails at shell-crossing
✅ Schrödinger method: can go beyond shell-crossing
blue S contours: Schrödinger method
red dotted Z line: Zeldovich solution (dust model)
Virialization
❌ even in extended models: no virialization
✅ Schrödinger method: bound structures like halos
FOUND
WANTED
theoretical
N-body
double
for fN
Features of Schrödinger Method
Schrödinger method
• Coarse-grained Wigner function, constructed from self-gravitating field
W
fcg
(x, p)
=
Z
3 0 3 0
d xd p
exp
3
( ⇥x ⇥ p )

(x
0 2
x)
2⇥x2
(p
0 2
p)
2⇥p2
Z
3

n̄(x) = exp
•
2
2 x
⇤
n(x)
C (0) = ln n,
C (n+2) =
C (1) = r
~2
rrC (n)
4
add coarse-graining on top of that
i
~
¯ 0 + x̃)
x̃)⇤(x
special p-dependence
⇥1
1
v̄(x) =
exp 2
amn̄(x)
higher cumulants given self-consistently
evolution equations fulfilled automatically
n exp
d x̃
i 0
0
exp
2
p
·
x̃
⇤(x
(2 ~)3
~
Cumulants
• lowest two: macroscopic density & velocity
⇥1
=
p
2
x
⇤
(nr ) (x)
closure of hierarchy
CU, Kopp & Haugg (2014, PRD 90, 023517)
Multi-streaming
•
•
higher cumulants encode multi-streaming effects
during shell-crossing: higher moments sourced dynamically
H1L
1
CH1L =
= uu
C
H2L
H2L
33
H3L
H3L
Out[112]=
u @10-3 cD
C
C
Out[112]=
u @10-3 cD
u @10-3 cD
Features of Schrödinger Method
C
C
0
-1
0.0
0.5
1.0
1.5
2.0
2.0
2.5
2.5
x @MpcD
a
Schrödinger method: cumulants at x = −0.5 Mpc:
all equally important after shell crossing
x @MpcD
Schrödinger method at a glance
f
f
Wigner
Vlasov
Features of Schrödinger Method
!
WANTED
N-body
double
fN
blue S: Schrödinger method!
red Z: Zeldovich solution
Virialization
after shell-crossing: all cumulants important
dust
H1L
1
C =u
C
3
C
H2L
H3L
0
-1
0.0
!
❌ even in extended models: !
no virialization
✅ Schrödinger method: !
bound structures!
fd
phase space density
Multi-streaming
!
❌ dust model: !
fails at shell-crossing!
!
✅ Schrödinger method:
beyond shell-crossing
W
0.5
1.0
1.5
2.0
2.5
FOUND!
!
!
WANTED!
!
!
theoretical
N-body
double
for fN
coarse grained
Vlasov
coarse grained
Wigner
fcg
W
fcg
Dark Matter power spectrum
with the
Coarse-grained dust model
Coarse-grained perturbation theory
f
Vlasov
fd
WANTED
dust
N-body
double
coarse grained
dust
d
fcg
fN
coarse grained
Vlasov
coarse grained
Wigner
fcg
W
fcg
Eulerian Perturbation Theory
Dust model
• express fluid equations in terms of = n 1 and ✓ = r · v /
• perturbative expansion: separation ansatz (fastest growing mode)
(⌧, k) =
1
X
an (⌧ )
n (k)
n=1
✓(⌧, k) = H(⌧ )
1
X
(no vorticity)
an (⌧ ) ✓n (k)
n=1
Correlation function
• 2-point correlation: excess probability of finding 2 objects separated by r
homogeneity & isotropy: ⇠(r) = ⇠(r)
dP = n[1 + ⇠(r)]dV
⇠(r)
distribution with
excess & defect
r
r+
r-
random
0
Poisson
distribution
r+
r-
r
Eulerian Perturbation Theory
Dust model
• express fluid equations in terms of = n 1 and ✓ = r · v =
• perturbative expansion: separation ansatz (fastest growing mode)
(⌧, k) =
1
X
an (⌧ )
✓(⌧, k) = H(⌧ )
n (k)
n=1
Density power spectrum
1
X
(no vorticity)
an (⌧ ) ✓n (k)
n=1
105
correlation function
=
FT of power spectrum
Pdd @Mpc-3 D
104
1000
⇠(r) =
100
10
0.001
•
•
linear
1-loop
0.01
k @Mpc-1 D
0.1
1
1
2⇡ 2
Z
dk k 2 P (k)
sin(kr)
kr
Eulerian Perturbation Theory
Coarse grained dust model
• consider only 𝜎x correction in Schrödinger method
• in 1st order: smoothing of input power spectrum
Density power spectrum
105
Pdd @Mpc-3 D
104
1000
finite resolution
effects
linear dd SPT
linear dd cgSPT 1Mpc
linear dd cgSPT 2Mpc
1-loop dd SPT
100
1-loop dd cgSPT 1Mpc
1-loop dd cgSPT 2Mpc
10
0.001
0.01
k @Mpc-1 D
0.1
1
trivial effect:
density power spectrum
gets smoothed
Eulerian Perturbation Theory
Coarse grained dust model
• similar to dust, but mass-weighted velocity
• large scale vorticity w̄ := r ⇥ v̄ 6= 0 ❗
nv
v̄ :=
n̄
Vorticity power spectrum Pww(k)
corresponding N-body data
Pww @HkmêHs MpcêhLL2 h-3 Mpc3 D
CU & Kopp (2015, PRD 91, 084010, arXiv: 1407.4810)
100
ww cgSPT 1Mpcêh
10
ww cgSPT 2Mpcêh
1
ww cgSPT 4Mpcêh
0.1
0.01
0.001
0.001
0.01
k @h Mpc-1 D
0.1
1
Hahn, Angulo & Abel (2014, arXiv:1404.2280)
Conclusion & Prospects
Schrödinger method
• models CDM using a self-gravitating scalar field
• analytical tool to access nonlinear stage of structure formation
• describes multi-streaming & allows for virialization
CU, Kopp, Haugg (2014, PRD 90, 023517, arXiv: 1403.5567)
Coarse-grained dust model
• mass-weighted velocity via Gaussian smoothing
• vorticity compatible with N-body
CU & Kopp (2015, PRD 91, 084010, arXiv: 1407.4810)
Prospects
• Schrödinger method in terms of a filtering of the distribution function
• approximate hierarchy closures instead of exact truncation
• comparison of moments 1D Schrödinger vs. 1D Vlasov-Poisson
• disentangle limitations of pressureless fluid & perturbation theory
•
•
consider the effect of varying phase-space resolution ~
understand universal density profiles of halos (NFW)
q
Thank You for Your Attention
Questions?