Motivation Kinetic Field Theory Results
A Kinetic Field Theoryfor Cosmic Structure Formation
Velocity Statistics
Carsten Littek
Supervisor: M. BartelmannCollaborators: F. Fabis, E. Kozlikin, R. Lilow, and more
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Why (another) Analytic Theory?
Large Scale Structure
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Basic Idea
position q
time t
momentum p
trajectories(p, q)(t)
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Advantages of Phase-Space
I Hamiltonian equations of motion for each particle
q =∂H∂p
, p = −∂H∂q
, x = (q, p)
I Solution given by a retarded Green’s function
x = G(t, 0)x (i)︸ ︷︷ ︸freemotion
−∫
dt ′G(t, t ′)K (t ′)︸ ︷︷ ︸interaction
I static space, and potential v
G(t, t ′) =
(1 (t − t ′)/m0 1
), K =
(0∇v
)A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Generating Functional
Statistical Mechanics
Zc =
∫dΓe−βH(x)
Total Energy
〈E 〉 = − ∂
∂βlnZc
Kinetic Field Theory
Z = eiSI∫
dΓ(i)ei∫dt〈J(t),x(t)〉
Momentum
〈~p〉 =δ
iδJpZ
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Generating Functional
Statistical Mechanics
Zc =
∫dΓe−βH(x)
Total Energy
〈E 〉 = − ∂
∂βlnZc
Kinetic Field Theory
Z = eiSI∫
dΓ(i)ei∫dt〈J(t),x(t)〉
Momentum
〈~p〉 =δ
iδJpZ
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Application to Cosmology
I Distribution in Phase-Space from CMB correlations
P[q,p] = NC(Cδδ,Cδp)exp
[−1
2p>Cppp
]I Adapt the Green’s function to expanding space
I (improved) Zel’dovich trajectories
I effective potential decreases in expanding Universe
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Full Correlations
1e+01
1e+02
1e+03
1e+04
0.001 0.01 0.1 1 10
Pδ(
k)
wave number k [h-1 Mpc]
Cosmic Emulatorquadratic Cpp
full Cpp
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Dark Matter and Gas
1e-12
1e-11
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.01 0.1 1 10 100
dens
ity
pow
er s
pect
ra
k [h Mpc-1]
free Pδ(k)Pδ(k), hydro
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Velocity Field - Divergence and Vorticity
1e-08
1e-07
1e-06
1e-05
1e-04
1e-03
1e-02
1e-01
1e+00
1e+01
1e+02
0.001 0.01 0.1 1 10
velo
city
pow
er s
pect
ra
wave number k [h-1 Mpc]
free divergencedivergence
free vorticityvorticity
A Kinetic Field Theory: Velocity Statistics
Motivation Kinetic Field Theory Results
Summary
I non-equilibrium statistical field theory
I Hamiltonian equations of motion with simple Green’s function
I Expansion parameter is deviation from unperturbedtrajectories
I numerical results are reproduced quite well already at 1st
order particle interactions
I inclusion of gas is possible
I vorticity is generated naturally
A Kinetic Field Theory: Velocity Statistics
