2 - INFN

Solving the collisionless Boltzmann equation
using N-body simulations
• Gravitational dynamics
• N-body simulations
The distribution of galaxies in the
local Universe. Distances
(redshifts) are measured using
Hubbles velocity-distance relation.
The galaxies are not distributed
randomly but are clustered along
filaments that connect together
the groups and clusters.
This clustering pattern is repeated
on larger scales and can be
quantified using correlation
functions, topology, fractal
dimensions, power spectrum
analysis….
cfa survey (1980)
1000 galaxies
1 00
Mp
c
2df survey (2001)
100,000 galaxies
0
0
10
M
pc
Globular clusters
Omega-Cen
A cluster of 10^4 – 10^6 stars roughly 1-10
parsecs across.
Galaxies are surrounded by a swarm of
globular clusters – the brighter the galaxy
then the more globular clusters.
Spectroscopy of the stars reveals that they
are moving at a few km/s.
d2 r
GM
acceleration =
=− 2
2
dt
r
Integrating this equation with the boundary condition r=R at t=0:
Tfreefall
1
1 R3
−
=
∝ (Gρ ) 2
2 GM
= 1/ 2 (10 x3.1e16)3 /(6.67e − 11x1.0e6 x 2.0e30)
= 0.0003Gyrs
ρstar = 250 stars/pc3
ρ proton = 104 cm −3
Tcollision = 1/ (π nσ v ) = 1015 yrs >> age of the universe
Globular clusters are fascinating objects for many reasons:
•Short dynamical time – undergo core collapse, perhaps sites where black holes form.
•Formation before the main galaxy – they tell us something about the initial conditions.
•They can be used as “test particles” to weigh the galactic systems.
Many body gravitating systems
Systems with N>2 gravitating objects (stars, protons etc) can’t be
solved analytically.
Two ways to proceed:
Statistical treatment of dynamics:
-determine the gravitational potential
-estimate/observe shape of orbits
-solve for the past and future evolution
Numerical simulations:
-direct summation of gravitational forces
-integrate equations of motion
-update positions, velocities
-repeat process on timestep dt
Equation of motion:
Many body systems
d 2x
m 2 = f ( x),
dt
Phase space:
dx
dv
= v, m = f ( x )
dt
dt
The flow of particles through phase
space is quite intricate. Within a small
volume dxdydz the phase space
3
3
density f ( x, y, z , vx , v y , vz , t )d x, d v
is conserved.
The evolution of the entire system can
be followed using the collisionless
Boltzmann equation and Poissons eq.
It’s a 6 dimensional equation that is
difficult to solve. Gravitating systems
do not have an equation of state that
relates pressure and density.
The origin of structure
in the Universe
The origin of cosmic
structures is are complex
non-linear problems best
solved by computational
methods
1
T
10
n
dy
1
10
^
0
10
^
0
N
10
Galaxy formation, the stability of
the solar system, the core
collapse of a globular cluster into
a black hole all require about
10^18 floating point operations.
The distribution function
Within a smooth potential the number of stars with positions in
volume d3x centered on x and velocities d3v centred on v is:
f ( x, v, t )d 3 xd 3v
is the distribution function (phase space density)
Given f ( x, v, t ) we can predict f ( x, v, to ) at any time in the future or past
Assume that f is a smooth continuous function such that particles move
smoothly through phase space. i.e. N >> 106
The coordinates in phase space:
w ≡ ( x, v) ≡ ( x, y, z , vx , v y , vz ) ≡ ( w1 , w2 , w3 , w4 , w5 , w6 )
The velocity of the flow
w& ≡ ( x& , v&) = (v, −∇Φ )
Therefore f ( w, t ) must satisfy a continuity equation analogous
to ρ (x,t) of an ordinary fluid. i.e. stars are conserved in the flow
and drift slowly through phase space.
In the most general definition, a Boltzmann Equation (BE) is a differential
equation for the evolution of a probability density during time furnished
with an initial condition. This means that there are BEs for problems with
discrete and continuous states allowing discrete and continuous time
evolution.
The Boltzmann equation appears in many areas of science:
Kinetic treatment of gas
Material science
Cell Biology
Fluid dynamics
Electron transport in nuclear reactors
Radiative transport in planetary atmospheres
Astrophysics
The fluid analogy
Rate of change of mass with time:
dm
∂ρ 3
= ∫ d x = − ∫ ρ v ⋅d 2 S
dt V ∂t
S
volume V
S
(v, ρ )
Use the divergence theorem:
⎛ ∂ρ
⎞ 3
∫V ∇ ⋅ fd x = ∫S fd S → V∫ ⎜⎝ ∂t + ∇ ⋅ ( ρ v) ⎟⎠d x = 0
This must hold for any volume V therefore:
∂ρ
(Continuity equation)
+ ∇ ⋅ ( ρ v) = 0
∂t
Now consider the analogous stellar phase space distribution function
3
2
6
∂ ( fw& α )
∂f
+∑
=0
∂t α =1 ∂wα
6
∂ ( fw& α )
∂f
+∑
=0
∂t α =1 ∂wα
Simplify this equation:
volume V
ρ
3
⎛ ∂vi ∂v&i ⎞
∂w& α
(
v
,
)
=
+
⎜
⎟
S
∑
∑
∂vi ⎠
α =1 ∂wα
i =1 ⎝ ∂xi
∂vi
= 0 since x and v are independent coordinates of phase space
but
∂xi
6
and since ∇Φ does not depend on velocity we have:
⎛ ∂Φ ⎞
⎜
⎟=0
⎝ ∂xi ⎠
Now the continuity equation becomes the CBE:
∂w& α 3
∂
=∑ −
∑
∂vi
α =1 ∂wα
i =1
6
6
3
⎡ ∂f ∂Φ ∂f ⎤
∂f
∂f
∂f
+ ∑ w& α
= 0 or
+ ∑ ⎢vi
−
⎥=0
∂t α =1 ∂wα
∂t i =1 ⎣ ∂xi ∂xi ∂vi ⎦
or
3
⎡ ∂f ∂Φ ∂f ⎤
∂f
+ ∑ ⎢vi
−
⎥=0
∂t i =1 ⎣ ∂xi ∂xi ∂vi ⎦
in vector notion:
∂f
∂f
+ v ⋅∇f − ∇Φ ⋅ = 0
∂t
∂v
This is the fundamental equation of stellar dynamics.
What does it mean?
Transform to Lagrangian coordinates that travel along with a star.
x = xo = const.
d
∂
= + w& ⋅∇ 6 →
dt ∂t
6
df ∂f
∂f
=
+ ∑ w& α
=0
dt ∂t α =1 ∂wα
The fluid analogy
volume V
S
(v, ρ )
As in the fluid case we can integrate the CBE to see that the
first term is the rate of change of stars in V and the second term is
the rate of change of outflow/inflow of stars from/into V .
df
is the rate of change of phase space density seen by an observer
dt travelling along with a star
df
= 0 → that the flow of stellar points through phase space is
dt
incompressible. The space density near the star is constant.
Example of incompressible flow: idealised marathon race where all the
runners run at a constant speed. At the start of the race they travel at a
wide variety of speeds. At the finish the density is low but the runners
going past have nearly the same speed.
∂f
In the absence of collisions then
= 0 i.e. phase space is conserved
∂t
along a given trajectory through 6-d space. This also implies that any maximum
of the distribution always remains the same. If the force is conservative, as
the gravitational force, then the 6 dimensional flow is incompressible, which
means that the phase space volume of a contour f ′ is always conserved.
The N-body technique
The collisionless Boltzmann equation is the fundamental
equation of a collisionless system
∂f
∂f
∂f
+ v ⋅ − ∇Φ ⋅ = 0
∂t
∂r
∂v
Where f = ( x, y, z, vx , v y , vz ) is the phase space density.
This can be combined with Poissons equation to completely
describe a gravitating system
∇ 2 Φ = 4π G ρ
Where ρ = ∫ fd 3v
Solving these equations using finite
difference techniques is impractical
therefore solve the equations of motions of
N particles directly. The N particles are a
Monte-Carlo realisation of the true initial
conditions.
dr
dv
= v,
= −∇Φ
dt
dt
Gm j
Φ = −∑
2
i≠ j
rj − ri + ε 2
(
)
1/ 2
To represent the mass distribution function f(r,v,t) one uses a set of N
bodies each possessing mass, position and velocity – i.e. one replaces the
continuous distribution function with a set of delta functions.
N
f (r , v) → ∑ miδ 3 (r − ri )δ 3 (v − vi )
i =1
For this substitution to work the expected mass of the bodies within any
phase space volume V must be equal to the integral of the distribution
function over that volume:
3
3
d
rd
vf (r , v) =
∫
V
∑
( ri ,vi )∈V
mi
Where the angle brackets indicate an average of statistically equivalent
realizations and the sum includes all bodies with phase space coordinates
within the volume V
The potential is calculated using the particle distribution so that
Poisson’s eq. looks like:
N
∇ Φ = 4π G ∑ miδ 3 (r − ri )
2
i =1
It is a standard trick to use orders of magnitude fewer particles than a
real system may contain.
i.e. air turbulence has a length scale of about 10cm. A cube of this size
would contain about 10^22 particles. The largest volume of air that could
be simulated exactly is 0.01mm^3.
Weather prediction may wish to resolve this 10cm length scale on the scale
of the earth which would require a grid of order 10^18 or 10^20 particles.
To make these problems manageable then use a macroscopic fluid
treatment where small scale turbulence is parameterised in viscosity and
diffusion terms.
For some cosmology problems we can use one particle per star – i.e. a
globular cluster with 10^6 stars can be followed exactly.
However other problems, such as the 10^70 dark matter particles in a
galactic halo can’t be treated exactly and we use “super massive” particles
to represent the phase flow – as long as particle-particle collisional effects
do not become important.
Constructing Initial Conditions
Before we discuss the numerical techniques of how one evolves N particles, first
we need to set up some initial conditions – i.e. a Monte Carlo representation of a
mass distribution using a set of points.
This is non-trivial. For example, assume we want to evolve a Globular Cluster of
stars to see if the central region will collapse to a black hole. We need to initialise
positions and velocities of particles such that our model globular cluster is in
equilibrium.
In other words, we have to assign velocities to particles as a function of radius
such that the flow of particles through any small volume is conserved. This ensures
that the model is in equilibrium and satisfies the collisionless Boltzmann equation.
The CBE can only be solved exactly for a few density-potential pairs that are
usually spherical systems with isotropic velocity dispersion profiles.
King model, Plummer sphere, Hernquist halo, Evans disks…
(The infinite isothermal sphere is also simple but needs to be an infinite model to
be stable.)
Usually we are trying to model a system with an observed density profile
therefore we wish to solve for the velocity distribution of particles.
Assigning positions to particles to create a random realisation of a 3d
density is reasonably easy. (Use accept/reject method or faster to draw
radii from the cumulative mass distribution.)
Assigning velocities is the tricky part. Use Poissons equation to relate the
observed density profile to the integral over the phase space density.
∇ 2 Φ = 4π G ρ = 4π G ∫ fd 3v
Jean’s theorem states that for a spherically symmetric system the
distribution function f is a function of energy alone. Thus:
∇ 2Φ =
1 d ⎛ 2 dΦ ⎞
⎜r
⎟ = 4π G ∫
2
r dr ⎝ dr ⎠
⎛1
⎞
f ⎜ v 2 + Φ ⎟d 3v
⎝2
⎠
Assume some form for the distribution function and solve Poisson’s
equation for the density profile – this will be a solution to the CBE.
Alternatively, can solve this equation directly for the energy dist.
Approximate the distribution function for all components using moments of
the CBE. For a spherical halo the 2nd moment is:
d ( ρ h vr2 ) ρ h ⎡ 2
dΦ
+
2vr − (vθ2 + vΦ2 ) ⎤ = − ρ h
⎦
dr
r ⎣
dr
If the halo is isotropic (random velocity vectors) then we can integrate:
∞
1
dΦ
v =
dr
ρ h (r )
∫
dr
ρ h (r ) r
2
r
This gives the velocity dispersion as a function of radius. We can now
draw velocities randomly from a function whose 2nd moment is v 2
r
i.e. a Gaussian:
⎛ 1 ⎞
F (v, r ) = 4π ⎜
2 ⎟
⎝ 2πσ ⎠
3/ 2
σ 2 exp(−σ 2 / 2vr2 )
This will have incorrect higher order moments!
More complex models - galaxies
ρ halo (r ) =
ρ bulge
M
2π
3
2
2
)
α exp(−r / rcore
rcore (r 2 + β 2 )
M
1
,
=
3
2
2π ac [t (1 + t ) ]
ρ disk ( R, z ) =
x2 + y 2 z 2
t =
+ 2
2
a
c
2
M
2
exp[
−
R
/
h
]
sech
( z / zo )
4π h 2 zo2
Approximate all the components using moments of the CBE, so that the
potential is the sum of the disk, bulge and halo components….
∞
1
dΦ
v =
dr
ρ h (r )
∫
dr
ρ h (r ) r
2
r
Nice example is to study the effects of the stellar bar
on the central dark matter distribution
Simulating N-Body systems
Simulating N-Body systems
N-Body codes calculate the gravitational
force between particles and update the
positions and velocities on a sequence of
discrete timesteps.
Simulating N-Body systems
Here we are simply
following the
Newtonian gravitational
force between N stars
as they orbit within
their collective
potential.
The hard part is doing
this quickly and
accurately for many
particles.
In 3d each two body interaction requires 10 flops.
Total number of operations for a small simulation:
T force = 10.N timesteps N particles ( N particles −1 )τ
where τ is the wallclock time per flop
N timesteps ≈ 1000
N particles ≈ 106
τ CrayT 3 E ≈ 10−11 sec
⇒ T force ≈ 14, 000hours=2 processor years
Galaxy formation, the stability of the solar system, the core collapse of a globular
cluster into a black hole all require about 10^18 floating point operations.
Hardware and software gains
8
Ghigna
7
6
Log(N)
Carlberg, Dubinski,
Warren etal
5
4
White
3
White
2
Holmberg
Moore’s law
Aarseth, van
Albada, Peebles
1
0
1940
1950
1960
1970
1980
1990
2000
2010
Year
Consider the increase in the resolution of collisionless simulations of clusters
The world’s highest resolution N-body simulation?
Andreas Adelmann, PSI (Paul Scherrer Institute)
1 particle per proton = 10^11 particle simulation
Ways to speed up the force calculation
Direct summation
Start with initial positions and velocities
Compute gravitational potential from particle distribution
Compute accelerations from potential gradients
Update positions and velocities for one step.
Goto step 1
Robust, accurate but expensive – scales as n^2
Can use special purpose hardware that does 1/r^2 on hardware PCI board
Field Expansion (PM)
Poissons equation can be expressed in terms of Fourier components.
Compute density at a point (interpolate onto a grid)
Compute potential on the grid by FFT
Compute accelerations by finite differencing the potential
Update particle positions and velocities
Goto step 1
Scales as 0(n)
Can use P3M where P-P forces are calculated within a cell
Ways to speed up the force calculation
Hierarchical methods (TREES)
The higher order multipoles of an objects gravitational field decays rapidly
with respect to the dominant monopole term.
The long range gravitational potential of a region can be approximated as
1/r therefore replace the sum over N-1 bodies by a sum over (0)LogN
regions.
Create a TREE data structure by recursively dividing the computational
volume.
OCT-Trees:
KD-Tree:
Split space into successively smaller octants by volume.
Each node has 8 children or branches.
Divide by one dimension at a time. (Fail for gravity since
one ends up with long cells that can’t be approximated with
even hexadecipole expansion.
Trees – building and walking
OCT-Trees:
Construct Tree from top down in time proportion to Nln(N)
Building
Root node is the entire volume of the simulation
Split volume in 8
Loop over all particles and sum mass in each quadrant
If octant has > 1 particles, create new node
If octant has =1 particle, create pointer
If octant has =0 particles, create null pointer
Repeat until no active quadrants with >1 particle
For a given particle, walk down the tree to determine the interaction list
Open cells according to theta criteria
Sum up the accelerations and multipoles
Walking
Each node of the Tree has:
Pointers to 8 children in 3D (node,
particle or null)
Pointer to the parent node
Position of the centre of mass and
physical centre
Mass
Higher order multipoles of the node
The active tree on a single particle
Hierarchical methods (TREES)
The force on a given particle is
calculated by moving up the tree,
opening cells according to a simple
opening angle criteria.
r
θ
d
If r/d<0.5 then open the cell and sum
the contributions to the force from
the particles within the subcells,
otherwise do not open the cell.
This is the entire
two dimensional tree
for a system
modelled with 1000
particles.
Next we will look at
the “active” tree for
a single particle.
Grids
Assign particles to the grid to determine the
density at grid points via interpolation
Compute gravitational potential at grid points by
solving Poisson’s equation on the grid (FFT)
Compute gravitational accelerations at grid points
by finite differencing the potential
Calculate accelerations at grid points using the
same interpolation method as in step 1.
Update positions and velocities using leap-frog
integrator.
Invented in Los Alamos in 1950’s – now used for:
Plazmas, Galactic dynamic, fluids, molecular
dynamics, weather…….
Interpolation techniques:
Nearest grid point (NGP)
Simple
Quick
Density discontinuous across the grid
Cloud in cells (CIC)
Most popular
Each particle is a cube of size the grid spacing
Most particles intersect 2^d grid cells
First derivative of the density is discontinuous,
but density is smooth across grid cells
Triangular clouds (TC)
Rarely used
Expensive
Each particle is a cube of (tri) linearly rising density
Density and derivative of density are continuous
Spherical Profiles (SP)
Each particle has some linearly rising density profile
Useful for P3M since automatically have a smoothing radius.
Determining the potential
N
Gα = ∑ Gαβ M β
β =1
M β is the mass in the β cell
Gαβ is the potential at the center of the α cell generated by unit
mass at the β cell
−1
Gαβ ∝
2
ε 2 + xα − xβ
Gαβ is the "Greens" function for the grid that needs to be calculated only once at t = 0
Then Φ ( xα ) can be estimated at each step using FFT's
In Fourier space Φ (k ) = G (k ) ρ (k )
(This is a discrete Fourier transform convolution)
Φ (α ,β ,γ )= ∑ G i,j,k ρ i , j ,k e 2π i(iα +jβ +kγ )/M
i,j,k
Perform FFT to get ρ (k)
Multiply each array element by G ( K ) to get Φ (k )
Reverse FFT to transform result Φ (k ) → Φ ( x ) in real space
FFT's can be executed in time ∝ N cell Ln(N cell ) steps
FFT's are periodic which is good for cosmological volumes
Determining the potential
Gα =
N
Gαβ M β
∑
β
=1
M β is the mass in the β cell
Gαβ is the potential at the center of the α cell
generated by unit mass at the β cell
−1
Gαβ ∝
2
ε 2 + xα − x β
Gαβ is the "Greens" function for the grid that needs to be calculated only once at t = 0
Then Φ ( xα ) can be estimated at each step using FFT's
In Fourier space Φ ( k ) = G ( k ) ρ ( k )
(This is a discrete Fourier transform convolution)
Φ ( α , β , γ )= ∑ G i,j,k ρ i , j , k e 2 π i(iα +jβ +k γ )/M
i,j,k
Perform FFT to get ρ (k)
M ultiply each array element by G ( K ) to get Φ ( k )
Reverse FFT to transform result Φ ( k ) → Φ ( x ) in real space
FFT's can be executed in time ∝ N cell Ln(N cell ) steps
FFT's are per iodic which is good for cosmological volumes
The Greens function
This plot shows the force errors
that occur when single particles
are placed randomly in the
computational box at a random
distance from the origin.
Error on scales <3 cells due to geometry and discretisation.
Error on box scale due to periodic boundaries
N
Gα = ∑ Gαβ M β
β =1
M β is the mass in the β cell
Gαβ is the potential at the center of the α cell
generated by unit mass at the β cell
Gαβ ∝
−1
ε 2 + xα − xβ
2
Gαβ is the "Greens" function for the grid that needs to be calculated only once at t = 0
Then Φ ( xα ) can be estimated at each step using FFT's
In Fourier space Φ ( k ) = G ( k ) ρ ( k )
(This is a discrete Fourier transform convolution)
Φ (α ,β ,γ )= ∑ G i,j,k ρ i , j , k e 2π i(iα +jβ +kγ )/M
i,j,k
Perform FFT to get ρ (k)
Multiply each array element by G ( K ) to get Φ (k )
Reverse FFT to transform result Φ ( k ) → Φ ( x ) in real space
FFT's can be executed in time ∝ N cell Ln(N cell ) steps
FFT's are periodic which is good for cosmological volumes
Determining the acceleration
Usually calculated by finite differencing the grid of potential energies:
acceleration ∝ ∇Φ i.e.
x
a i,j,k
= −(Φ i +1, j ,k − Φ i −1, j ,k ) / 2, similar for a y and a z
This is a simple 2 point scheme.
Finally interpolate the accelerations back onto the locations of particles.
Important to use the same interpolation scheme used to place ρ on the grid.
Last step is to update ( x, v) using leap frog.
Resolution poor within 3 grid cells
Memory limits calculation < 1000 grid cells
Can use P3M code that evalulates P-P forces directly in cells that contain
many particles
AP3M codes use grids within grids
Simulating N-Body systems
Several numerical
parameters are
important during
these simulations.
•Force softening
•Discreteness noise
•Force
approximations
•Time-stepping
•Integration errors
•Machine round-off
errors
Gravitational force softening
dri
= vi
dt
and
m j (rj − ri )
dvi
=∑
2
2 3/ 2
dt
j ≠i ( r − r
+ε )
j
i
The particles represent mass points moving in phase space.
The force becomes infinite at zero separation which does
not occur in real systems.
Forces are softened to prevent scattering (collisional
processes) during close encounters. This is particularly
important when we are trying to follow collisionless systems.
(In some situations, such as to follow the core collapse and
black hole formation, star-star encounters are important and
need to be followed accurately.)
Timestepping
The orbits are calculated using discrete timesteps. One must
choose these shorter than the local dynamical time. A large
speedup can be obtained using variable timesteps – each
particle moves on a locally determined timestep.
However this destroys the nice symplectic nature of the
leapfrog integrator.
Variable timesteps can be based on various local quantities:
∂t (v, ρ , −∇Φ, ε )
Consider solar system: Mercury orbits every 88 days but Pluto
orbits every 250 years. For Keplerian systems can save
factors of 1000 by scaling timesteps with velocity.
Multistepping increases the complexity of the parallel domain
decomposition – end up with few active particles.
Timestepping and integrators
Recall that an 2nd order differential equation can be reduced to a set of 1st
order ODE’s
Equation of motion:
d 2x
m 2 = f ( x),
dt
Phase space:
dx
dv
= v, m = f ( x )
dt
dt
The Euler method
Split orbit into a series of timesteps ∂t
xi +1 = xi + ∂t. f ( xi , ti )
Taylor expansion of f ( x) → error ∝ O∂t 2
i+1
i
dt
Timestepping and integrators
Runga-Kutta
∂x1 = ∂t. f ( xi , ti )
∂x2 = ∂t. f ( xi + ∂xi / 2, ti + ∂ti / 2)
∂x3 = ∂t. f ( xi + ∂x2 / 2, ti + ∂t2 / 2)
∂x4 = ∂t. f ( xi + ∂x3 , ti + ∂t3 )
xi +1 = xi + ∂x1 / 6 + ∂x2 / 3 + ∂x3 / 3 + ∂x4 / 6 + Ο∂t 5
Leapfrog
Offset positions and velocities by ∂t / 2
xi +1 = xi + vi +1/ 2 ∂t
vi +3/ 2 = vi +1/ 2 + f ( xi +1 )∂t
Error of the order O∂t 2
But it is time reversable → can replace ∂t=-∂t
Time reversibility is important since it guarantees conservation of energy,
angular momentum – any conserved quantity.
Consider again the problem of a simple elliptical solution of the
gravitational two-body problem. Imagine that our integrator makes an
energy error of +E as it integrates the system forward through 1 orbital
period. Now imagine reversing the integration. You might guess that the
energy error in the reverse integration would be -E , but this is not the
case. In any system where the equations of motion are unchanged by time
reversal (and, specifically, in any case where the function f depends only on
the coordinates), the time-reversed orbit is itself a solution of the original
ODE (with v simply replaced by -v), so the energy error is still +E. But if
our integration scheme is time-reversible, we know that the final energy
error is zero (because we return to our starting point). The only possible
way that this can occur is if E=0 (i.e. energy is exactly conserved!).
As illustrated below, even the small error (exaggerated here by choosing a
timestep 5 times greater than in the other two integrators) made by
Runge-Kutta-4 is systematic, leading to a long-term drift in the orbital
parameters, the energy error in the Leapfrog scheme has no such longterm trend. There is a periodic error over the course of an orbit, at the
same level as the error in the Mid-point scheme, but the errors incurred
over the outgoing portion of the orbit exactly cancel those produced on
the incoming segment, so no net error results. The Leapfrog method is only
second-order accurate, but it is very stable.
Force approximations
The direct and most exact approach calculates forces between all N
particles. The calculation scales as N^2. The largest N^2 calculations
approach 10^5 particles.
Special purpose hardware (pci “grape boards”) can be used to do 1/r^2
calculations in hardware.
Treecodes, mesh codes or particle-mesh codes all attempt to speed up
the force calculations by approximating long range forces.
This introduces additional errors – but they can be quantified and
controlled. These codes scale as order Nl.n(N) or even as order N.
Discreteness effects
In many cases we can’t use one particle per real star. For example, a dark
matter halo surrounding a galaxy may contain 10^70 particles! We can fit
at most 10^10 particles in the memory of a parallel supercomputer.
Discreteness leads to collisional processes and sets a limit to the phase
space density that can be resolved.
One can esimate the “relaxation time” of a system of N particles
analytically or by following the actual energy changes of individual
particles.
What we really need to simulate the Universe, is the processing
power of a single human brain.
•Each cell is an elaborate chemical computer with power management,
read only and random access memory.
•Each cell has a backup of the genetic code of the entire body.
•Cell reproduction alone requires terabytes/second of DNA copying.
•The 10^10 neurons in the celebral cortex can each exhange
information via several mechanisms leading to a processing power of
about 10^14 bits/second.
•The power consumption of a human is only about 30 Watts.
Let’s build a supercomputer!
The zBox can process 10^12
operations per second and
transfer information to the
server at 10^9 bits/second.
This is about 1% of a human
brain. But it also consumes and
dissipates 1000 times as much
power….
zBox: (Stadel & Moore)
288 AMD MP2200+ processors, 144 Gigs ram
Compact, easy to cool and maintain
Very fast Dolphin/SCI interconnects - 4 Gbit/s, microsecond latency
A teraflop supercomputer for $500,000
Roughly one cubic meter, one ton and requires 40 kilowatts of power
The beginning…
The horizon project
A 15 Teraflop
2000 processor
Petabyte disk store
3d SCI network
Unizh/PSI/CSCS
The largest supercomputer in Europe
Astrophysics – computation biology – computational chemistry
Need to make the codes scale in parallel to thousands of processors
The power spectrum of density fluctuations in three different dark matter models
CMB
Large scales
(galaxy clusters)
Small scales
(dwarf galaxies)
How does dark matter change P(k)?
Most models for the generation of fluctuations predict a scale free
spectrum of perturbations. i.e. P(k) proportional to k.
This is imprinted on the mass distribution but the linear growth of
fluctuations alters the primordial power spectrum through various
physical “filtering” processes.
The final processed power spectrum is curved due to two main
phenomenon: horizon crossing and free streaming.
(Also see wiggles on large scales due to the coupled baryon-photon
plasma between matter-radiation equality and recombination.)
Free streaming
When the particles “freeze out” from equilibrium with the radiation then they
may be moving relativistically. If the particles were within some density
perturbation as it came through the horizon then they would move out of the
fluctuation in some random direction.
This effectively damps out fluctuations below some mass scale.
The comoving distance which a free-streaming particle can travel by time t:
t
v(t ')
dt '
R(t )
0
rFS = ∫
Split this into terms that come from relativistic era and to t equality
t EQ
t NR
v(t ')
v(t ')
dt ' + rFS = ∫
rFS = ∫
dt '
R (t )
R(t )
t NR
0
The first term is the horizon scale over the time the particle is relativistic and
the second term is the period over which the streaming velocities are damped
due to the expansion of the Universe (vpeculiar ∝ R −1 )
m
⎞
rFS ≈ 28Mpc ⎛⎜ ν
⎟
⎝ 30eV ⎠
−1
−2
m FS
m
⎞ M
≈ 10 ⎛⎜ ν
⎟
⎝ 30eV ⎠
15
Creating cosmological initial conditions
Need to model the correct power spectrum of the linear density field
Start with an initial 3-d distribution of particles that have zero power on
all scales: GRID or a GLASS
Perturb particles from there initial conditions so that they have a
clustered distribution that represents the P(k).
Use non-linear perturbation theory in comoving (Lagrangian) coordinates
Zeldovich approximation:
x = R(t )r + b(t ) p (r )
x = proper coordinate r = comoving coordinate
The first term is the uniform expansion of the background model
The second term represents the perturbation of a particles position about
its Lagrangian or comoving coordinate r
b(t ) is the growth rate of the displacements of particles from their initial
displacements p (r )
Goals:
Model the development of large scale structure
Predict the structure of dark matter halos surrounding
galaxies and galaxy clusters. (Sizes, masses, shapes,
density profiles, substructure)
Make predictions for direct and indirect detection
experiments that are sensitive to the phase space
structure of dark matter halos on parsec scales.
Constrain the nature of particle dark matter
candidates.
N=700 (White 1976)
This overmerging problem
was part of the motivation
for White & Rees (1978)
and the beginning of semianalytic galaxy formation
models.
The formation of large scale structure in the universe
The colours show
the local density
of dark matter
plotted using
comoving
coordinates that
expand with the
Universe.
This volume is a
substantial
fraction of the
observable
universe.
We are able to
reproduce the
same clustering
pattern as
observed in the
galaxy
distribution.
100Mpc
A theoretical prediction for the mass distribution in the Universe on very large scales
Large scale distribution of matter does not test
the real nature of the dark matter particle.
(Apart from hot dark matter, the DM particle
may have a diverse range of properties that
would not affect the development of large scale
structure, but may change the structure of nonlinear objects. i.e. WDM, a particle that
interacts weakly with photons, a particle that
interacts with some strong interaction with
itself, etc….)
Want to look at individual objects at a resolution
probed by the stars – i.e parsec-kpc scales
within a 100Mpc volume.
Lets look at the
formation of a single
cluster of galaxies.
This movie shows
the density of dark
matter in physical
coordinates so that
you can see the
expansion of the
universe.
This is worlds
highest resolution
calculation of
structure formation
that took several
months running in
parallel on hundreds
of processors.
Structure
formation in
“comoving
coordinates”
that expand
with the
universe.
The power spectrum of density fluctuations in three different dark matter models
CMB
Large scales
(galaxy clusters)
Small scales
(dwarf galaxies)
Next class will confront these and other dark matter candidates with observations
z=3
z=0
HDM
WDM
CDM