N-Body ICs

From Eric's Wiki

Contents 1 N-Body Codes 2 Initial Conditions 3 Zel'dovich Approximation 4 Expectations 4.1 In an Ideal World 4.2 In an Almost Ideal World 4.3 In the World I Thought I Lived In 4.4 The World I Find Myself 5 The problem 5.1 Enzo 6 A simple test 6.1 PM 6.2 Hydra 6.3 Gadget 6.4 Gadget with createcosmo

N-Body Codes

• Simulate the evolution of matter subject to gravitational forces only
• Move lots of little bits of matter (particles) around

Initial Conditions

• The initial distribution of particle positions
• Laid out to emulate a density distribution, ρ(x)
• Good for a given time (redshift, z_init) at which the simulation is to start
• A series of delta functions trying to emulate a continuous function.

Zel'dovich Approximation

• Generates cosmological ICs from a given density distribution
• Uses the simplification that growth is linear for small perturbations
• Displaces uniformly distributed particles (z_init = ∞) and gives them a velocity
• The velocity is proportional to the displacement.

Expectations

In an Ideal World

• z_init can be very very large
• N-body code will evolve particle positions accurately and quickly

In an Almost Ideal World

• N-body code takes time
• Zel'dovich approximation saves us time by doing the evolving the particles through the linear growth phase

In the World I Thought I Lived In

• Bad things happen if Zel'dovich approximation is pushed too far: end up in non-linear growth regime using linear growth
• Bad things happen if Zel'dovich approximation is not pushed far enough: numerical precision cannot resolve displacements
• There's exists a happy sweet spot in the middle

The World I Find Myself

• There is no happy place

The problem

Enzo

First there was a problem with generating ICs for Enzo. I needed at 10Mpc volume and had a couple of prescriptions as to what z_init should be. Unfortunately the results at z=12 (when I wanted to turn my radiation source on) weren't even close.

Enzo: 10Mpc volume at z=12. zinit=35

Enzo: 10Mpc volume at z=12. zinit=380

A simple test

Push the initial conditions generator to the limits to find the sweet spot.

Expectation

• At low z_init, Zel'dovich pancakes will be forming (shell crossing)
• At high z_init, numerical errors will eliminate small-scale structure.
• In the middle will be a swath of z_init's that produce the same final distribution: The sweet spot

PM

I suspected my z_init prescriptions were wrong. So I tried a test with Martin White's PM code: Evolve to the redshift of z=10 from initial conditions generated at z_init = 10, 20, 50, 100, 200, 500.

With every increment in z_init:

• there is an increase in the amount of small-scale substructure
• large-scale structure is increasingly distorted.

Note, a further test showed the results to be independent of precision: double compared with single.

Hydra

Not knowing if the problem was with PM, I tried the tests with hydra (Couchman, Thomas, & Pearce).

• Again too much small-scale structure.
• Regular small-scale clumping.
• Large-scale structure fine.

The two overdensity figures chart the:

• Maximum overdensity in the ICs as a function of the z_init
• Maximum overdensity in the final evolved z=10 state, as a function of the z_init of the IC.

The IC overdensity figure show that the ZA is pushed into the non-linear regime by z_init < 100.

The final overdensity figure illustrates the growth of small-scale structure for z_init > 10000.

The results of the overdensity figures might lead one to think that 100 < z_init < 10000 is a valid range. But the actual particle distribution would seem to still suffer enhanced small-scale structure (on a disturbingly regular grid) all the way down to z_init < 100.

hydra: 10Mpc volume (Original). Slices at z=10

hydra: 10Mpc volume (Original). Maximum overdensities of the ICs

hydra: 10Mpc volume (Original). Maximum overdensities at z=10

25 Mpc volume

After discussions with Frazer Pearce and Chris Power, we agreed to try the simulations with the common parameters:

• 25Mpc volume
• σ₈ = 0.9
• Powerlaw spectrum with n=-1

We couldn't seem to agree on the number of particles, but there you go. All my results are with 128³.

At a first, and second glance, there doesn't seem to be any problem with hydra running these conditions, with good agreement for 50 < z_init < 20000, and only a hint of the enhanced growth of small-scale structure from z_init > 20000.

hydra: 25 Mpc volume. Slices at z=10

hydra: 25 Mpc volume. Maximum overdensities of the ICs

hydra: 25 Mpc volume. Maximum overdensities at z=10

10 Mpc volume revisited

Confused by why hydra magically seemed to work after just talking to Frazer (he must have a magic touch), I returned to the 10 Mpc volume which originally led me to believe there was a problem. But this time I used exactly the same IC parameter file as used in the 25 Mpc tests, varying only the volume size.

First thing to note was that the runs were unable to evolve to z=10 in a reasonable time (5 hours), so I looked only at the z=20 dumps.

Again: No apparent problem. Clearly for too low z_init the ZA has gone well beyond its limits. But for 100 < z_init < 50000 the results are quite reasonable.

That we couldn't evolve to z=10 gives us a hint: the power normalisation is much larger.

hydra: 10 Mpc volume. Slices at z=20

hydra: 10 Mpc volume. Maximum overdensities of the ICs

hydra: 10 Mpc volume. Maximum overdensities at z=20

10 Mpc volume, low σ₈

So let's lower the power. In the following, σ₈=0.09.

Back to the problems, but not quite as seriously so. Visually, the happy zone is 50 < z_init < 200, not particularly wide.

Note that the power, as illustrated in the IC max overdensity figure, is still not as low as in the original 10 Mpc run. In the original, the power spectrum used was createcosmo's "CDM". Apparently it has very low power. and it is at low power that the happy zone is pinched between the non-linear regime and a the domain where hydra creates artificial forces that lead to clumping on a regular grid.

The max IC overdensity figure shows the ICs aren't safely in the linear regime until z_init > 200, by which point small-scale structure is starting to be enhanced. So the happy zone is limited to z_init &simeq; 200. Any lower the power normalisation, and the zone would be pinched out of existence, as has happened for the CDM 10 Mpc volume tested originally.

Of course, my description of the "pinching" makes no sense, since power can always be boosted by moving to lower z_init.

hydra: 10 Mpc volume σ8=0.09. Slices at z=10

hydra: 10 Mpc volume σ8=0.09. Maximum overdensities of the ICs

hydra: 10 Mpc volume σ8=0.09. Maximum overdensities at z=10

Gadget

Volker Springel's Gadget produces some interesting results.

For the generation of the IC's, I used createcosmo, which is the IC generator for hydra, and N-GenIC, made available by Volker via the DEISA Benchmark Suite.

Let's concentrate on N-GenIC, since there were interesting problems with createcosmo's ICs.

The input parameters to N-GenIC are not quite exactly the fiducial: h=0.65 instead of 0.7, and the input spectrum is Eisenstein & Hu, since powerlaw is not an option to N-GenIC.

Quite differently from hydra, there is an increasing lack of growth of power at all scales as z_init is increased. The power in the ICs is low to begin with. However, if we consider that ICs for z_init < 100 are too far into the non-linear regime, then we do see a small increase in growth of power as z_init increases, but there is a sweet spot between 200 < z_init < 500. Is the gradual decline in the final power for 500 < z_init < 20000 an error due to the ZA being pushed further into the linear regime, or is it an indication that other errors are being overcome as the ZA is allowed to evolve into it's regime of validity?

I don't understand why IC's with ZA pushed to too low z_init produce less substructure in Gadget but more in hydra. Could it be simply a function of the form of the initial power spectrum? We aren't comparing apples and oranges: a power-law vs Eisenstein & Hu.

hydra: Gadget 10 Mpc volume at z=10, evolved from a variety of zinit.

hydra: Gadget initial maximum overdensities in a 10 Mpc volume at a variety of zinit. ICs generated by N-GenIC.

hydra: Gadget final maximum overdensities at z=10 in a 10 Mpc volume from a variety of zinit ICs generated by N-GenIC.

Gadget with createcosmo

Gadget was run on a 10 Mpc volume with IC's generated using createcosmo, the tool used to generate IC files for hydra. The IC's weren't quite fiducial, instead matching the N-GenIC parameters (notably, h=0.65 instead of 0.7). The spectrum was </tt>createcosmos</tt> "CDM", which, similarly to N-GenIC's spectrum, has much less power than a power-law with n=0.9.

The results are comparable to the N-GenIC runs, with decreased structure growth from larger z_init, but growing amounts of structure growth as z_init gets very large. The apparent "sweet spot" is in the range 100 < z_init < 10⁴ .

But again, why is there less structure as z_init increases compared with the hydra results?

Gadget 10 Mpc volume at z=10, evolved from a variety of zinit. The IC files were generated with a version of createcosmo that scaled the velocities by 100-1.

Gadget initial maximum overdensities in a 10 Mpc volume at a variety of zinit. ICs generated by with a version of createcosmo that scaled the velocities by 100-1.

Gadget final maximum overdensities at z=10 in a 10 Mpc volume from a variety of zinit ICs generated by a version of createcosmo that scaled the velocities by 100-1.

For comparison, this is what hydra does with the same IC's (though without the velocities scaled by 100^-1.

Hydra 10 Mpc volume at z=10, evolved from a variety of zinit generated with a CDM spectrum.

Hydra initial maximum overdensities in a 10 Mpc volume at a variety of zinit. ICs generated by createcosmo using a CDM spectrum.

Hydra final maximum overdensities in a 10 Mpc volume at a variety of zinit. ICs generated by createcosmo using a CDM spectrum.

From where did this velocity scaling come? I compared the velocity distributions of the ICs generated by N-GenIC with those produced by createcosmos. They differed by a factor of approximately 100 times. For example, at z_init = 20000:

method	mean(v)	σ(v)	max(v) (all km s-1)
createcosmos	1.0x104	1.21x104	4x104
N-GenIC	100	110	400

Note that the spectra are not the same, so it is not obvious that these are comparable numbers.

Ack! Gadget scales it's internal velocities by &sqrt;a. That wasn't included in createcosmo. Here are the results of the patched version of createcosmo:

Gadget 10 Mpc volume at z=10, evolved from a variety of zinit. The IC files were generated with a new version of createcosmo with proper velocity scaling.

Gadget initial maximum overdensities in a 10 Mpc volume at a variety of zinit. ICs generated by with a new version of createcosmo with proper velocity scaling.

Gadget final maximum overdensities in a 10 Mpc volume. ICs generated at a variety of zinit with a new version of createcosmo with proper velocity scaling.