First simulation

We first spotted an unusual effect in the 100 Mpc/h simulation with 5123 particles which we received in July 2007. The ratio of the real-space power spectrum to the redshift-space power spectrum for a variety of redshifts is shown in this figure. The power spectra are calculated by fourier transforming the correlation function using A.J.S. Hamilton's fftlog code.

The linear theory value for the ratio here is approximately 0.537 for all redshifts. This appears not to hold, but the oddest feature is the dip at intermediate scales which doesn't seem to be expected in perturbation theory.

The effect can also be seen directly in the correlation function, for example here for the z=6 output. The linear theory ratio is shown as the dotted line. Plotting the correlation functions directly (rather than plotting their ratio) an artifact at the mean inter-particle separation can still be seen at z=6, as seen here.

 

Millennium Simulation

I decided to check this result in the Millennium Simulation. I sparsely sampled the particle distribution before calculating the correlation function, then used fftlog as before. The ratio is fit using the fitting function shown in the plot. For the four-parameter fit, the function tends to the linear theory ratio on large scales. For the five-parameter fit, the large-scale asymptote is a free parameter. I used the same function to fit the results of the earlier simulation, where it seemed to do a pretty good job. It seems to favour that the ratio goes like k2 on small scales, in agreement with previous work. The factor in the denominator allows the dip on intermediate scales.

Here are plots for three different redshifts:
Four parametersFive parameters
z=9.278 View View
z=6.712 View View
z=3.060 View View

There doesn't seem to be a pronounced dip in this simulation. Because it ran to redshift zero, it was possible to verify that by redshift three, the large-scale asymptote seemed to conform to linear theory.

 

New Simulations

The table below shows similar figures for the recent 100Mpc/h and 400Mpc/h simulations. The power spectra are calculated in two different ways: firstly by gridding, fourier transforming the grid and then calculating ⟨|δk|2⟩ in bins; and secondly by the same procedure as above. In the former case, a correction is made for the power introduced by the gridding, but it's small on the relevant scales and for the TSC gridding used here.
fftlog ξ(r) P(k) from grid FFT
Four parametersFive parametersFour parametersFive parameters
L=100 Mpc/hz=9 View View View View
z=6 View View View View
z=3 View View View View
L=400 Mpc/hz=9 View View View View
z=6 View View View View
z=3 View View View View

I don't really understand why the ratio from the FFT'ed grid seems to be so much less stable than the one from the fftlog'ed correlation function. Certainly if you plot the power spectrum itself rather than the ratio, the two methods agree very well, though it's easier to calculate the correlation function over a wide range of scales since one can choose logarithmic bins in r, rather than being restricted to a grid which effectively bins linearly in r.

 

Small timesteps

A 100Mpc/h simulation with the same initial conditions but using smaller timesteps at high redshifts was run. A simulation in which the initial conditions were generated with an extra flag in N-GenIC switched on was also run. A comparison of the ratio of the real- and redshift-space power spectra in the simulations with normal and small timesteps, and with different settings for the initial conditions generator, is shown in this figure.