Nonlinearities in modified gravity cosmology I: signatures of modified gravity in the nonlinear matter power spectrum

Abstract

A large fraction of cosmological information on dark energy and gravity is encoded in the nonlinear regime. Precision cosmology thus requires precision modeling of nonlinearities in general dark energy and modified gravity models. We modify the Gadget-2 code and run a series of N-body simulations on modified gravity cosmology to study the nonlinearities. The modified gravity model that we investigate in the present paper is characterized by a single parameter \zeta, which determines the enhancement of particle acceleration with respect to general relativity (GR), given the identical mass distribution (\zeta = 1 in GR). The first nonlinear statistics we investigate is the nonlinear matter power spectrum at k < 3h/Mpc, which is the relevant range for robust weak lensing power spectrum modeling at l < 2000. In this study, we focus on the relative difference in the nonlinear power spectra at corresponding redshifts where different gravity models have the same linear power spectra. This particular statistics highlights the imprint of modified gravity in the nonlinear regime and the importance to include the nonlinear regime in testing GR. By design, it is less susceptible to the sample variance and numerical artifacts. We adopt a mass assignment method based on wavelet to improve the power spectrum measurement. We run a series of tests to determine the suitable simulation specifications (particle number, box size and initial redshift). We find that, the nonlinear power spectra can differ by ~30% for 10% deviation from GR (|\zeta-1| = 0.1) where the rms density fluctuations reach 10. This large difference, on one hand, shows the richness of information on gravity in the corresponding scales, and on the other hand, invalidates simple extrapolations of some existing fitting formulae to modified gravity cosmology. Comment: 13 pages, 6 figures. V2, version for publication in PRD

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arXiv:1001.5184v2 [astro-ph.CO] 26 May 2010
Nonlinearities in modified gravity cosmology I: signatures of modified gravity in the
nonlinear matter power spectrum
Weiguang Cui1,∗Pengjie Zhang1,†and Xiaohu Yang1‡
1Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory,
the Partner Group of MPA, Nandan Road 80, Shanghai, 200030, China
A large fraction of cosmological information on dark energy and gravity is encoded in the nonlinear
regime. Precision cosmology thus requires precision modeling of nonlinearities in general dark energy
and modified gravity models. We modify the Gadget-2 code and run a series of N-body simulations
on modified gravity cosmology to study the nonlinearities. The modified gravity model that we
investigate in the present paper is characterized by a single parameter ζ, which determines the
enhancement of particle acceleration with respect to general relativity (GR), given the identical
mass distribution (ζ = 1 in GR). The first nonlinear statistics we investigate is the nonlinear
matter power spectrum at k<
∼3h/Mpc, which is the relevant range for robust weak lensing power
spectrum modeling at ℓ<
∼2000. In this study, we focus on the relative difference in the nonlinear
power spectra at corresponding redshifts where different gravity models have the same linear power
spectra. This particular statistics highlights the imprint of modified gravity in the nonlinear regime
and the importance of including the nonlinear regime in testing GR. By design, it is less susceptible
to the sample variance and numerical artifacts. We adopt a mass assignment method based on
wavelet to improve the power spectrum measurement. We run a series of tests to determine the
suitable simulation specifications (particle number, box size and initial redshift). We find that, the
nonlinear power spectra can differ by ∼ 30% for 10% deviation from GR (|ζ−1| = 0.1) where the rms
density fluctuations reach 10. This large difference, on one hand, shows the richness of information
on gravity in the corresponding scales, and on the other hand, invalidates simple extrapolations of
some existing fitting formulae to modified gravity cosmology.
PACS numbers: 98.65.Dx,95.36.+x,04.50.+h
I.INTRODUCTION
One of the biggest challenges of modern cosmology and
physics is the existence of the dark universe. Assuming
the validity of general relativity (GR), cosmological ob-
servations lead to the discovery of dark matter and dark
energy, which account for ∼ 96% of the total matter and
energy budget of the Universe (e.g. [1]). However, since
we do not have independent tests of GR at relevant scales,
the same set of observations could imply another pos-
sibility, the failure of general relativity at galactic and
cosmological scales. This possibility, which serves as an
alternative to dark matter/dark energy, has become an
area of active research. Discriminating between the dark
matter/dark energy and modified gravity (MG) models,
testing GR at cosmological scales and probing dark mat-
ter and dark energy through cosmological observations,
are thus an entangled task, of crucial importance for both
cosmology and physics.
Challenges exist in both the observation side and the-
ory side. Although there are numerous and potentially
powerful observations suitable for this task [2, 3], their
precision measurements are challenging. On the other
hand, much of the cosmological information is encoded
in the nonlinear regime. Modeling the nonlinearities to
∗Electronic address: wgcui@shao.ac.cn
†Electronic address: pjzhang@shao.ac.cn
‡Electronic address: xhyang@shao.ac.cn
the required ∼ 1% accuracy is challenging too, even for
the simplest case, the standard ΛCDM cosmology with
only gravitational interaction (e.g. [4–7]). Cosmologies
based on dynamical dark energy or MG are facing similar
requirements. References [6–11] have performed N-body
simulations for dynamical and coupled dark energy mod-
els [12].
Comparing to the dark matter/dark energy cosmology,
understanding the evolution of the Universe in MG mod-
els is often more difficult, due to the intrinsically nonlin-
ear feature of gravity in these models or the existence of
extra dynamical fields. Despite these difficulties, the ex-
pansion history of the Universe and the structure growth
to the first order have been robustly understood for many
of the MG models such as TeVeS [13, 14], DGP(short
for Dvali,Gabadadze and Porrati) [15, 16] and the f(R)
gravity [17–21]. People have also achieved success in
understanding the nonlinear evolution through analyti-
cal and semianalytical methods (e.g. [22–25]). Recently,
self-consistent gravity solvers for f(R) [26–29] and DGP
gravity [30] models have been developed and led to sig-
nificantly improved understanding of the nonlinear evolu-
tion. Simulations with extra scalar fields and interaction
with dark matter have also been performed (e.g. [31]).
Since deviations from GR in general lead to nonlinear
differential equations of gravity, in principle we have to
develop the suitable N-body codes for each viable MG
model and run the corresponding simulations. However,
since we do not have the final theory of gravity based
on the first principles, there are in principle infinite MG

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models to be investigated. One possibility to circumvent
this problem is to choose a suitable parameterization for
the MG models and run a finite number of simulations
to sample the relevant parameter space. We are then
able to interpolate/extrapolate the simulation results to
explore the whole relevant parameter space.
The statistics we focus on is the matter power spec-
trum. It determines the lensing power spectrum and is
also highly relevant to the 2D galaxy clustering, both
will be measured to high precision by ongoing and planed
imaging surveys, such as DES, LSST, JDEM/SNAP, Eu-
clid/DUNE, and KDUST. As we will show later, the
density evolution is determined by a single parameter
of gravity, ζ, which quantifies the ability of mass con-
centration to distort the space-time metric. In princi-
ple, ζ can be both scale, time, and environmental de-
pendence. Comprehensive investigation on general ζ is
beyond the scope of the current paper. Instead, we will
adopt a highly simplified form of ζ and run a series of
N-body simulations to quantify the nonlinear evolution
of the Universe.
As shown in Heitmann et al. 2008, (hereafter H08,
[5]), however, to run simulations and model nonlinear
matter power spectrum to 1% accuracy to k ∼ 1hMpc
is very challenging, requiring Gpc or larger simulation
box size, 10243or more particles, and beyond. Being
aware of these difficulties and limited computation re-
source, we take a modest goal, to quantify the influence
of MG on the nonlinear matter power spectrum with re-
spect to the standard ΛCDM to ∼ 1% accuracy. Namely,
the statistics that we will focus on is the relative differ-
ence between the nonlinear matter power spectra in the
given MG model and in ΛCDM. We will choose the right
redshifts of simulation output such that the linear matter
power spectra in the given MG models are equal to the
ones in the corresponding ΛCDM. This particular statis-
tics has a number of attractive features. First, it isolates
and highlights the role of MG in nonlinear evolution. Sec-
ond, it reduces much of the numerical artifacts by taking
ratios. Similar tricks have been adopted in many pre-
vious simulations (e.g. [8–10]). Third, it can improve
the efficiency to understand nonlinearities in MG mod-
els, which can now be reduced to two separate ingredi-
ents: the nonlinear evolution in ΛCDM and the relative
difference between MG models ΛCDM.
The current paper only analyzes a very limited set of
simulations. Nonetheless, it robustly show that, even
after scaling out the difference in the linear evolution,
gravity still leaves significant features in the nonlinear
power spectrum. In subsequent studies, we will run sim-
ulations covering larger parameter space to better un-
derstand these features and hopefully develop a general
fitting formula. Furthermore, we will study the peculiar
velocity power spectrum and the redshift distortion (3D
galaxy clustering), based on these simulations. The ongo-
ing spectroscopic redshift surveys like BOSS, LAMOST
and WiggleZ, and planned spectroscopic redshift surveys
like BigBOSS, JDEM/ADEPT, Euclid/SPACE and SKA
are able to measure these statistics to unprecedented ac-
curacy. We will also investigate the halo statistics, one
of the key scientific goals for galaxy and cluster surveys.
This paper is organized as follows.
present the MG parameterization adopted for the simu-
lations, the precision requirements and the code specifi-
cations. In Sec. III, we test the accuracy of the simula-
tions. We present major simulation results in Sec. IV ,
discussion in Sec. V and more results in the Appendix.
In Sec.II, we
II. THE SIMULATION LAYOUT
A.The ζ parameterization on modified gravity
There are several existing parameterizations and gen-
eral guidances on modified gravity [3, 22, 32–38]. What
we adopt in this paper is the ζ parameterization. It is a
condensed version of the Geff-η parametrization [3, 36],
which quantifies two key aspects of gravity.
To understand this point, we begin with the structure
formation in GR, for which the central issue is to deter-
mine the particle acceleration given the mass distribu-
tion. The scalar perturbation of the space-time metric is
described by two potentials, ds2= −(1+2ψ)dt2+a2(1+
2φ)dx2. The usual Poisson equation, k2φ = 4πGa2δρ
(in Fourier space), relates the potential φ to the matter
distribution ρ. However, φ is not the potential directly
responsible for the structure formation in our N-body
simulations of nonrelativistic cold dark matter particles.
The contribution to the particle acceleration from this
potential is suppressed by a factor (v/c)2≪ 1, compar-
ing to the contribution from the other potential ψ. Thus
for the nonrelativistic cold dark matter particles that our
simulations deal with, their acceleration is determined
solely by ψ, d(av)/dt = ikψ, where v is the proper mo-
tion. GR predicts ψ = −φ, if dark energy anisotropic
stress is negligible. Now, given an initial mass distribu-
tion ρ, we obtain φ from the Poisson equation, with the
coupling constant G. Then through the relation ψ = −φ,
we obtain ψ and then the acceleration. Thus given the
initial positions (density) and velocities of particles, we
can move particles in each simulation time step and then
have a closed procedure to simulate the evolution of the
Universe under gravity.
A natural parametrization of modified gravity is thus
to replace the Newton’s constant G by the effective
Newton’s constant Geff and the relation ψ = −φ by
η ≡ −φ/ψ. Now, given the mass distribution, the ac-
celeration is solely determined by the combination ζ [72],
ζ(k,z) ≡Geff(k,z)/G
η(k,z)
.(1)
This is the quantity that enters into the ψ-ρ relation,
k2ψ = −ζ4πGδρ . (2)
GR has the value ζ = 1. Clearly, if the two universes
have the identical initial conditions, identical expansion

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rate and identical ζ(k,z), the statistics of the density and
velocity fields would be identical [73].
Thus, instead of running a series of simulations on
a 2D grid of Geff-η parameter space, we just need to
run a series of simulations on a 1D grid of ζ parameter
space. It significantly reduces the amount of simulations
required. This is the major reason that we adopt this sin-
gle parameter parametrization on modified gravity. Be-
sides it, there are a number of attractive features of this
parametrization.
First of all, many MG models, such as the DGP grav-
ity model [16] and the f(R) gravity model [18, 19] (in
the linear regime), the Yukawa-like MG model and the
γ-index MG model [32] fit into this parameterization.
Second, such parameterization requires minimum mod-
ification in the N-body gravity solver, does not require
extra computation time, and thus is suitable for fast ex-
ploration of the vast parameter space of MG models. In
fact, there already exists a number of simulations on
Yukawa-like gravity [39, 40].
hence ζ), can be measured in a rather model independent
manner, by combining imaging surveys and spectroscopic
surveys [36, 41]. This links theories and observations di-
rectly. Furthermore, the reconstruction accuracy can be
improved by including all available data and performing
a multiparameter fitting [42, 43].
Clearly, this parameterization does not capture all fea-
tures of MG, such as the environmental dependence of
gravity, as found in f(R) gravity [44] and the DGP model
[45]. Nevertheless, the simulations based on this param-
eterization serve as an useful step toward better under-
standing of MG cosmology. The simulation results can
be used as templates to understand more complicated
MG models. A close analogy is the scale-free simula-
tions. Although the real CDM(cold dark matter) trans-
fer function is certainly not scale-free (power-law), these
scale-free simulations do significantly improve our under-
standing of the nonlinear evolution of structure forma-
tion. They are helpful in developing fitting formula like
that of Peacock-Dodds (hereafter PD96, [46]) and Smith
et al. 2003 (hereafter halofit, [47]). We hope that sim-
ilar procedure applies to the case of MG models. For
example, the formalism proposed by [22] relies on the
interpolation between the nonlinear power spectrum in
GR and the one in MG without environmental depen-
dence. Understanding the nonlinearities in MG models
without environmental dependence thus serves as a natu-
ral step to understand nonlinearities in more complicated
MG models.
Modifying existing N-body codes to incorporate the ζ
parameterization is straightforward. The only modifica-
tion is to change the particle acceleration? a to ζ×? a. In the
simulation setup, we fix the expansion rate identical to
that of the flat ΛCDM cosmology [74]. In addition, we do
not aim to explore the whole space of ζ(k,z). Rather, we
will focus on very special cases of ζ and postpone the gen-
eral investigation for future studies. The ζ(k,z) adopted
in our simulations is scale independent (ζ(k,z) = ζ(z)).
Third, Geff and η (and
The success of CMB (cosmic microwave background) and
BBN (big-bang nucleosynthesis) implies that GR is likely
valid in the early Universe. For this reason, we adopt a
step function in z, such that ζ = 1 at z ≥ zMG and
ζ =constant?= 1 at z < zMG. Throughout this paper, we
have adopted zMG= zi= 100, where ziis the initial red-
shift of simulations. Since we have GR valid at high red-
shift (z ≥ 100), the transfer function at zi= 100 adopted
in the MG models is identical to that in GR. For the
adopted MG parameterization, the linear density growth
factor D(k,z) is scale independent D(k,z) = D(z). Thus
the linear power spectrum for modified gravity models
only differs from ΛCDM by the linear density growth
factor D(z). In a companion paper, we will explore MG
models with other redshift dependence.
B.The precision requirements
All MG simulations begin with the identical initial con-
dition at zi= 100. Since the adopted ζ is scale indepen-
dent, the linear density growth factor D(z,ζ) is scale in-
dependent, as can be seen from the equation at z < zMG,
δ
′′
m+ δ
′
m
?
3
a+H
′
H
?
− ζ ×3
2
Ω0H2
H2a3
0
δm
a2= 0 . (3)
Here,
matter density in unit of the critical density. H0 and
H are the present day Hubble constant and the Hubble
parameter at z = 1/a − 1. δm is the linearly evolved
matter over-density and D ∝ δm is the linear density
growth factor. Thus, given a redshift zSin the standard
ΛCDM, we can find the corresponding redshift zζin the
MG universe, such that
′≡ d/da and
′′= d2/da2. Ω0 is the present day
D(zS,ζ = 1) = D(zζ,ζ) .(4)
Here, the subscript S denotes the standard ΛCDM cos-
mology. Since all the simulations begin with the identical
initial condition, the above relation means that,
PL(k;zS,ζ = 1) = PL(k;zζ,ζ) .
Here PLis the linear matter power spectrum. Through-
out this paper, we use the subscript “L” for the linear
statistics and the subscript “NL” for the nonlinear statis-
tics.
Modifications in GR change the structure growth his-
tory. The structure grows faster in a universe with big-
ger ζ. The primary quantity that we want to measure
through the simulations is
ǫ(k;zζ,ζ) ≡
PNL(k;zζ,ζ)
PNL(k;zS,ζ = 1).(5)
ǫ ?= 1 has a number of implications. (1) If the nonlinear
power spectrum is completely determined by the linear
one, independent of the expansion and structure growth

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history and the underlying gravity, then ǫ = 1. A number
of fitting formulae applicable to GR have been extended
to study the nonlinear evolution in MG models, based
on this assumption. Thus ǫ provides a direct test on the
applicability of these fitting formulae to MG models. Pre-
cision cosmology requires that, only if |ǫ − 1|<
the relevant k range, may the systematical error induced
by these fitting formulae be subdominant. Otherwise,
significant modifications shall be made. (2) ǫ ?= 1 also
means that there is extra information of gravity encoded
in the nonlinear matter power spectrum, which does not
show up in the linear power spectrum at the same epoch.
This helps to test GR at nonlinear regimes. Such in-
formation is complementary to those in the linear power
spectrum at the same epoch and those in the deeply non-
linear regime where gravity reduces to GR through en-
vironmental dependence mechanisms like the chameleon
mechanism and the Wainshtein mechanism [48].
∼10−2in
Much of the cosmological information in weak lens-
ing surveys come from the lensing power spectrum mea-
surement at ℓ<
∼2000 of source galaxies at zs ≃ 1.
Since the lensing kernel peaks at half way between the
source and the observer, the peak contribution comes
from k ≃ ℓ/[χ(zs)/2]<
∼2h/Mpc. At ℓ = 2000, the sta-
tistical error in the lensing power spectrum measurement
can reach below 1% for the planning of wide surveys. Un-
der the Limber approximation, the lensing angular (2D)
power spectrum is linearly proportional to the 3D nonlin-
ear matter power spectrum. Thus, to match the observa-
tion accuracy, we set a goal to model ǫ to ∼ 1% accuracy
at z ∼ 0.5 and k < 3h/Mpc.
Since the simulations run from the identical initial con-
dition, the cosmic variances in the resulting power spec-
tra PNL of different MG models are highly (positively)
correlated. Since the simulations are run by the same
code, with the same time steps, errors induced by the
numerical artifacts into PNLshould also be highly (pos-
itively) correlated. When taking the ratio of two power
spectra to evaluate ǫ, much of the errors in PNLcancels.
We thus expect higher accuracy in ǫ than in PNL. Thus,
once we control the error in PNLto ∼ 1% accuracy, we
are likely able to measure ǫ to 1% accuracy.
We run a set of N = 5123particle N-body simula-
tions using the GADGET-2 code, on the 32-CPU Ita-
nium server at the Shanghai astronomical observatory.
All the simulations that we use to calculate ǫ adopt
L = 300h−1Mpc. Adopting a smaller box size allows
us to go deeper into the nonlinear regime. However, a
smaller box size can cause numerical artifacts, due to the
missing of power at k < 2π/L, which affects the nonlin-
ear evolution through mode coupling [5]. Another reason
that we do not adopt a smaller box size is that, we plan to
use the same simulations for velocity and halo statistics,
which prefer a larger box size.
C. The GADGET-2 simulation specifications
We adopt a parallel GADGET-2 N-body code [49, 50]
to run the simulations. With a TreePM algorithm, where
only short-range forces are computed with the “tree”
method while long-range forces are determined by par-
ticle mesh (PM) algorithm, GADGET-2 combines high
efficiency with high resolution.
The background expansion history is fixed as the one
in a flat ΛCDM cosmology with the matter density
Ω0= 0.276 and the cosmological constant ΩΛ = 0.724.
The transfer function is fixed by the above parameters,
the baryon density Ωb = 0.046 and the dimensionless
Hubble constant h = 0.703. The amplitude of the ini-
tial fluctuations is fixed such that, if linearly evolved to
z = 0 in the adoption of ΛCDM cosmology, the rms den-
sity fluctuation within a sphere of radius 8h−1Mpc is
σ8= 0.811.
We use 5123PM mesh grids through all the simula-
tions. The force softening length γ depends on the mean
inter particle separation, with γ = 0.022L/N1/3, where
L is the box size and N is the particle number. For simu-
lations performed with 5123particles in the 300h−1Mpc
box, γ = 12.89h−1kpc. In GADGET-2, the adaptive
time step is set by ∆t =
?2ξγ/|a|, where ξ controls time
for all the simulations. With the adopted small softening
length, the number of total adaptive time steps for our
ΛCDM simulation is about 4000. Fig. 13 of H08 shows
that, for 3000 time steps in total, the resulting difference
in the power spectra is less than 0.04%. We thus believe
that, the time stepping we adopt suffices for the purpose
of this paper.
step accuracy and a is the acceleration. ξ is fixed at 0.5%
III. SIMULATION TESTS
In this section, we present steps to control the robust-
ness of simulation results. We adopt the Daubechies mass
assignment method to improve the accuracy of power
spectrum measurement. We run a number of tests to
justify that the adopted simulation specifications (parti-
cle number, simulation box size and initial redshift) are
adequate to constrain the nonlinear power spectrum out
to k = 3hMpc−1with ∼ 1% accuracy. Finally, we show
that the modified GADGET-2 code reproduces the cor-
rect linear evolution in the linear regime.
A.Calculating the matter power spectrum
Usually people use the fast Fourier transform (FFT)
to calculate the matter power spectrum. This requires
assigning simulation particles to uniform grids first. For
commonly used mass assignment methods, the resulting
power spectrum is biased by the smoothing and aliasing
effects, even at scales well below the Nyquist frequency
(e.g. [51]). To reach the required accuracy, we must

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FIG. 1: The top panel shows the matter power spectra for
simulations of different particle numbers. The bottom panel
shows the ratios with respect to 5123particle simulation.
All the power spectra are calculated through FFT on 5123
meshes. The dotted vertical line shows the scale of 0.7kNy
for our FFT power spectrum measurement. The black long
dashed line is the linear power spectrum. At z = 0, nonlinear
correction becomes significant at k > 0.2hMpc−1.
correct for these biases. Reference [51] proposes an iter-
ative method to perform such task. Alternatively, [52]
adopts the Daubechies wavelet transformation for the
mass assignment. The scale function of the Daubechies
wavelets transform has compact top-hat like support in
the Fourier space, which avoids the sampling effect and
allows computationally efficient mass assignment onto
grids. Using this scale function to do the mass assignment
allows for robust measurement of the power spectrum to
k = 0.7kNy [52]. Throughout this paper, we will adopt
this method to calculate the matter power spectrum.
B. Particle number
The particle number in GADGET-2 controls the mass
resolution, force resolution and the time step. A larger
particle number is necessary to avoid errors from dis-
creteness effects at small scales of interest. As pointed
out by Sirko [53], although simulations can probe the
evolution of structures beyond the particle Nyquist fre-
quency, kNy,p = πN1/3/L, it is unclear whether or not
the shot noise term beyond this frequency already in the
initial condition will impact power at the wavenumbers
of interest. The issue may be made moot merely by us-
ing negligible values of V/N in simulations. How many
particles are required to sufficiently sample the density
field and calculate the matter power spectrum robustly
to k = 3h/Mpc? To answer this question, we run three
simulations with identical initial conditions and a box
size of 300h−1Mpc, but with 1283, 2563and 5123par-
ticles, respectively. Fig. 1 shows the nonlinear power
FIG. 2: The impact of initial redshift on the matter power
spectrum. In the top panel, we show the power spectra of
the two simulations with starting redshift zi = 49( red line)
and zi = 100 (the black line). The bottom panel shows the
relative difference. The black horizontal dotted line shows
the 1% precision requirement. The red vertical dotted line is
0.7kNy. The same as Fig. 1, the black long dashed line shows
the linear power spectrum.
spectra calculated by the Daubechies’ mass assignment
method. We see clearly the impact of particle number on
the simulated power spectrum in the nonlinear regime.
The relative difference between the 2563and 1283results
at 1h/Mpc<
∼k<
∼3h/Mpc is ∼ 4%, implying a minimum
error of 4% in the 1283particle simulation, due to the
resolution limitation. But the relative difference reduces
to below 1-2% between the 5123and 2563ones, showing
that the resolution induced error in the 2563particle sim-
ulation is reduced significantly. This trend of convergence
implies that the resolution induced error in the 5123sim-
ulation is likely below ∼ 1%. We then speculate that, if
the Daubechies’ mass assignment method was adopted,
nonlinear power spectrum in the 5123particle simulation
can attain O(1%) accuracy out to k ∼ 3hMpc−1. To ro-
bustly test it, higher resolution simulations (e.g. ones
with 10243particles or more) are required.
shall be performed in future works.
This test
C.Initial redshift
Testing the effect of changing the starting redshift in
simulations is also important. Since the initial condition
is generated under the Zel’dovich approximation [54], the
initial redshift zican not be too low, otherwise higher or-
der corrections can be non-negligible. However, it is not
automatically the case that higher ziis better, because
numerical errors (most obviously suppression of power by
limited force resolution) have more time to accumulate
in that case [8]. Our initial redshift tests are started at
zi= 49 and zi= 100 respectively, both with 5123par-