Power Spectra to 1% Accuracy between Dynamical Dark Energy Cosmologies
Mon. Not. R. Astron. Soc. 000, 1–9 (2007) Printed 31 October 2018 (MN LATEX style file v2.2)
Power Spectra to 1% Accuracy between Dynamical Dark
Energy Cosmologies⋆
Matthew J. Francis1†, Geraint F. Lewis1 and Eric V. Linder2
1 School of Physics, University of Sydney, NSW 2006, Australia
2 University of California, Berkeley Lab, Berkeley, CA 94720, USA
ABSTRACT
For dynamical dark energy cosmologies we carry out a series of N-body gravitational
simulations, achieving percent level accuracy in the relative mass power spectra at
any redshift. Such accuracy in the power spectrum is necessary for next generation
cosmological mass probes. Our matching procedure reproduces the CMB distance to
last scattering and delivers subpercent level power spectra at z = 0 and z ≈ 3. We
discuss the physical implications for probing dark energy with surveys of large scale
structure.
Key words: methods:N-body simulations — methods: numerical — dark matter —
dark energy — large-scale structure of Universe
1 INTRODUCTION
The mass power spectrum plays a central role in understand-
ing the large scale structure of the Universe and the cosmic
expansion affecting the development of structure. Cosmolog-
ical probes of the growth history or the expansion history
such as weak gravitational lensing, baryon acoustic oscilla-
tions, or galaxy/cluster abundances depend crucially on ac-
curate knowledge of the mass power spectrum for physical
interpretation of the data from large scale surveys.
Most commonly the mass power spectrum is approxi-
mated by the ‘Halofit’ form of Smith et al (2003), and this is
used to determine cosmological parameters from the data (or
estimate future precision of parameter extraction). However,
the Smith et al (2003) formula is calibrated only on ΛCDM
models, and for these has a precision of ∼10%. This will be
insufficient for future large structure surveys that aim to ex-
plore the acceleration of the cosmic expansion and the prop-
erties of dark energy responsible for it. Huterer & Takada
(2005) estimate that for weak gravitational lensing surveys,
for example, 1% accuracy in knowledge of the mass power
spectrum will be required.
N-body simulations provide a well-tested technique for
calculating the dark matter power spectrum at the percent
level (Heitmann et al 2005). While this treats purely gravi-
tational forces, leaving out baryonic effects including heat-
ing and cooling, this should be a sufficient approximation for
wavemodes k < 3hMpc−1 (Jing et al. 2006; Zhan & Knox
⋆ Research undertaken as part of the Commonwealth Cosmology
Initiative (CCI: www.thecci.org), an international collaboration
supported by the Australian Research Council
† Email: mfrancis@physics.usyd.edu.au
2004; White 2004), or the scales larger than galaxies, that
are of most relevance for large surveys. However, carrying
out simulations for every possible cosmological model is ob-
viously impractical. If one could devise a mapping procedure
that matched models with the same key physical quantities,
ideally to a single class of cosmologies like ΛCDM, then this
would greatly aid the study of the cosmological information
carried by the distribution and growth of large scale struc-
ture.
In Section 2 we outline the approach to such a mapping
procedure and compare to previous work. Section 3 describes
the details of the simulations performed and tests carried
out. The qualitative physical consequences of the mapping
are interpreted in Section 4. We present the computational
results in Section 5 and identify several interesting features
in the wavemode and redshift dependence of the power spec-
trum. Physical interpretation of these results are discussed
in Section 6, with conclusions and future directions sum-
marised in Section 7.
2 DARK ENERGY AND COSMIC
STRUCTURE
While the influence of dark energy on the linear growth fac-
tor of matter density perturbations can be calculated simply
(see below), the full, nonlinear mass power spectrum requires
N-body simulations. Such simulation studies of the non-
linear power spectrum that exist for dark energy other than
a cosmological constant have tended to be for a constant
dark energy pressure to density ratio, or equation of state ra-
tio w (Ma et al. 1999; White & Vale 2004; Linder & White
2005; McDonald et al 2006).
c© 2007 RAS
http://arxiv.org/abs/0704.0312v1
2 Francis, Lewis & Linder
A few simulation studies have considered the effects
of dynamical dark energy on the power spectrum, either
through a parameterized time (or scale factor a) dependent
form w(a) or a specific scalar field potential (Klypin et al
2003; Macciò et al 2004; Ma 2006).
Efforts to calibrate non-linear power spectrum fitting
formulas include early work by Ma et al. (1999), which was
much improved upon with the advances in computing power
by McDonald et al (2006). Both of these studies looked at
modifying existing fits in the case of constant w cosmologies.
Linder & White (2005) (LW) investigated the effects of w on
the non-linear power spectrum, searching for key physical
quantities, and discovered a simple matching prescription
for calculating the non-linear mass power spectrum to within
one to two percent.
This work extends the study of the dark energy ef-
fects on the full, non-linear mass power spectrum to models
with dynamical dark energy, utilizing the model indepen-
dent, physically motivated (Linder 2003) evolving equation
of state w(a) = w0+wa(1−a). At the same time, we employ
the approach of seeking central physical matching quanti-
ties that incorporate CMB data through agreeing on the
distance to the last scattering surface.
In studying the non-linear power spectrum of mass fluc-
tuations, a natural place to start is with the linear power
spectrum. The effects of dark energy on the linear mass
power spectrum can be calculated through the relation
P (k, a) =
D2(a)
D2(ai)
P (k, ai) (1)
(see e.g. Coles & Lucchin (2002)) with the growth factor
D(a) given by the formula (e.g. Hu (2002); Linder & Jenkins
(2003))
1 +X(a)
1 +X(a)
= 0 (2)
with derivatives with respect to scale factor a, and where
X(a) is the ratio of the matter to dark energy densities,
given by X(a) = Ωm
d ln a′w(a′), with Ωm the di-
mensionless present matter density.
The non-linear power spectrum cannot be written in
terms of a simple differential equation and requires the use
of large volume, high resolution, N-body simulations. These
are computationally expensive and therefore accurate semi-
analytic fitting formulas derived from simulation results
are a valuable tool. The most widely adopted current for-
mula, sometimes called Halofit, was presented in Smith et al
(2003). This formula is motivated by the halo model of struc-
ture growth with free parameters in the function set by fit-
ting to a large suite of simulations. All these simulations,
however, were of cosmological constant, w = −1, cosmolo-
gies. McDonald et al (2006) produced a fitting formula as a
multipolynomial series for constant w models, intended to
be used to modify the Smith et al result. This modification
was estimated to be accurate to within a few percent in the
range of cosmologies encompassed by the simulation grid.
Taking a different approach, LW demonstrated that
when the linear growth factors between different w =const
models were matched at a high redshift point as well as at
z = 0, by compensating with other cosmological parameters,
the non-linear power spectrum from N-body simulations
also matched to much better than a percent at those red-
shifts, as well as matching to one to two percent at any inter-
mediate redshift. Additionally, LW also found that the dis-
tance to the surface of last scattering, dlss, closely matched
when their growth matching criteria was implemented, pre-
serving CMB constraints. With this formalism the power
spectrum for a dark energy model can be matched to, say,
a ΛCDM cosmology. Hence one can either use an appropri-
ately matched Halofit result or carry out a vastly reduced
suite of only ΛCDM simulations to achieve the desired ac-
curacy on the mass power spectrum.
This article concentrates on developing accurate match-
ing of the non-linear mass power spectrum for dynamical
dark energy models. We employ a somewhat different match-
ing procedure from LW, explicitly matching the distance to
CMB last scattering dlss and the mass fluctuation amplitude
σ8 at the present and studying the effect on the growth. In
this respect, our approach is essentially the converse of the
LW approach. The geometric factor of the distance to CMB
last scattering suffices to incorporate substantially the CMB
constraints on the dark energy parameters. Since dark en-
ergy had a negligible density in the early universe (except in
special, early dark energy models, e.g. see Wetterich (2004);
Doran & Robbers (2006); Linder (2006)), the physical size
and nature of features in the CMB at the surface of last scat-
tering is largely insensitive to the properties of dark energy.
However, the angular size of such features is set through
the angular diameter distance, which does depend on the
properties of dark energy, since it relates to the expansion
history of the universe a(t). Therefore, dark energy mod-
els giving the same distance to the last scattering surface
are largely degenerate with respect to the CMB (some dif-
ferences relating to secondary anisotropies such as the ISW
effect remain, see Hu & Dodelson (2002)). For a given dy-
namic dark energy model (w0, wa), there is a corresponding
constant equation of state weff , say, that gives the same dlss
as the dynamical model, holding all other cosmological pa-
rameters (such as the physical matter density Ωmh
2) fixed.
This article examines the relation between the non-linear
mass power spectra of the dynamical and the weff models.
Once a tight correspondence is established, one can then
either employ a constant w fitting formula such as from
McDonald et al (2006), carry out only a suite of constant
w simulations, or adjust the other cosmological parameters
such that one chooses weff = −1 and requires only ΛCDM
simulations. We discuss these alternatives further in §7.
3 SIMULATION DETAILS
The simulations were performed using the GADGET-2 N-
body code (Springel 2005), modified to incorporate the back-
ground evolution a(t) appropriate for dynamical dark energy
cosmologies with w(a) = w0 + (1 − a)wa. Fiducial simula-
tion runs use 2563 dark matter particles in a 256 h−1Mpc
periodic box with a 5123 force grid; the initial redshift was
z = 24 and the force softening was set to a constant co-
moving length of 60h−1Kpc. In order to check numerical
convergence, runs were also performed with combinations
of box size and particle number a factor of 2 greater and
smaller than the fiducial. In addition, runs checking conver-
gence were performed for numerical parameters including
the start time, softening length, PM grid spacing, time and
c© 2007 RAS, MNRAS 000, 1–9
Power Spectra to 1% Accuracy 3
force accuracy and tree update frequency. The ratio of power
between the different dark energy models were largely insen-
sitive to these parameters, changing by a small fraction of a
percent out to k < 3hMpc−1.
The linear matter power spectra used to cre-
ate the initial conditions were calculated using CAMB
(Lewis, Challinor & Lasenby 2000). Initial conditions were
generated from the power spectrum using part of
the GRAFIC program within the COSMICS package
(Ma & Bertschinger 1995).
For each set of distance-matched runs, the same in-
put power spectrum, generated by CAMB using the w =
constant model, was used for each model. In order to match
the amplitude of linear growth today (identical σ8 at z = 0)
for simulations of different cosmologies, the initial density
and velocity perturbations of the particles were scaled in
the Zel’dovich approximation using the linear growth fac-
tor ratio D(astart)/D(a = 1) for the different models. This
ansatz for initial conditions is robust as long as the dark
energy does not change the shape of the linear power spec-
trum at zstart, i.e. the dark energy plays little role in the
very early universe. We have verified this to high accuracy
using a version of CAMB modified for (w0, wa) models with-
out dark energy perturbations. Note that in the presence of
dark energy perturbations, the initial power spectrum over
our range of k = 0.1 − 3hMpc−1 is affected by less than
1% for constant w models that are not distance matched.
We expect that distance-matched (w0, wa) models with per-
turbations will show less effect but future work will address
this.
The calculation of the power spectrum in simulations
outputs used the ‘chaining the power’ method described in
Smith et al (2003) utilising the cloud in a cell assignment
scheme. No correction was made for shot noise, as the quan-
tity of interest was the ratio of the power between different
models. See McDonald et al (2006) for an extended discus-
sion of the usefulness of taking power spectrum ratios to
eliminate many numerical errors in this type of study. See
also White (2005).
All simulations in this paper used the best fit cosmo-
logical parameters from Spergel et al (2006) of Ωm = 0.234,
h = 0.74, Ωb = 0.0407 and σ8 = 0.76 in a flat ΛCDM
universe. For each set of simulations, a constant equation of
state weff is selected and several values for the parameters w0
and wa that maintained the same dlss were calculated. One
consequence of this methodology is that these w(a) models
cross the value w = −1 at some point in cosmic history. De-
bate exists surrounding the physical validity of crossing be-
tween the phantom regime, defined as w < −1, and w > −1.
This will eventually be settled by a microphysical theory
for dark energy, rather than merely a phenomenological de-
scription. With this in mind we do not consider the issue of
phantom energy and phantom crossing further.
We select three values, weff = −0.9,−1,−1.1, as
the foundations for our comparison of w(a) cosmologies.
This range is in accord with constraints on constant w
from current cosmological data sets (Spergel et al 2006;
Seljak, Slosar, & McDonald 2006) and provides a reasonable
variety for testing the matching procedure. For each con-
stant w model, simulations were carried out for four more
w(a) models with matching distance to the LSS. The dark
energy models used are summarized in Table 1.
weff = −0.9 weff = −1.0 weff = −1.1
w0 wa w0 wa w0 wa
-1.1 0.620 -1.2 0.663 -1.3 0.707
-1.0 0.319 -1.1 0.341 -1.2 0.363
-0.8 -0.336 -0.9 -0.359 -1.0 -0.381
-0.7 -0.686 -0.8 -0.732 -0.9 -0.778
Table 1. Distance Matched Models. Simulations were carried out
for five models (including w = weff) for each of three values of
weff , where all five models in a column had identical distances to
CMB last scattering.
Figure 1. Dark energy equation of state vs. a for the models
matched to the CMB last scattering surface distance for w = −1.
4 THE CONSEQUENCES OF DISTANCE
MATCHING
If the simple distance matching procedure outlined in this
paper is to succeed in producing a good match in the mat-
ter power spectrum, it might be expected to keep a range
of physical conditions similar through cosmic history. This
is indeed what is found. In particular, a variety of physical
quantities exhibit pivot or crossover points, indicating not
only near equality at that epoch, but a tendency toward
agreement of quantities integrated over cosmic history. For
instance, Figure 1 plots w(a) for the four models with match-
ing distance to the w = −1 model; there is a clear epoch at
a ≈ 0.7 where the values all cross w = −1.
The linear growth factors of the various distance
matched models also closely track each other. While the lin-
ear growth is matched at a = 1 by construction, there is an
additional epoch at high redshift where the linear growth of
all associated models closely matches. For the weff = −1 set
of models shown in Figure 2, the matching point is a = 0.24
or z = 3.12. The other two sets of models match at a similar
value and show a similar trend. This behaviour illustrates
the converse of what was found in LW, where dlss was found
to closely match when the linear growth was matched at
some high redshift point. The crossover is important, since
rather than all models diverging from the z = 0 match,
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4 Francis, Lewis & Linder
Figure 2. Ratio of the linear growth factor D̄(a) ≡ D(a/D(1)
relative to the central w = −1 model for the w(a) models with
matched dlss.
Figure 3. Ratio of ΩDE(a) relative to the central w = −1 model
for the w(a) models with matched dlss.
the curves track each other relatively closely. Since the non-
linear power spectrum is intimately tied to the linear be-
havior, this provides hope that a mapping of the full power
spectrum between models can be realized.
The growth crossover behaviour results from a change
in sign of the relative rate of growth at a ≈ 0.5. In the linear
regime the growth of fluctuations can be seen as a balance
between the mutual gravitational attraction of the overden-
sity, which is amplified by higher mean matter density, and
the expansion rate of the universe characterised by H(a),
which acts like a frictional term, suppressing growth the
higher the expansion rate. In this picture, greater relative
matter domination at a particular epoch will produce a more
rapid growth rate at that time compared to a less matter
dominated model. With this in mind it is worth comparing
the matter domination history of our suite of cosmologies.
Figure 3 shows the relative dark energy density Ωde(a) for
the w(a) models matched to w = −1.
Comparing figures 2 and 3 we see that where the rela-
tive dark energy density is higher, the relative rate of linear
growth is lower. For instance, the long dashed model has a
Figure 4. Ratio of the non-linear mass power spectrum at z=0
relative to the w = −1 model for models with matched dlss.
more negative slope in the region a < 0.5 in figure 2 than the
other models, corresponding to a higher relative dark energy
density in this region as shown in figure 3. The magnitude of
growth is greater initially in the long dashed model in order
to achieve the match at z = 0, however the growth initially
grows more slowly in comparison to the other models. The
change in sign of the relative growth rates at a ≈ 0.5 in fig-
ure 2 corresponds to the crossover point in relative matter
domination in Figure 3. The energy density, like the linear
growth, exhibits a striking crossover point, again keeping
physical conditions similar between models throughout cos-
mic history.
In a chain of related conditions, the crossover point of
the equation of state w(a) (see Figure 1) causes the con-
vergence of the dark energy density Ωde(a) (see Figure 3,
and the crossover of Ωde(a) leads to the convergence of the
growthD(a) (see Figure 2), which then creates a crossover in
the growth at higher redshift. This in turn will keep the non-
linear power spectrum closely matched between the models
over the entire range z = 0− 3.
5 RESULTS
The ratio of power measured in the simulation outputs at
z = 0 relative to the central weff model for each of the
sets of simulations is given in Figures 4, 5 and 6. The most
outstanding result is the excellent agreement between the
mapped power spectra, at the 0.1% level for k < 1hMpc−1
and . 1% for k < 3hMpc−1 (. 0.5% at the higher k for
the less rapidly varying dark energy models). These figures
show very similar trends, regardless of the fiducial weff model
chosen. Since the trends are similar, the remaining figures
will show the results for the w = −1 central models only, for
brevity.
The simulations shown in these figures started with an
identical realisation of initial conditions, albeit scaled with
respect to the linear growth factor to produce matched linear
growth at z = 0. As the deviations between models are
small, it is important to take care to ensure that any features
are real and not the results of a spurious numerical effect.
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Figure 5. As Fig. 4, for the central model w = −0.9.
Figure 6. As Fig. 4, for the central model w = −1.1.
The convergence tests described in Section 3 addressed the
effects of numerical parameters 1.
There are also two other potential sources of error, the
limited volume of the simulation box and the error sam-
pling error in calculating the power spectrum of the simula-
tion snapshots. Care must be take that these effects are not
causing spurious results. Figure 7 shows the results for the
w = −1 central model simulations with rms sampling errors
from the FFT power spectrum calculation plotted. For clar-
ity these have been omitted from the other plots, however
the errors are similar in all cases.
From this plot the deviation between models is roughly
1 Changing particle resolution did cause a slight systematic shift
in features seen in the power spectrum ratios. The onset of the
dispersion between the models seen in figures 4-6 at k ∼ 1hMpc−1
shifted to lower k with reduced particle resolution and higher k
with an increase. This shift was of order 0.1 in log k for factors of
two differences in particle resolution. We cannot fully account for
this numerical effect, however the difference in power at a given
k-mode due to the shift is at most ∼ 0.1% and since subpercent
effects are beyond the ability of N-body simulations to probe ac-
curately, we do not believe this effect is of significant consequence.
Figure 7. As Fig. 4, with rms sampling errors included, shown
by thin lines of the same line style as each model.
Figure 8. The effect on the power spectrum due to different real-
isations. Displayed are the power in three realisations of a single
cosmological model, w = −1, as a ratio to the power in a fourth
realisation. The rms errors for each power spectrum calculation
are also shown. As expected the finite volume error decreases as
k increases due to the greater number of modes present at higher
wavenumbers.
a factor of 1−2 that of the rms error. In order to verify that
the effects seen are genuinely due to the difference in dark
energy models, another three sets of simulations with the
same parameters but different realisations were performed.
The scatter in the calculated power spectrum due to dif-
ferent realisations is shown in Figure 8. The scatter in this
figure clearly demonstrates the inability to accurately deter-
mine the absolute power with the box sizes and number of
realisations used in this study due to finite volume errors.
From the figures shown it is clear that the difference be-
tween realisations for a single cosmological model is greater
than the difference within a single realisation for the differ-
ent models. This makes accurate modeling of the absolute
value of the power spectrum an extremely challenging task,
requiring larger boxes, many more realisations and highly
detailed consideration of sources of numerical error. Instead,
we are interested in the effect of dark energy models relative
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6 Francis, Lewis & Linder
Figure 9. The effect of different realisations on the ratio of a
dynamical dark energy power spectrum to the associated w =
const model. Shown are 4 realisations of the ratio between the
strongly time varying (w0, wa) = (−1.2, 0.663) model and the
w = −1 model at z = 1, plotted as a ratio of the main realisation
used in the paper. This is a typical example of the magnitude of
the variation due to different realisations.
to one another and therefore what is important is how much
the ratios between models (such as shown in Figures 4-6)
are affected by different realisations. Fortunately the effect
of different realisations is vanishingly small as seen in Fig-
ure 9, which shows a typical example of the variation in
power ratios across the four realisations used.
From Figure 9 we can be confident that the computed
non-linear power spectra ratios are not visibly affected by
spurious finite volume errors or effects due to the FFT cal-
culation of the power spectrum.
For cosmological structure probes, we are interested not
just in how well we can predict the power spectrum at z = 0
but across all redshifts. In the simulations performed, data
was output at a number of times. From Figure 2, two epochs
are of particular interest. The first is the crossover in linear
growth at a = 0.24 and the second is at a = 0.5 where the
linear growth is most varied between models. The latter cor-
responds to z = 1, which is extremely relevant to a number
of forthcoming cosmological surveys. Hence an accurate es-
timation of power here, provided by the distance matching
scheme, is of great importance for understanding possible
constraints on dark energy cosmologies.
The ratio of power measured in the simulations boxes
at these epochs are shown in Figures 10 and 11.
We can do even better, however, by realizing that much
of the difference in power, particularly at a ≈ 0.5, can be
accounted for by the difference in linear power. Scaling this
out via Equation 1, the results are as shown in Figures 12
and 13.
From these figures we can see that the combined dis-
tance and growth matching procedure is generally accurate
to better than 1%. The greatest deviation found in all sim-
ulation outputs is 2% for k ≈ 3 hMpc−1 at a = 0.5.
Since the results shown thus far display a good match
for distance matched models, it is worth considering how
much improvement this matching achieves compared to ar-
bitrary dark energy models that are not distance matched.
Figure 10. As Fig. 4, but at a = 0.24 where from Fig. 2 the
linear growths closely match (slightly lower than for w = −1).
Figure 11. As Fig. 4, but at a = 0.5 where from Fig. 2 the linear
growths are most divergent. Much of the difference in power comes
from this difference in linear power.
Figure 12. As Fig. 10, but with the linear growth difference
scaled out. Note the reduced y-axis scale relative to Fig. 10.
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Figure 13. As Fig. 11, but with the linear growth difference
scaled out. Note the reduced y-axis scale relative to Fig. 11.
Figure 14. The ratio of the non-linear power in several non-
distance matched cosmologies to a ΛCDM model at z = 1. The
amplitude of linear growth at z = 0 is the same for all models
as in previous figures. Note that the divergence between models
increases with k but that the divergence begins at a lower k and
is significantly greater than with the distance matched models
shown previously
In other words, how much of a role do the values of the
dark energy parameters play in structure formation, if all
other parameters are kept fixed? Figure 14 shows the ra-
tio of power at z = 1 between a ΛCDM model and several
non-distance matched models with the same linear growth
amplitude at z = 0. The divergence between these mod-
els is significantly greater than what is seen in the distance
matched models (and of course they will disagree on the
CMB), illustrating the improvement achieved by this sim-
ple scheme.
6 EVOLVING DARK ENERGY AND
STRUCTURE GROWTH
The matching prescription used in this article produces a
mapping between the matter power spectra of dark energy
cosmologies accurate to . 1% over a wide range of wave-
modes and a wide range of cosmic history. This agreement
1) indicates that simple physical quantities determine the
nonlinear power spectrum over this range, leading to the
prospect of understanding structure formation on a funda-
mental level even in dynamical dark energy cosmologies, 2)
points the way to significant advances in computational effi-
ciency by reducing the dimension of the grid of simulations
necessary to produce accurate estimations of power spectra
required for interpretation of cosmological probes such as
weak gravitational lensing, baryon acoustic oscillation, and
other large scale structure surveys, and 3) identifies a degen-
eracy that makes it difficult to distinguish between models
lying on a particular subsurface of the cosmological param-
eter space.
To try to ameliorate the degeneracy, we note that an
evolving equation of state does imprint a small but system-
atic effect on the non-linear matter power spectrum. The
general trend shown by Figures 4, 5 and 6 is that dark en-
ergy with a less negative value today but more negative value
at high redshift (i.e. negative wa) gives greater non-linear
power at k & 1hMpc−1 than its dlss-matched weff model.
Similarly, more negative equations of state today with posi-
tive wa possess less power in the same range. This deviation
is however relatively small, remaining less than 2% out to
k = 3hMpc−1. Even so, this partial degeneracy is not too
worrying since it can readily be broken by other cosmolog-
ical dependencies (e.g. the geometric distance dependence
entering with the mass power spectrum into the weak lens-
ing shear power spectrum or the baryon acoustic scale) or by
complementary cosmological probes. Thus the model map-
ping technique does not appear to have any real drawbacks
to detract from its physical and computational advantages.
Elaborating on the physical import of the mapping, a
striking feature is the marked difference at z = 0 between
k < 1hMpc−1 where the power spectra between models
match extremely closely and k > 1hMpc−1 where they di-
verge. This seems to suggest a transition between the lin-
ear region at low k where by design the power should be
matched, and the fully non-linear region at high k where
differences in cosmic evolution have imprinted a different
signature on the growth of structure on small scales, per-
haps reflecting the effect of dark energy on conditions when
structure formed at high redshift.
Those models that show a greater non-linear growth are
those that in the early universe had a greater contribution
of the matter density relative to the dark energy density;
these correspond to the models with today w0 > weff and
in the recent universe had lower matter density relative to
dark energy density. This suggests that non-linear growth
is more sensitive to conditions (including the effects of dark
energy) in the early, matter dominated universe than it is
to conditions in the later, accelerated era of dark energy
domination.
Carrying this forward, one conjecture is that the “tran-
sition” in the behavior of the z = 0 power spectrum at
k = 1 hMpc−1 might be related to early, rather than z = 0,
non-linear effects. While the non-linear scale at z = 0 should
be near k . 0.2, the power spectrum remains well matched
here, possibly because the non-linear growth was already
matched as a result of the model mapping, i.e. “pinned
down” by the agreement at a = 0.24. So the greatest dif-
c© 2007 RAS, MNRAS 000, 1–9
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ference in non-linear growth, arising from times earlier than
the a = 0.24 matching, might appear at the non-linear scale
associated with a < 0.24, or k & 1hMpc−1, rather than
the z = 0 non-linear scale. In any case, the accurate ap-
proximation of the power spectrum utilizing the matching
prescription indicates that reasonably simple physics lies be-
hind even the non-linear mass power spectrum.
7 CONCLUSION
The mass power spectrum lies at the foundation of many cos-
mological observables, such as the weak lensing shear statis-
tics of galaxies, the large scale structure clustering distri-
bution (including baryon acoustic oscillations), and cluster
abundances. To utilize any of these cosmological measure-
ments as next generation probes of large scale structure,
cosmology, or dark energy requires clear and accurate un-
derstanding of the mass power spectrum over the range of
models under consideration, e.g. dynamical dark energy not
just ΛCDM.
The main results of this article (see Figs. 4,12 and 13)
demonstrate that non-linear mass power spectra of dynam-
ical dark energy models with smooth equation of state evo-
lution can be determined to percent accuracy by calculating
the power for the constant equation of state cosmology that
gives a matching distance to the CMB last scattering sur-
face. By varying other parameters as well, such as the matter
density Ωm and h keeping Ωmh
2 constant (see LW), one can
envision mapping a wide variety of dark energy models to
ΛCDM models, resulting in significant gains in computa-
tional efficiency. This can also alleviate concerns regarding
phantom crossing.
Finding the distance matched models as described in
this paper is a trivial task numerically requiring the inte-
gration of a differential equation and a one dimensional pa-
rameter search. This simple procedure however provides a
mapping that is accurate to a percent. Of course the ac-
curacy of the resultant power spectrum estimation is ulti-
mately only as good as the accuracy in the model being
mapped to. For that model we can then utilise fitting for-
mulas, such as Halofit, for a rough, ∼ 10% accuracy or more
generally perform N-body simulations with the desired pa-
rameters. However, the distance matching scheme in this
case allows a much reduced grid of simulations to be carried
out while still maintaining a high degree of accuracy.
The physics behind the matching prescription involves
a chain of consequences from the crossover in the behavior
of one variable (e.g. equation of state) to the convergence in
another, then leading to matching in the large scale struc-
ture growth. Further physics, not yet fully elucidated, points
to the full nonlinear power spectrum being dependent not
on the linear growth, but the linear growth history , where
the conditions (e.g. matter density or growth factor) at one
epoch directly manifest in the nonlinear behavior at a later
epoch.
Future work should pursue this further, as well as inves-
tigating the prospects for discerning the signatures of more
complicated equations of state or perturbations. While the
prescription here for mapping the mass power spectrum to
percent accuracy between cosmologies is useful in itself and
for computational gains, the most exciting prospects are for
improved analytic fitting formulas and deeper physical un-
derstanding.
ACKNOWLEDGMENTS
MF acknowledges the support of a Science Faculty UPA,
thanks Chris Power and Jeremy Bailin for helpful discus-
sions, advice and pieces of code, and thanks LBNL and
SNAP for hospitality and support during much of the writ-
ing of this article. We thank Martin White for pointing out
the nice method of getting high resolution FFT’s without
needing large arrays and for other useful discussions. This
work has been supported by the Australian Research Coun-
cil under grant DP 0665574 and in part by the Director,
Office of Science, US Department of Energy under grant
DE-AC02-05CH11231.
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	Introduction
	Dark Energy and Cosmic Structure
	Simulation details
	The Consequences of Distance Matching
	Results
	Evolving Dark Energy and Structure Growth
	Conclusion
