A&A 487, 63-74 (2008)
DOI: 10.1051/0004-6361:20077688
C. Wagner - V. Müller - M. Steinmetz
AIP - Astrophysikalisches Institut Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany
Received 20 April 2007 / Accepted 15 May 2008
Abstract
Context. The measurement of the scale of the baryon acoustic oscillations (BAO) in the galaxy power spectrum as a function of redshift is a promising method to constrain the equation-of-state parameter of the dark energy w.
Aims. To measure the scale of the BAO precisely, a substantial volume of space must be surveyed. We test whether light-cone effects are important and whether the scaling relations used to compensate for an incorrect reference cosmology are in this case sufficiently accurate. We investigate the degeneracies in the cosmological parameters and the benefits of using the two-dimensional anisotropic power spectrum. Finally, we estimate the uncertainty with which w can be measured by proposed surveys at redshifts of about z=3 and z=1, respectively.
Methods. Our data is generated by cosmological N-body simulations of the standard CDM scenario. We construct galaxy catalogs by ``observing'' the redshifts of different numbers of mock galaxies on a light cone at redshifts of about z=3 and z=1. From the ``observed'' redshifts, we calculate the distances, assuming a reference cosmology that depends on
.
We do this for
,
and -1.2 holding the other cosmological parameters fixed. By fitting the corresponding (an)isotropic power spectra, we determine the apparent scale of the BAO and the corresponding w.
Results. In the simulated survey we find that light-cone effects are small and that the simple scaling relations used to correct for the cosmological distortion work fairly well even for large survey volumes. The analysis of the two-dimensional anisotropic power spectra enables an independent determination to be made of the apparent scale of the BAO, perpendicular and parallel to the line of sight. This is essential for two-parameter w-models, such as the redshift-dependent dark energy model
w=w0+(1-a) wa. Using Planck priors for the matter and baryon density and
for the Hubble constant, we estimate that the BAO measurements of future surveys around z=3 and z=1 will be able to constrain, independently of other cosmological probes, a constant w to
and
(68% c.l.), respectively.
Key words: cosmology: cosmological parameters - cosmology: large-scale structure of Universe
There are two main observational indications of the existence of an unknown
energy component in the Universe, which is usually referred to as dark energy (DE). First,
observations of distant supernovae Ia (Perlmutter et al. 1999; Riess et al. 1998) favor an accelerating
expansion of the Universe and therefore imply an energy component of
negative pressure P. In particular, the parameter w of the equation of state
(EOS)
must obey w<-1/3. Second, measurements of the anisotropies
in the cosmic microwave background (CMB) (Spergel et al. 2007,2003) in combination with observations of
the large-scale structure of the Universe argue for a
spatially flat Universe. Matter (baryonic and dark) contributes however less
than 30% to the critical density. Hence, about 70% of the present-day
energy density of the Universe appears to be in an unknown form of energy.
The simplest way to account for this missing energy and the accelerating
expansion is to introduce a cosmological constant
in Einstein's
equations, which has a redshift-independent EOS parameter w=-1. So far
all observations appear to agree with the model of a cosmological constant.
There is however at least one theoretical drawback. The observed value of
,
which is interpreted as vacuum energy, is highly inconsistent with
current predictions by particle physics, a discrepancy commonly referred to as
the cosmological constant
problem (for a review, see Carroll 2001). This has motivated consideration
of more general dark energy models that have a redshift-dependent EOS
parameter w(z). The measurement of the parameter w can therefore
help to distinguish between not only a simple cosmological constant and other dark
energy models, but potentially also between these different models.
Several possible methods to constrain the EOS parameter w are summarized by
the Dark Energy Task Force Report (Albrecht et al. 2006).
The method that we consider here uses the expansion history
of the Universe. To measure this precisely, we require a
standard candle or a standard ruler. For redshifts up to
,
supernovae Ia can be observed and calibrated to be standard candles, with which one can measure
the luminous distance
.
Eisenstein et al. (1999,1998)
proposed that the baryon acoustic oscillations (BAO) imprinted in the galaxy power
spectrum could be utilized as a standard ruler. The BAO have the same origin as the acoustic
peaks in the anisotropies of the CMB. Before recombination, baryons and photons
were tightly coupled. Gravitation and radiation pressure
produced acoustic oscillations in this hot plasma; during the expansion of the
Universe, the plasma then cooled and finally nuclei and electrons recombined. The released
photons propagated through the expanding Universe and are observed by
ourselves as
the highly redshifted radiation of the CMB; in contrast, the baryons followed
the clustering of the dark matter and eventually collapsed to form
galaxies. Since the dark matter did not participate
in the acoustic oscillations, the oscillatory feature in the galaxy power
spectrum is far less pronounced than for the CMB photons
(Bond & Efstathiou 1984; Holtzman 1989; Peebles & Yu 1970; Sunyaev & Zeldovich 1970; Eisenstein & Hu 1998; Hu & Sugiyama 1996).
When the physical scale of the BAO has been calibrated using precise CMB measurements, it can be applied as a standard ruler for measuring the angular
diameter distance
and Hubble parameter H(z).
In contrast to supernovae Ia, the scale of the BAO is a more reliable
standard ruler at high redshifts. As the number of unperturbed peaks
and troughs corresponding to the BAO in the power spectra increases,
the wavelength of the BAO can be determined more accurately.
On scales where structure growth is already nonlinear, the oscillations
cannot be easily discerned (Springel et al. 2005; Gottlöber et al. 2006; Eisenstein et al. 2007b). Hence, with
decreasing redshift the uncertainty in the observed scale of the BAO
increases. To have good statistics for the first peaks a large volume has to
be surveyed. The BAO have been detected in the present-day largest galaxy
redshift surveys (Cole et al. 2005; Eisenstein et al. 2005). Data sets studying larger volumes and higher redshifts are
however required to achieve tight
constraints on dynamical DE models. Galaxy redshift surveys at about
z=3, 1, or 0.5, like HETDEX (Hill et al. 2004), the WFMOS BAO survey (Bassett et al. 2005; Glazebrook et al. 2005), and BOSS (SDSS-III Collaboration 2008), are designed for measuring
the scale of the BAO and the EOS parameter w with a precision to
a few percent and thereby constrain a variety of DE models.
Many studies presented methods for extracting the scale of the BAO and estimated the accuracy of its measurement achievable by future surveys. In these analyses, Monte Carlo simulations (Glazebrook & Blake 2005; Blake et al. 2006; Blake & Glazebrook 2003), Fisher matrix techniques (Seo & Eisenstein 2003; Matsubara 2004; Hu & Haiman 2003; Seo & Eisenstein 2007; Linder 2003; Hütsi 2006b), N-body simulations (White 2005; Angulo et al. 2008; Koehler et al. 2007; Angulo et al. 2005; Huff et al. 2007; Seo & Eisenstein 2005), and observational data (Hütsi 2006a,2005,2006c; Eisenstein et al. 2005) were used.
In this article, we attempt to include all important (physical and observational) effects for measuring the BAO and deriving constraints on the EOS parameter w. Our perspective is that of an observer, i.e. the starting point for the BAO measurement should be a galaxy catalog that provides celestial coordinates and redshifts of the galaxies. Since we do not have in hand the type of real observations that we require, we have first to generate mock catalogs; we achieve this by completing ``observations'' on the data products of N-body simulations. Using these ``observations'', we are able to study for the first time in detail the light-cone effect and the accuracy of the scaling relations used to compensate the cosmological distortion that arises by assuming an incorrect reference cosmology. A crucial point for all BAO measurements is the fitting method. We develop a method which uses only the oscillatory part of the power spectrum in a way that produces unbiased and robust results. Further, we compare the results of fitting the angle-averaged one-dimensional power spectrum and anisotropic two-dimensional power spectrum. Finally, we predict the uncertainty with which proposed surveys will be able to measure the EOS parameter w by assuming two different w-models.
The paper is structured as follows. In Sect. 2, we review how the EOS parameter w is measured using BAO. In Sect. 3, we explain how we generate the ``observed'' data from N-body simulations, and in Sect. 4, we describe the power spectrum calculation and our fitting method. In Sect. 5, we present our results, and finally we provide our conclusions in Sect. 6.
Knowledge of the true comoving scales, both parallel (
)
and transverse (
)
to the line of sight, of an observed physical property at a given
redshift z, in the case of BAO statistical properties of large-scale
structure, enables us to derive the Hubble parameter H(z) and angular diameter
distance
from the measured quantities (redshift
and
angle
):
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One problem of using the scale of the BAO as a standard ruler is that the BAO appear in a
statistical quantity. We cannot measure
of the BAO, but we can measure
the redshifts of the galaxies and then, assuming a reference cosmology,
reconstruct their positions and derive their power spectrum. From this power
spectrum, we can determine the apparent scale of the BAO and compare this
with its true value. If they agree, we have used the correct cosmology. In
principle, for every trial cosmology we have to recalculate the distances and
recompute the power spectrum. A more efficient method is to scale
appropriately the power spectrum derived for the reference cosmology using
the following approximations (Glazebrook & Blake 2005; Seo & Eisenstein 2003)
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Figure 1:
The scaling factors
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To derive realistic samples of large galaxy surveys, we first completed an N-body
simulation in a large box (
)
and constructed
the corresponding dark matter distribution on a light cone by interpolating
between about 20 snapshots. We calculated the redshifts, which would
be observed including the effect of peculiar velocities. Assuming a certain
cosmology, we converted these redshifts into distances. Finally, we selected a
certain number of particles by applying a simple bias scheme and defining them as
galaxies. In the following subsections, we describe each step of this procedure.
Our principal N-body simulation consists of 5123 dark matter particles of a mass
of 2
in a
box. The
initial power spectrum was produced by CMBfast (Seljak & Zaldarriaga 1996). Starting from
a glass distribution, the particles were displaced according to second order
Lagrangian perturbation theory by using the code of Sirko (2005). As
cosmological parameters, we chose
,
,
h=0.7,
,
,
and w=-1.0.
The simulation was performed with GADGET-2 (Springel 2005) using a softening
length of comoving
.
The starting redshift was z=20.
We also completed twelve 2563 dark matter particles simulations with the same cosmological parameters but different realizations of the initial conditions. We used these simulations to investigate systematic effects. Especially, we tested if our fitting procedure provides unbiased results.
An observer at the present epoch t0 (z(t0)=0) identifies the galaxy
distribution on his past light cone. He receives photons emitted at t<t0that have traversed the comoving distance
.
To construct a light-cone survey, we follow an approach used by
Evrard et al. (2002). We identify for each particle two consecutive
snapshots between which it crosses the light cone and interpolate between
them to find the position and velocity of that particle
on the light cone. Expressed in formulas, the interpolated position is given by
where
is the position
of the particle in snapshot i and
,
and
is determined by requiring that
with
being the time step between the two snapshots. After a Taylor
expansion for the last term, we can solve for
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For the galaxy sample at redshift z=3, we place the center of the simulation
box at redshift z=3, i.e.
comoving distance away from the virtual observer.
The orientation of the box is such that the line connecting the center of the
box with the observer is parallel to the z-axis.
The
box then extends from redshift 2.1 to 4.7, i.e. from 5.3 Gpc to 7.6 Gpc in
comoving distances. For this redshift range, we have 17 snapshots at different
times, i.e. of different expansion factors:
.
The
box centered at redshift z=1 ranges from redshift 0.6 to 1.7. For this
interval, we use 27 snapshots with
for the light
cone construction.
Our virtual observer is at redshift z=0 and the simulation box
is centered on redshift z=3 and z=1, respectively. To derive a redshift for
each particle, we compute the comoving distance
to the observer
and solve the following equation numerically for the expansion factor a,
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We next convert the observed redshifts to distances by assuming different reference cosmologies, where the dark energy EOS parameter w has the values
,
-1.0, and -1.2. In this case, the Hubble parameter becomes
.
Since the mass resolution of our simulation is too low for identifying galaxy-sized
friends-of-friends halos, we use a simple bias scheme
to compile mock galaxy samples from the dark matter distribution (Cole et al. 1998; Yoshida et al. 2001).
Using the initial density field, we define a probability function by
if
is above the threshold
and zero otherwise. The dimensionless variable
is given
by the density contrast normalized by its root-mean-square on the
grid
,
i.e.
.
The density contrast
of each particle is computed in the following way. First, we assign the
particles with the cloud in cell (CIC) scheme to a 5123 grid
(Hockney & Eastwood 1988) to obtain the density
on the grid points. Then, we
calculate the density contrast
on the grid points and interpolate it to the positions of the particles.
In the next step, we Poisson sample the dark matter particles according to the probability function and track these ``galaxies'' throughout the snapshots and the constructed light cones.
As parameters, we use (
,
)
for a strongly
biased sample and (
,
)
for a more weakly biased
sample. We calculate the corresponding galaxy bias to be the square root of
the ratio of the galaxy power spectrum to the dark matter power spectrum
in real space:
.
The bias of the mock galaxies decreases with decreasing redshift (Fig. 2)
as expected in the model. We select primarily particles in high-density
regions of the initial density field, which occupy less
prominent structures at later times due to the further development of gravitational
clustering. The bias has a mild scale dependence due to the schematic
procedure of grid-based density estimation and the specific probability
selection function.
The parameters
and
were chosen such that
the strong bias sample at redshift z=3 is consistent with the expected bias
of the target galaxies of HETDEX (Hill et al. 2004). The bias of this sample is very similar to the
friends-of-friends halo sample with a halo mass higher than
4
obtained from the Mare Nostrum
simulation (Gottlöber et al. 2006). As observations of the target galaxies of
HETDEX, namely Lyman-
emitting galaxies (LAE), suggest
(Gawiser et al. 2007; Gronwall et al. 2007), this expected value for the bias might be too
optimistic. Therefore, we also use the weaker bias sample at z=3,
which matches the measured bias of LAE observed by the MUSYC
collaboration (Gawiser et al. 2007; Gronwall et al. 2007).
For z=1, we use the ``weaker'' bias sample, which is fairly consistent with the expected bias of the target galaxies of WFMOS (the KAOS Purple Book 2003; Bassett et al. 2005). The bias of the strongly biased sample at z=1 is too high for these galaxies and we do not use this sample in the rest of the paper.
For the survey centered on z=3, we generate several mock catalogs of one million galaxies from the strongly and weakly biased light cones in redshift space using the entire
box, which has a volume of
.
These numbers are at the upper limit of the current baseline of the HETDEX project.
For the mock catalogs at z=1, we use only the galaxies on the light cone
around z=1 in redshift space with a bias of
.
As a number
of tracers, we choose one and two million galaxies in the entire box.
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Figure 2: Galaxy bias for the two different parameter sets at redshift z=3 and z=1. To construct this plot, we used 15 million galaxies to reduce the shot noise. |
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We analyze the mock catalogs first by calculating the power spectrum and then by fitting the extracted BAO. In the following two subsections, we describe how we calculate the power spectrum and provide details about the fitting procedure.
Since we use an FFT to complete the Fourier transformation, we first have to assign
the particles to a regular grid. We consider a 10243 regular grid and
select the CIC scheme to complete the mass assignment. After converting the density field
to the density contrast
and
performing the FFT to evaluate the Fourier transform (FT)
,
we
compute the raw isotropic (one-dimensional) and anisotropic (two-dimensional)
power spectrum by averaging
over spherical shells
and rings
,
respectively. These raw
power spectra are related to the true power spectrum in the following way (Jing 2005):
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We estimate the error in the power spectrum by counting the number of independent modes used in the calculation of
,
which corresponds approximately to
and
(Feldman et al. 1994)
for the anisotropic and isotropic power spectrum, respectively. Here,
denotes the thickness of the rings and shells we averaged over.
For the anisotropic power spectrum, there is additionally the parameter
which is the range in the values of the cosine of the angle between the wave vector and the line
of sight. The shot noise is denoted by
.
All the power
spectra in this paper have independent bins with a bin width of
.
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Figure 3:
Power spectrum of the dark matter distribution at z=3
(lower curves) and z=1 (upper curves). The dashed lines are the linearly
evolved initial power spectrum. In the case of redshift space,
the linear power spectrum was multiplied in addition by
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Figure 4:
Fractional difference of the light-cone power spectrum
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Figure 5:
Fractional difference of the galaxy light-cone power
spectrum
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In Fig. 3, we show the isotropic dark matter power spectra at
redshift z=3 and z=1 obtained by assuming the correct cosmology with
.
If not stated otherwise, we always use as the reference cosmology the cosmology of
the simulation, i.e. we assume that
.
The linear redshift distortion
(Kaiser 1987) amplifies the power by
with
,
where f(z) is the linear growth rate and b(z)
the bias parameter. In the case of dark matter we have b=1.
For scales shown, the nonlinear evolution is still mild. The
deviations in the real-space power spectra from the linearly evolved initial
power spectra (plotted as dashed lines) start around
and
for z=3 and z=1, respectively. The
nonlinear redshift distortions (``finger of God'' effects) in redshift space
lead to a suppression of power on small scales, which almost balances
the increase in power due to nonlinear clustering at z=1 (filled squares in Fig. 3). The BAO can be seen as tiny wiggles in
the power spectrum. Their amplitudes are
of P(k).
We do not show the light-cone power spectra, since they lie almost exactly
on the corresponding snapshot power spectra. The fractional differences in
the power spectra derived from the light cone and the corresponding snapshot
are instead plotted in Fig. 4. On linear scales, the fractional
differences are
and
at z=3 and z=1, respectively.
The reason for these differences is that the value of the growth function at
the mean redshift is not equal to the mean growth function.
The stated numbers can be understood by averaging the square of the
redshift-dependent growth function multiplied by an appropriate geometrical
factor over the survey box (Yamamoto et al. 1999).
Numerical calculations for our survey designs are shown as dashed (real space)
and solid (redshift space) lines. The deviations at larger k are due
to nonlinear effects. The question of whether these light-cone effects alter
the fitting of the BAO is addressed in Sect. 5.
In Fig. 5, we compare the biased galaxy power spectra derived from the light cones with those evaluated for the corresponding snapshots. Qualitatively, we observe the same behavior as in the dark matter case. The larger scatter is due to higher shot noise and the means are shifted slightly, since, as happened before for the growth function, the bias at the mean redshift is not equal to the bias averaged over the appropriate redshift range.
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Figure 6: The left upper panel shows the oscillatory part of the power spectrum at redshift z=3 obtained from real space and its best-fit function including the suppression factor (dashed line). The dotted line corresponds to the oscillatory part of the initial power spectrum. The solid line is the best-fit of the reconstructed data. At the bottom left, we present the same in redshift space. On the right the corresponding plots for z=1 are shown. |
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Our fitting method attempts to remove all the information
apart from the BAO themselves. This is easily achieved by dividing the power
spectrum by a smoothed version of itself. The advantage of this is that one does
not need to model all the physical processes that alter the power spectrum
such as redshift distortion, nonlinear evolution, and galaxy bias or uncertainties
in cosmological parameters, such as ,
,
and massive neutrinos,
which affect the overall shape of the power spectrum but have little effect
on the BAO.
The extracted BAO of the reference power spectra were then fitted to the
extracted BAO, allowing the scaling factors
or
to vary.
In an additional fitting attempt, our free parameter is instead the EOS parameter w and we derive the scaling factors
or
by applying Eq. (5).
The fitting parameters are determined by Monte Carlo Markov chain (MCMC) techniques. We use the Metropolis-Hastings algorithm (Metropolis et al. 1953; Hastings 1970) to build up the Markov chain.
In the following, we describe each step of the fitting procedure.
There are previous proposals to use only the oscillatory part of the power
spectrum for measuring the scale of the BAO
(Glazebrook & Blake 2005; Hütsi 2006a; Koehler et al. 2007; Hütsi 2005; Angulo et al. 2008; Blake & Glazebrook 2003; Percival et al. 2007b).
All of them divide the power spectrum by (or subtract from it) a non-oscillating
fit. The methods used to derive the smooth fit range from deriving a semi-analytic
zero-baryon reference power spectrum, fitting with a quadratic polynomial
in log-log space, or using a cubic spline, to fitting with a non-oscillating
phenomenological function. In this paper, we generate a smoothed power spectrum in an
almost parameter-free way, by computing for each point the arithmetic mean
in log space of its neighbors in a range of
in k. In the
two-dimensional case, we smooth radially in the direction of |k|. This
smoothing length is the only parameter in our smoothing method, which, for all input power spectra, we select to have the same value of approximately equal to the wavelength of the BAO.
By dividing the measured power spectrum by its smoothed version, we derive the
purely oscillatory part of the power spectrum
(see Fig. 6).
In place of fitting the extracted BAO by a (modified) sine function
(Glazebrook & Blake 2005; Blake & Glazebrook 2003) or using a periodogram (Hütsi 2005)
to measure the scale of the BAO, we compare the BAO with a range of different oscillatory reference
power spectra
produced from thousands of linear
power spectra generated with CMBfast, which differ in the cosmological
parameters
,
,
,
and
.
We prefer this method since the BAO are not exactly harmonic (Koehler et al. 2007; Eisenstein & Hu 1998) and the uncertainties in the aforementioned cosmological parameters can easily be included. As priors on these cosmological parameters we use the predicted uncertainties for the Planck mission (The Planck Bluebook 2005). For the Hubble constant H0, we combine the Planck priors with constraints provided by large-scale structure (Percival et al. 2007a) and include measurements from the HST Key project (Freedman et al. 2001):
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We determine the scaling factors
or
by fitting a scaled
or
to
.
Since nonlinear structure growth diminishes the amplitudes of the wiggles
(Eisenstein et al. 2007b), we can improve the fitting by adding a suppression factor
to our fitting function
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If the density field is known accurately, i.e. the shot noise is small and the
galaxy bias is known, it is possible to use reconstruction techniques
(see e.g. Narayanan & Croft 1999, for a comparison of methods) to undo, at least
in part, the suppression of the wiggles (Eisenstein et al. 2007a). We applied the PIZA
method (Croft & Gaztanaga 1997) to the dark matter distributions. We attempted a similar
technique for the mock catalogs but due to shot noise and galaxy
bias our results were unsatisfactory. Further efforts are required to achieve
this goal. In Fig. 6, we indicate the BAO extracted from the data by
crosses. The dashed lines show the best-fit functions and the dotted lines depict
the corresponding non-damped reference
.
We observe that the suppression of the BAO is higher in redshift space and
increases with time. Additionally, the results of the reconstructed density
fields are plotted (data as filled circles and the best-fits as solid lines).
The amplitude of the BAO could not be reestablished completely but to a
significant fraction.
The k range over which we use the data for our fitting plays an important role.
In the fitting procedure we use a minimum wave number
,
since for the points with
the smoothing
is not well defined, and the error due to sample variance is high. As the
maximum wave number, we select
,
for redshift z=1 (3).
Although nonlinear evolution has already started on these scales, the BAO
are still clearly visible (see Fig. 6).
The most important measurements in our analysis are the marginalized probability distribution functions (PDF) of the scaling factors and the EOS parameter w. To determine their values, we marginalize the joint PDF produced via the MCMC technique over all other fitting parameters. From these functions, we can derive the best-fit values and the accuracy of both the scaling factors and the EOS w.
With the twelve 2563 simulations we assess the robustness of our fitting
method and the presence of systematic effects. We determine the best-fitting
scaling factors, both parallel and transverse to the line of sight, for the dark matter power
spectrum in real and redshift space for all twelve simulations at redshift
z=10, 3, and 1. For all redshifts, we find that the mean value
of the twelve simulations is clearly less than
(standard deviation)
away from unity. Hence, our fitting method provides unbiased results within the
margins of error (see Table 1).
Table 1:
Results for the scaling factors
and
obtained from the dozen 2563 simulations.
The differences between results for real and redshift space in each simulation
do not show a systematic trend. The mean difference is in fact consistent with zero. The
error in the scale factor parallel to the line of sight
is
larger in redshift space, since the damping of the BAO is stronger in redshift
space (see Fig. 6).
To understand the degeneracies in the scaling factor for the
cosmological parameters included in our method, we complete the fitting
by keeping all cosmological parameters fixed to the value of the simulation
apart from the parameter under consideration, for which we calculate
the PDF of
for different values of this parameter.
The corresponding
confidence lines for the dark matter power
spectrum at z=3 (solid) and at z=1 (dotted) are shown in
Fig. 7
.
Since the sound horizon imprinted in the power spectrum is redshift
independent, we expect that the derived degeneracies are also
redshift independent. This is indeed the case for our fitting method.
The only difference with respect to z is the larger uncertainty in
at lower redshifts due to the suppression of the
BAO by nonlinear evolution.
The interval of cosmological parameters plotted in Fig. 7
was chosen to be centered on the values of the simulation and in the range of
.
In this way, we can immediately assess the degree of degeneracy.
We find that if we alter one of the cosmological parameters by
the scaling factor changes by
for
,
for
,
for H0, and
for
.
These numbers agree well with those given by the fitting
formula for the sound horizon by Eisenstein & Hu (1998):
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Figure 7:
Dependence of the fitted
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More interesting are the dependences of the equation of state w on the
cosmological parameters, in particular the dependence on
and H0,
which enter the relation between the scaling factors and w. For a constant w model (w=w0), we indicate the derived correlations in Fig. 8, where
the solid line corresponds to z=3 and the dotted line to z=1. In contrast
to the previous case, these correlations are redshift dependent, for example
at z=1 the dependence of both the sound horizon and the Hubble parameter H(z) on
almost exactly cancel each other, whereas at z=3 the
effect originating in the Hubble parameter H(z) is stronger.
Among the cosmological parameters the largest uncertainty in w0 originates in
H0, in particular at lower redshifts.
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Figure 8: Same as in Fig. 7 but for w0. |
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We have already mentioned that the power spectra derived from the light cone output
and corresponding snapshot at the mean redshift are almost
on top of each other. The more evolved parts of the light-cone sample
compensate almost precisely the less evolved parts, such that its power spectrum
is almost identical to that at the mean redshift (see Fig. 4).
In Fig. 9 we show the corresponding PDFs of the scaling factors. The lines show the
and
confidence levels obtained
from the dark matter snapshot (dashed) and light-cone (dotted) power spectra
at redshift z=3 (left) and z=1 (right) in real (top) and redshift space
(bottom). The differences in the error ellipses derived from the light-cone and snapshot
data are overall small. Hence, light-cone effects in this survey volume and at these redshifts are unimportant to our fitting.
The error ellipses for the reconstructed samples are shown in Fig. 9 as solid lines. We observe that for data at z=1 in redshift space the reconstruction of the BAO shrinks the error ellipses by a factor of three. For this reason, it would be desirable to develop a reconstruction method that can be applied to noisy and biased density fields. For surveys at redshift z=3, this effect is less important.
The errors in the scaling factors originate in two different sources. One source is
the errors in the power spectrum; the other source is the
uncertainties in the cosmological parameters
and
,
which produce an uncertainty in the sound horizon, i.e. the scale of the BAO.
If the principal error could be attributed to the uncertainty in the physical
sound horizon, the orientation of the ellipse would be approximately in the
direction of the line defined by
.
For a low signal-to-noise
ratio power spectrum for which the scale of the BAO in the power spectrum
cannot be determined accurately, the orientation of the error ellipse is
instead in the direction of the line of
.
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Figure 9: The marginalized probability distribution functions (PDF) of the scaling factors at redshifts z=3 ( left) and z=1 ( right) in real space ( top) and redshift space ( bottom) derived from the light cone (dotted), snapshot (dashed), and ``reconstructed'' light cone (solid). |
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We assess if the approximations to compensate for an incorrect reference
cosmology i.e. the scaling relations given in Eq. (4), are sufficiently
accurate for our simulated surveys. We calculate the distances
assuming three reference cosmologies that differ only in terms of
and compute the corresponding power spectra.
As an example, the oscillatory parts of the dark matter power spectra at z=3
in real space are shown in Fig. 10. We observe
that the scale of the oscillations is compressed and stretched for
and
,
respectively.
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Figure 10:
The oscillatory part of the real space power spectra at redshift z=3 derived for three
different cosmologies and its best-fit function. For display purposes, the
power spectrum for
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We scale the one-dimensional power spectra according to the scaling
relation, i.e.
,
where the factor
is due to the scaled volume element.
The fractional difference of these scaled power spectra with respect
to that of the correct cosmology, i.e. w=-1, is shown in Fig. 11.
The data points for w=-1.2 (squares) and w=-0.8 (plus signs) are shifted
by
and
,
respectively, to center them on the zero line.
These small shifts have a similar origin as in the light-cone versus snapshot
comparison; we calculate the scaling factor at the mean redshift, which is
not equal to the mean scaling factor. The noise in the scaled power
spectra is substantially smaller on large scales than the error, which is
already implicit in the power spectrum due to cosmic variance (dashed line)
and additional shot noise (solid line). To determine if this
(additional) noise impairs the fitting of the BAO and
with it the measurement of the equation of state w, we fit the corresponding data of the
twelve 2563 simulations by assuming a constant w. We find that the best-fit
value for w0 varies among the different reference cosmologies for a
single simulation by not more than
,
and is typically
.
This
should be compared with the standard deviation for a single w0 measurement,
which is about
.
All PDFs scatter about the mean value of
w0=-1, no systematic effects are noticeable. We conclude that the scaling
relations do not introduce a noticeable bias or enlarge the error in w.
For real data, a consistency check would be
to recalculate the distances and corresponding power spectrum and
redo the analysis using the measured w as the reference
value for the assumed cosmology.
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Figure 11: The fractional difference of the scaled real-space power spectra at redshift z=3 is shown for the three different reference cosmologies. Note that the data points for w=-1.2 and w=-0.8 are shifted vertically by +0.0025 and -0.0040, respectively, to enhance the visibility of the scatter. The lines show the intrinsic error in the power spectrum due to cosmic variance (dashed line) plus shot noise (solid line). |
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The two-dimensional power spectrum provides the possibility to measure
and
,
instead of the combination
only, as in the case of the one-dimensional power spectrum.
Since both scaling factors depend on w in different ways, we expect that wis more accurately constrained by the two-dimensional
power spectrum. In the case of a simple constant w, this is not the case:
both the mean values and the errors are similar to those derived from the
one-dimensional power spectra. For more complex models of w, the measurement of
and
is very helpful as we see from
Fig. 12, where we applied the model
w=w0+(1-a) wa.
For display purposes, we performed a coordinate transformation of the
parameters to the ``pivot'' system
where we chose
as discussed below.
The left (right) panel indicates the constraints obtained from the dark
matter light-cone power spectrum around z=3 (z=1) in real space. One sees
that for z=3 the contour lines are open towards negative wa even for the
two-dimensional case. Nevertheless, in both redshift cases the use of the
anisotropic power spectrum tightens the constraints substantially.
For a prediction of how well upcoming observations will enable w to be measured, we analyze the mock galaxy catalogs introduced above, namely the weakly and strongly biased one million galaxy catalogs at redshift z=3 and the one and two million galaxy catalogs at z=1. In Fig. 13, we present typical fitting results of the scaling factors for each type of catalogs. For the solid lines, we used the aforementioned priors on the cosmological parameters, whereas for the dashed lines we fixed these parameters.
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Figure 12:
This plot shows the ![]() ![]() |
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Figure 13:
Typical contour plots of the joint PDF of
![]() ![]() |
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Since the error in the scaling factors is dominated by the errors in the power spectrum, improving the accuracy of the cosmological parameters to higher than that of the assumed priors does not significantly reduce the error in the scaling factors. However, it tightens the constraints on w (see Fig. 14 for constant w): the uncertainty in H0, in particular, degrades the measurement of w.
For the strongly biased sample at z=3 of one million galaxies in
a volume of
,
the mean
uncertainties in the Hubble
parameter (
and angular diameter
distance (
)
are
and
(
c.l.), respectively. This corresponds to an error of
for a constant w. By keeping the cosmological parameters fixed, the
uncertainty in w is lowered to
.
The corresponding numbers for the two million galaxy mock catalog at z=1
are
and
for the Hubble parameter and angular diameter
distance, respectively. In this case, we derive an accuracy of
and
for a constant w with Planck priors and fixed cosmology, respectively.
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Figure 14: PDFs for constant w derived from the one million strongly biased galaxy sample at z=3 ( left) and from the two million galaxy at z=1 ( right). |
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To translate the results for the scaling factors into constraints
of the redshift-dependent w-model
w=w0+(1-a)wa, we combine the data
of the strongly biased one million galaxy catalog at z=3 and two
million galaxy catalog at z=1 with 192 SN Ia observations
(Astier et al. 2006; Wood-Vasey et al. 2007; Davis et al. 2007; Riess et al. 2007). In addition we include constraints from
CMB measurements by holding the ratio of the distance to the last
scattering surface and the sound horizon fixed:
.
The confidence contours for w0 and wa are shown in the upper panels of Fig. 15. In the lower panels, the confidence contours are presented in the pivot variable
,
where we choose
such that the error ``ellipse'' from the combined datasets (solid line) is least tilted; we find
.
One sees that the intersection angle
of the BAO contours with respect to the error ellipse derived from SN data
varies with redshift (see also Fig. 16).
For z=3, the orientation is very similar to the CMB contours,
whereas for z=1, it is more aligned with the SN ellipse. Measurements at both
redshifts would therefore constrain the parameters of this model significantly
more than two (or twice a good) measurements at a single redshift.
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Figure 15:
The ![]() ![]() ![]() |
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In Fig. 16, we show the constraints on w derived alone from BAO measurements at two different redshifts, namely z=1 (dashed lines) and z=3 (dotted lines), and the combination of both measurements (solid lines). The redshift-dependent shape and orientation of the contour lines is clearly evident in the pivot system (right panel).
For all confidence contours in Figs. 15 and 16, the above stated priors for the cosmological parameters
,
,
,
and H0 were used.
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Figure 16:
The ![]() ![]() ![]() |
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We have simulated large redshift surveys by ``observing'' mock galaxies on a light
cone obtained from an N-body simulation with 5123 dark matter
particles in a
box. By fitting the apparent
scale of the BAO in the power spectrum, we have derived constraints on the
EOS parameter w.
Our fitting method uses only the oscillatory part of the power spectrum and is therefore insensitive to changes in the power spectrum due to nonlinear evolution, redshift distortions, and scale-dependent galaxy bias. Fitting methods that use the overall shape of the power spectrum suffer in general from these complicated physical effects (unless they are accurately modeled) and tend to provide biased results (Smith et al. 2007).
The drawback of fitting only the oscillatory part is that it is insensitive to information provided by the overall shape of the power spectrum. Determining the growth function would be interesting in particular for dark energy constraints (Sapone & Amendola 2007; Amendola et al. 2005).
It is unclear how to extract the BAO from the power spectrum in the most appropriate and accurate way. Our simple smoothing method works fairly well. It is essentially parameter-free and produces unbiased and robust results.
We analyzed the degeneracies in the EOS w with other cosmological parameters. Given the expected accuracies in the cosmological parameters involved, we found that more accurate measurements of H0 would have the largest effect on lowering the uncertainties in w, especially for observations at lower redshifts.
Comparing the results of the light-cone power spectrum with those of the power spectrum of the snapshot at the corresponding redshift, we did not find evidence for substantial differences. For the surveys under consideration, light cone effects therefore do not play a role in determining w with our fitting method. This is in addition true, when we include redshift-dependent galaxy bias. Fitting methods, sensitive to the overall shape of the power spectrum, in particular to the growth function, must include explicitly the light-cone effects to produce unbiased results.
The PIZA technique (Croft & Gaztanaga 1997), which is used to reconstruct the linear density field and thereby the amplitude of the BAO, is effective if the density field is known to sufficient accuracy. In the case of substantial shot noise and unknown galaxy bias, more sophisticated reconstruction techniques are required.
To investigate the cosmological distortion due to an incorrect reference
cosmology, we adopted three different reference cosmologies with different values for
,
-1.0, and -1.2. Employing the usual scaling relations
(see Eq. (5)), we found in all three cases similar fitting
results within the margins of error. We conclude that even for this large survey
volume the simple scaling of the power spectrum is fairly accurate.
Nevertheless, for real data we propose to use an iterative scheme;
after measuring w for an assumed reference cosmology, one would then repeat
the analysis for the updated reference cosmology, using the measured value of w for
.
By fitting the two-dimensional power spectrum, we can determine independently both scaling factors. For a constant w, the two-dimensional fitting does not improve the constraints on w significantly. For models of w that are not so tightly constrained, independent measurements of the Hubble parameter (
and the angular diameter
distance (
)
will however be very helpful. For the model
w=w0+(1-a)wa, this is especially true at lower redshifts.
Our estimates for future surveys show that BAO measurements around redshifts
z=3 and z=1 in combination with SN and CMB data tighten the constraints on
dynamical dark energy substantially and that these redshift surveys deliver
complementary results. For a constant w-model the BAO measurements from our
two mock catalogs at z=3 and z=1 alone will constrain w0 to an accuracy of
and
,
respectively.
To achieve more realistic predictions, the survey geometry, i.e. the window function, has to be taken into account. A more realistic galaxy bias scheme, tailored in particular for the target galaxies, should also be developed. For such a study, large-volume simulations of far higher mass resolution are needed.
Acknowledgements
We are grateful to Jochen Weller for advice concerning the Monte Carlo Markov chain sampling method and dark energy models. We thank Karl Gebhardt and Eiichiro Komatsu for helpful discussion and useful comments on the draft. We acknowledge useful and constructive remarks by the anonymous referee.