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Year: 2009
How to evade the sample variance limit on measurements of
redshift-space distortions
McDonald, P; Seljak, U
McDonald, P; Seljak, U (2009). How to evade the sample variance limit on measurements of redshift-space
distortions. Journal of Cosmology and Astroparticle Physics, 10:007 .
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Journal of Cosmology and Astroparticle Physics 2009, 10:007 .
McDonald, P; Seljak, U (2009). How to evade the sample variance limit on measurements of redshift-space
distortions. Journal of Cosmology and Astroparticle Physics, 10:007 .
Postprint available at:
http://www.zora.uzh.ch
Posted at the Zurich Open Repository and Archive, University of Zurich.
http://www.zora.uzh.ch
Originally published at:
Journal of Cosmology and Astroparticle Physics 2009, 10:007 .
How to evade the sample variance limit on measurements of
redshift-space distortions
Abstract
We show how to use multiple tracers of large-scale density with different biases to measure the
redshift-space distortion parameter β ≡ b−1f ≡ b−1d ln D/d ln a (where D is the growth factor and a the
expansion factor), to, as the signal-to-noise (S/N) of a survey increases, much better precision than one
could achieve with a single tracer (to arbitrary precision in the low noise limit). In combination with the
power spectrum of the tracers this would allow a more precise measurement of the bias-free velocity
divergence power spectrum, f2Pm, with the ultimate, zero noise limit, being that f2Pm can be measured
as well as would be possible if velocity divergence was observed directly, with maximum rms
improvement factor ~ [5.2(β2+2β+2)/β2]1/2 (e.g., simeq 10 times better than a single tracer with β =
0.4). This would allow a determination of fD as a function of redshift with an error as low as ~ 0.1%
(again, in the idealized case of the zero noise limit). The ratio b2/b1 can be determined with an even
greater precision than β, potentially producing, when measured as a function of scale, an exquisitely
sensitive probe of the onset of non-linear bias. We also extend in more detail previous work on the use
of the same technique to measure non-Gaussianity. Currently planned redshift surveys are typically
designed with S/N ~ 1 on scales of interest, which severely limits the usefulness of our method. Our
results suggest that there are potentially large gains to be achieved from technological or theoretical
developments that allow higher S/N, or, in the long term, surveys that simply observe a higher number
density of galaxies.
arXiv:0810.0323v1 [astro-ph] 2 Oct 2008
How to measure redshift-space distortions without sample variance
Patrick McDonald
∗
Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON M5S 3H8, Canada
Uroˇs Seljak
Physics and Astronomy Department and Lawrence Berkeley National Laboratory,
University of California, Berkeley, California 94720, USA and
Institute for Theoretical Physics, University of Zurich, Switzerland
(Dated: October 2, 2008)
We show how to use multiple tracers of large-scale density with different biases to measure the
redshift-space distortion parameter β ≡ b
−1
f ≡ b
−1
d ln D/d ln a (where D is the growth rate and a
the expansion factor), to a much better precision than one could achieve with a single tracer, to an
arbitrary precision in the low noise limit. In combination with the power spectrum of the t racers
this allows a much more precise measurement of the bias-free velocity divergence power spectrum,
f
2
P
m
– in fact, in the low noise limit f
2
P
m
can be measured as well as would be possible if velocity
divergence was observed directly, with rms improvement factor ∼
ˆ
5.2
`
β
2
+ 2β + 2
´
/β
2
˜
1/2
(e.g.,
≃ 10 times better than a single tracer for β = 0.4). This would allow a high precision determination
of fD as a function of redshift with an error as low as 0.1%. We find up to two orders of magnitude
improvement in Figure of Merit for the Dark Energy equation of state relative to S tage II, a factor
of several better than other proposed St age IV Dark Energy surveys. The ratio b
2
/b
1
will be
determined with an even greater precision than β, producing, when measured as a function of
scale, an exquisitely sensitive probe of the onset of non-linear bias. We also extend in more detail
previous work on the use of the same technique to measure non-Gaussianity. Currently planned
redshift surveys are typically designed with signal to noise of unity on scales of interest, and are
not optimized for this technique. Our results suggest that this strategy may need to be revisited as
there are large gains to be achieved from surveys with higher number densities of galaxies.
PACS numbers: 98.65.Dx, 95.35.+d, 98.80.Es, 98.80.-k
I. INTRODUCTION
Growth of structure in the universe has long been recognized as one of the most powerful ways to learn about the
nature of dark energy and other properties of our universe. Currently the most promising method is weak lensing
tomography, which traces the dark matter directly and can measure the growth of structure by splitting the source
galaxies by their (photometric) redshift. For example, the Dark Energy Task Force concludes that of all proposed
next generation ex periments weak lensing holds the best promise to succeed in improving our knowledge of dar k
energy [1]. However, weak lensing is not necessarily the ideal method: it measures the structure projected over a
broad window in redshift and its ability to probe rapid changes in growth rate is limited. Moreover, one is measuring
a 2-dimensional projection only as opposed to the full 3-dimensional information, drastically reducing the amount of
available information. In addition, for all of the methods proposed so far the requirements on systematic error exceed
what is achievable today, so it is worth pursuing multiple methods until we have a better co ntrol of systematics.
Galaxy clustering has been the favorite method of measuring large scale str uctur e in the universe and it is likely
this will continue also in the future. The main reaso n is that galaxies are easily observed and that by measuring
their redshift one can reconstruct the 3-dimensional clustering information, in contrast to weak lensing or cosmic
microwave background anisotropies which only measure a 2-dimensional projection. The r e lation betwee n g alaxy and
dark matter clustering is however not straight-forward. In the simplest model of linear bias g alaxies tra ce dark matter
up to an overall constant called linear bias . The bias cannot be predicted from the theory and as a result ga laxy
clustering alone cannot measure the growth of structure with reds hift. I f one could determine bias with sufficiently
small error then galaxy clustering would be come the le ading method due to its higher informa tion co ntent.
There are several methods pro po sed to determine the bias. One is to use weak lensing, specifica lly galaxy-galaxy
lensing which measures the cross-corre lation betwee n galaxies and dark matter. This is proportional to bias b a nd in
combination with the galaxy auto-correlation function which sc ales as b
2
one can eliminate the dependence on bia s
∗
Electronic address: pmcdonal@cita.utoronto.ca
2
to measure Ω
2
m0
P
m
(k), where Ω
m0
is the matter density pa rameter today and P
m
(k) is the matter power spe c trum
at a given redshift [2, 3]. Alternatively, one can also measure the halo mass dis tribution with weak lensing which,
in connection with the theoretical bias predictions, can determine the bias and thus P
m
(k) [4]. A second method to
determine the bias a nd thus P
m
(k) is to measure the three-point function [5]. A third method, and the one we foc us
here, is to meas ure the redshift space distortion parameter β ≡ b
−1
f ≡ b
−1
d ln D/d ln a (where b is the bias of the
galaxies, D is the linear theory growth factor, and a is the expansion factor) [6, 7, 8, 9]. In combination with the galaxy
power spectrum this gives f
2
P
m
. Note that these methods give somewhat different dynamical measurements once
the bias is eliminated, so to some extent they are complementary to each other. However, none o f these methods is
presently competitive in terms of derived cosmological constraints, as they all have rather large statistical errors from
the current data, although this may change in the future as data improve and new analysis methods are developed
[10, 11, 12, 13].
In this paper we focus on redshift spa c e distortion parameter β as a way to determine the bias. Clustering of
galaxies along the line of sight is enhanced relative to the transverse direction due to peculiar motions and this allows
one to determine β. Current methods require one to compare the clustering strength as a function of angle relative
to the line of sight, but this method is only applicable on large scales where linear theory holds. As a result, the
sampling variance limits its statistica l precis ion. Recently, [14] proposed a new method for measuring the primordial
non-Gaussianity parameter f
NL
, by comparing two sets of galaxies with a different bias, and a different sensitivity to
f
NL
[15], which allows one to eliminate the sample var iance. Optimization of this technique was investigated by [16].
Here we apply this technique to the measurement of the redshift-space distortion parameter, which in turn improves
the measurement of the velocity divergence power spectrum P
θθ
≡ f
2
P
m
(k). As we show here, this approach can
in principle measure β perfectly and P
θθ
as well as if we observed velocity divergence directly. If this promise were
realized fro m the data it would allow for a much higher statistical power than weak lensing o r other 2-dimensional
projections. We also show that our method, and large-scale structure surveys in general, become even more powe rful
when additional cosmology dependence, such as the Alcock-Paczy´nski effect [17], is included.
We begin by presenting the basic method, followed by an a nalysis of the expected improvement as a function of
survey parameters. This is fo llowed by predictions fo r some of the existing and future surveys in terms of expected
improvement of dark energy parameters. On small scales the deterministic linear bias model eventually b e c omes
inaccurate and for this reason we often quote results as a function of the maximum usable wavenumber, k
max
. While
we have some idea wha t its value should be, we leave a more detailed analysis with numerical simulations for the
future. One should note, however, that the multiple-tracer appr oach is likely to actually help in disentangling non-
linear bias effects on quasi-linear sc ales, making those s c ales potentially much more useful than they would otherwise
be. We conclude with a summary and a discus sion of future directions.
II. METHOD AND RESULTS
The density perturbation, δ
gi
, for a type of galaxy i, in the linear reg ime, in redshift space, is [18]
δ
gi
=
b
i
+ fµ
2
δ + ǫ
i
(1)
where b
i
is the galaxy bias, µ = k
k
/k, δ is the mass density perturbation, and ǫ
i
is a white noise variable which
can represent e ither the standard shot-noise or other stochasticity. All equations are understo od to apply to the real
or imaginary part of a single Fourier mode unless otherwise indicated. In this paper we will consider two types of
galaxies, type 1 with bias b, and type 2 with bias αb. The perturbation equations can then be written
δ
g1
= f
β
−1
+ µ
2
δ + ǫ
1
, (2)
and
δ
g2
= f
αβ
−1
+ µ
2
δ + ǫ
2
. (3)
We are denoting β = f/b (the equivalent distortion parameter for the 2nd type of galaxy is β/α).
The traditional method to determine β is to look a t the angular dependence of the two-point correlation function
or its Fourier transform, the power spec trum. The correlations will be enhanced along the line of sight (where µ = 1)
relative to the direction perpendicular to it (wher e µ = 0), but to observe this enhancement one must average over
many independent modes to beat down the sampling (or cosmic) var iance. This is because each mo de is a random
realization of a Gaussian field and there will be fluctuations in the mea sured power even in the a bsence of noise. By
combining a measurement of β with that of the galaxy power spectrum one can determine f
2
P
m
, which no longer
depends on the unknown bias of the galaxies. This method has bee n applied to the data, most recently in [6, 7, 8, 9],
3
and is limited by the accuracy with which we can determine β. For example, for the analysis in [8], which currently
has the highest signal to noise measurement of β, the error on the overall amplitude of galaxy power spectrum P
gg
is about 1% adding up all the modes up to k = 0.1 h Mpc
−1
, while the erro r on β is about 12%, so the error on
reconstructed f
2
P
m
= β
2
P
gg
is entirely dominated by the error on β. Predictions for the future surveys with this a nd
related methods can be found in [12, 13].
To understand the main point of the new method we are proposing here let us consider the situation w ithout the
noise, which would apply if, for example, we have a very high density of the two tracers sampling the field, and no
stochasticity. In that case we can divide equation 3 by equation 2 above to obtain
δ
g2
δ
g1
=
αβ
−1
+ µ
2
β
−1
+ µ
2
. (4)
This expressio n has a sp e cific angular dependence, allowing one to extract α and β separately. Note that there is no
dependence on the density field δ. The random nature of the density field is therefore not affecting this method and
we can determine β exa ctly in the absence of noise. More generally, the pr e c ision with which we can determine β is
controlled by the (shot) nois e , i.e. density of tracers, rather than the sampling variance . In order to address the gains
in a realistic case we must perform the full analysis, which we turn to next.
Generally, the noise variables can be correla ted with e ach other, although they would not be for standard shot-noise.
For example, [19] showed that one generically expects non-linear structure forma tion to generate a full covariance
matrix for the noise at some level, so this will probably be the ultimate limit for this kind of measurement. The
covariance matrix of the perturbations is
C ≡
δ
2
g1
hδ
g1
δ
g2
i
hδ
g2
δ
g1
i
δ
2
g2
=
P
θθ
2
"
β
−1
+ µ
2
2
β
−1
+ µ
2
αβ
−1
+ µ
2
β
−1
+ µ
2
αβ
−1
+ µ
2
αβ
−1
+ µ
2
2
#
+
N
2
(5)
where P
θθ
≡ 2f
2
δ
2
and N
ij
≡ 2 hǫ
i
ǫ
j
i (no te that P
θθ
and N are the usual power spectrum and noise – the factor
of 2 comes from the fact that δ and ǫ are only the real or imaginary part of a Fourier mode). If we assume that
the noise matrix is known then the covariance matrix for two types of galaxies is a function of three parameters:
the velocity divergence power spectrum amplitude, P
θθ
, the redshift-space distortion parameter, β, and the ratio of
biases α. We work with α instead of a second β parameter b e c ause the ratio of biases will generally be deter mined
substantially more precisely than β, which means that the mea surement of the second β par ameter would be almost
perfectly correlated with the first. The Fisher matrix fo r the measurement of these parameters is
F
λλ
′
=
1
2
Tr
C
,λ
C
−1
C
,λ
′
C
−1
(6)
where C
,λ
≡ dC/dλ and λ are the parameters.
For any single mode, the inverse of the Fisher matrix is singular, i.e., we can only constrain two parameters, not
three. However, adding Fisher matric e s for modes with different µ breaks this dege neracy. Generally, the total Fisher
matrix will be an integral over modes with all ang le s. As a simple example, we assume first a pair of modes with
µ = 0 and µ = 1 and compute the error on the parameters in the small noise limit:
σ
2
α
α
2
= X
11
− 2X
12
+ X
22
, (7)
where X
ij
= N
ij
/b
i
b
j
P
m
,
σ
2
β
β
2
=
h
α
2
(1 + β)
2
+ (α + β)
2
i
X
11
− 2
h
α
2
(1 + β)
2
+ α (1 + β) (α + β)
i
X
12
+ 2α
2
(1 + β)
2
X
22
β
2
(α − 1)
2
, (8)
and
σ
2
P
θθ
P
θθ
2
= 1 . (9)
The key points are that the only lower limit on the errors on α and β from this single pair of modes is set by the
achievable noise-to-signal ratios on the tracers, X
ij
, and the error on P
θθ
is the error one would obtain for a simple
power spectrum measurement from two modes, with no degradation due to degeneracy with the bias parameters (this
is of course only true to leading order in the small noise limit).
![FIG. 5: Improvements in constraints on various parameters, vs. zmax, for the case including BAO, kmax(z) = 0.1 [D (z) /D (0)]−1 hMpc−1, and S/N = 10 for the Pθθ measurement. Lines are: black (solid): w0, green (long-dashed): w′, red (dot-short-dashed): Ωk, blue (dashed): ΩΛ, magenta (dot-long-dashed): ωm, cyan (dotted): logA.](/figures/fig-5-improvements-in-constraints-on-various-parameters-vs-2uoxr891.webp)
![FIG. 1: Projected fractional error on the normalization of Pθθ ≡ f 2Pm, for 30000 square degrees, in redshift bins with width dz = 0.2. The upper and lower green (short-dashed) lines show the constraints from single tracers with b = 2 and b = 1, respectively. Black (solid) lines show the two tracers together, each with, from top to bottom, S/N=1, 3, 10, 30, 100 at k = 0.4 hMpc−1. The red (long-dashed) line shows the case where both tracers are perfectly sampled. For the left panel we assume kmax(z) = 0.1 [D (z) /D (0)] −1 hMpc−1, while the right assumes kmax(z) = 0.05 [D (z) /D (0)] −1 hMpc−1.](/figures/fig-1-projected-fractional-error-on-the-normalization-of-2smo7ffl.webp)
![FIG. 11: Improvement in DETF FoM when all of the information in the redshift-space power spectrum is used rather than just BAO and f2Pm constraints, for kmax(z) = 0.1 [D (z) /D (0)] −1 hMpc−1 (relative to Planck+Stage II, and for 30000 sq. deg., as usual). Green (dashed) lines show single S/N=1 (at k = 0.4 hMpc−1) tracers with b = 1 (upper) and b = 2 (lower). The black curves show both types of galaxy together, each with (bottom to top) S/N=1, 3, 10, 30, 100, 1000 (the constraining power increases without bound as the S/N improves). Blue (dotted) shows BAO alone, with the lower curve using the above kmax and the upper curve using smaller scales as described in [22] (the rest of the curves effectively use the weaker version). The red (long-dashed) curves include weak lensing constraints, with the lower two (horizontal) lines including no redshift survey (lowest is approximately as SNAP weak lensing, 2nd lowest is LSST weak lensing), while the upper two include the b = 2-only (weakest) redshift survey.](/figures/fig-11-improvement-in-detf-fom-when-all-of-the-information-2tsgskp1.webp)
