Unified galaxy power spectrum measurements from 6dFGS, BOSS

Prepared for submission to JCAP

Uniﬁed galaxy power spectrum

measurements from 6dFGS, BOSS,

and eBOSS

Florian Beutler,

Patrick McDonald

Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edin-

burgh EH9 3HJ, UK

Lawrence Berkeley National Lab, 1 Cyclotron Rd, Berkeley CA 94720, USA

E-mail: ﬂorian.b[email protected], p[email protected]

Abstract. We make use of recent developments in the analysis of galaxy redshift surveys to

present an easy to use matrix-based analysis framework for the galaxy power spectrum multi-

poles, including wide-angle eﬀects and the survey window function. We employ this framework

to derive the deconvolved power spectrum multipoles of 6dFGS DR3, BOSS DR12 and the

eBOSS DR16 quasar sample. As an alternative to the standard analysis, the deconvolved

power spectrum multipoles can be used to perform a data analysis agnostic of survey speciﬁc

aspects, like the window function. We show that in the case of the BOSS dataset, the Baryon

Acoustic Oscillation (BAO) analysis using the deconvolved power spectra results in the same

likelihood as the standard analysis. To facilitate the analysis based on both the convolved

and deconvolved power spectrum measurements, we provide the window function matrices,

wide-angle matrices, covariance matrices and the power spectrum multipole measurements for

the datasets mentioned above. Together with this paper we publish a Python-based toolbox

to calculate the diﬀerent analysis components. The appendix contains a detailed user guide

with examples for how a cosmological analysis of these datasets could be implemented. We

hope that our work makes the analysis of galaxy survey datasets more accessible to the wider

cosmology community.

arXiv:2106.06324v2 [astro-ph.CO] 27 Oct 2021

Contents

1 Introduction 1

2 The survey window function 3

2.1 Including wide-angle eﬀects 4

2.2 Window function convolution as matrix multiplication 7

2.3 Bin averaging the window function 9

2.3.1 The observational binning ∆k

2.3.2 The theoretical binning ∆k

3 Wide-angle eﬀects as matrix multiplication 11

4 Deconvolution as matrix multiplication 12

4.1 Wide-angle compression 15

5 Datasets 16

5.1 6dFGS DR3 16

5.2 BOSS DR12 17

5.3 eBOSS DR16 QSO 18

6 Data analysis 19

6.1 Setup of power spectrum measurements 19

6.2 Deconvolution 19

6.3 Deconvolution and BAO 21

7 Conclusion 23

A User guide 30

B Window function pre-factors 33

C Derivation of the 2D window function 34

C.1 Excluding wide-angle terms and consistency with [1] 34

C.2 Including wide-angle terms and consistency with [2] 35

D Analytic calculation of the odd power spectrum multipoles 36

E Deconvolution results for all datasets 37

1 Introduction

In the last two decades the analysis of galaxy redshift survey datasets has become one of

the most powerful tools to constrain cosmological models [3, 4]. The next generation of

galaxy redshift surveys, such as the Dark Energy Spectroscopic Instrument (DESI [5]) and

the Euclid space mission [6], aim to observe 20 to 40 million galaxy redshifts, increasing the

largest current dataset (BOSS [7], 1 million galaxies) by more than an order of magnitude.

While observations of the Cosmic Microwave Background (CMB) have reached the sample

– 1 –

variance limit for some observables, galaxy surveys are far away from this limit. Extracting

cosmological information from galaxy surveys does pose signiﬁcant challenges. Most modes

which can be accessed by galaxy surveys are contaminated by non-linear contributions along

with redshift-space distortions and a complicated relation between galaxy density and matter

density. Many diﬀerent avenues are currently investigated to tackle these challenges (e.g. [8–

11]). Here we will ignore these issues, but focus on the analysis formalism itself.

Extracting cosmological information from the galaxy power spectrum comes with techni-

cal complications, such as wide-angle eﬀects (e.g., [2, 12, 13]) and the survey window function,

which convolves the measured power spectrum. While the survey window is usually known,

this convolution can add signiﬁcant modeling challenges. The current standard analysis de-

rives the survey window function from the data and convolves any power spectrum model with

the survey window before comparing it with the power spectrum measurement [1, 14, 15]. In-

stead of convolving the model we can also deconvolve the measured power spectrum. Past

attempts of deconvolution were done on a mode by mode basis [16] or assumed the global

plane parallel approximation (often called distant observer approximation [17–19]). Here we

present a deconvolution formalism, which can be applied to a wide-angle multipole represen-

tation of the power spectrum. Recently, [20] advocated deconvolution in the context of going

back to the quadratic estimator formalism originally derived in [21, 22]. For these estimators

one can naturally quote results convolved with a window, or deconvolved, or something in

between [23, 24]. Note that there is a technical diﬀerence between that formalism, which

assumes the k bands in which the quadratic averaging is performed are the same as the k

bands used to model the theory power, so that the window matrix is automatically square

and invertible, and the formalism in this and other recent papers, in which the theory bands

can have arbitrarily ﬁne k resolution, independent of the observational averaging band width.

First we express the power spectrum analysis as two matrix multiplications, one matrix

accounting for wide-angle eﬀects and a second matrix accounting for the window function.

We derive these matrices for some of the largest galaxy redshift surveys currently available

(6dFGS DR3, BOSS DR12 and the eBOSS DR16 QSOs sample) and make these matrices

available together with the power spectrum measurements

. The appendix of this paper

provides a step-by-step guide for a galaxy survey analysis, including Python-based examples.

The aim is to make galaxy redshift survey datasets more easily accessible for the wider

cosmology community.

As an example application we perform a BAO analysis on the deconvolved power spec-

trum of BOSS DR12. We show that with the products derived in this paper, the likelihood

derived from the convolved and deconvolved power spectrum multipoles is identical.

This paper is organized as follows. We start with a review of the current formulation

of the survey window function, which we turn into a matrix multiplication in section 2.

In section 3 we derive a similar matrix accounting for wide-angle eﬀects. In section 4 we

deriving our deconvolution procedure. In section 5 we introduce the 6dFGS, BOSS and

eBOSS datasets followed by an application of our matrix based deconvolution procedure in

section 6. We conclude in section 7. Appendix A provides a detailed user guide with Python-

based examples. In appendix B we discuss the correlation matrices needed for the window

function calculations. We also show consistency between the equations used in this paper with

the equations in [1] and [2] in appendix C. In appendix D we derive analytic equations for

wide-angle eﬀects. Finally, appendix E contains a summary of the convolved and deconvolved

https://fbeutler.github.io/hub/deconv_paper.html

– 2 –

power spectrum measurements for the diﬀerent datasets.

Throughout the paper we use a (ﬁducial) ﬂat ΛCDM cosmology with Ω

= 0.31 when

transferring observables (RA, DEC, z) into cartesian coordinates (x, y, z). When analysing

mock datasets we use the cosmological parameters of the underlying simulations, which are

diﬀerent for each of the galaxy samples studied here (see section 5). The Python code used

to calculate the wide-angle and window function matrices discussed in this paper is available

at https://github.com/fbeutler/pk_tools.

2 The survey window function

Most studies involving the galaxy power spectrum include the survey window function by

convolving the model for the power spectrum before comparing it to the measurement. Any

asymmetry in the survey window distributes power between the multipoles. This can bias

measurements of anisotropic observables like redshift-space distortions, if not taken into ac-

count correctly. The convolution of the power spectrum multipoles with the window function

multipoles has been laid out in [1] and is given by

conv

(k) =

(k, k

true

) −

(k)

(0)

(0, k

true

) , (2.1)

where W

(k, k

) describes the contribution of multipole `

to multipole ` due to the survey

window

. The second term on the right hand side describes the integral constraint correc-

tion [25] and includes the 1D survey window function multipoles in Fourier space Q

(k). While

the treatment of the integral constraint in eq. (2.1) is a good approximation, [26] showed that

there are additional contributions depending on how the reference (random) catalog has been

generated.

While [1] calculated W

(k, k

) through pair counting (see eq. 33 of that reference), re-

cently it was pointed out that this quantity can be obtained using a double Bessel integral [27]

(k, k

) =

(−i)

ds s

(ks)j

(s) , (2.2)

where the matrices C

are given in section 3.1 of [27]. The 1D window function multipoles

in conﬁguration-space are deﬁned as

(s) =

(2L + 1)

dΩ

4π

¯n

)¯n

(s + s

(

s ·

d) , (2.3)

with ¯n

(x) = ¯n(x)w(x), where ¯n(x) is the local mean density of galaxies (i.e., expected

number after selection eﬀects), and w(x) represents a weight applied to the density measured

at x (completeness and signal to noise weight). The normalization factor A here is the same

number that appears in the power spectrum estimate, e.g., eq. (3) of [15]. However, we do

not use the standard value of A given in eq. (13) of [15]. Instead we set A to the value

necessary to enforce Q

(s → 0) ≡ 1. The same A value is then also used to normalize the

power spectrum estimate, as required for consistency (as recently emphasized by [26]). We

use this deﬁnition because we ﬁnd it gives much closer to the desired unit normalized window,

i.e., window where integration over theory k gives 1. This is desirable because it makes the

Note that [1] uses a slightly diﬀerent nomenclature with |W (k, k

≡ W

(k, k

)

– 3 –

convolved power close to the true power, i.e., just smeared slightly by the window instead of

adding an oﬀset in the overall normalization. Note that, at the continuum limit of eq. (2.3),

our normalization convention can be written as

A =

dx ¯n

(x) . (2.4)

However, the traditional deﬁnition as represented by eq. (13) of [15] is also intended to

represent this equation! In the traditional calculation the integral over position and the ¯n(x)

factors are represented by a sum over randoms times ¯n(z), where ¯n(z) is supposed to be a

redshift dependent average of ¯n(x). This is an approximation that in practice only achieves

the goal of Q

(s → 0) = 1 to ∼ 10%. We guarantee Q

(s → 0) = 1 by computing Q

(s)

ﬁrst and using it directly to ﬁx A. For comparison with past results we give the relative

normalization between the two methods in Table 1. To be clear, if the normalisation between

the power spectrum and window function are consistent, it will not impact any likelihood

analysis, so it is essentially cosmetic.

In the following section we will demonstrate how our window function formalism can be

extended to include wide-angle eﬀects before moving on to develop a matrix based convolution

and deconvolution procedure.

2.1 Including wide-angle eﬀects

Any measurement of the anisotropic power spectrum has to make a choice regarding the

line-of-sight (LOS) direction for a galaxy pair (see, e.g., [2, 12, 13]). Based on this choice the

triangle conﬁguration between the observer and the galaxy pair is reduced to a separation

amplitude s and a multipole dependent weighting given by the Legendre polynomials L

This geometric simpliﬁcation introduces wide-angle eﬀects in the power spectrum multipoles

` > 0 as well as the associated window function multipoles. Crucially, wide-angle eﬀects can

break the symmetry between the galaxy pair and hence can introduce odd multipoles, like

the dipole and octopole. These wide-angle eﬀects can be absorbed into the window function

formalism described above. Based on [2] we can extend eq. (2.2) to get

(n)

(k, k

) =

(−i)

ds s

(ks)j

(n)

(s) , (2.5)

where the index n describes the order in the wide-angle expansion and the C

(n)

matrices are

given in appendix B. Eq. (2.5) includes new window function multipoles Q

(n)

at each order

of the wide-angle expansion n given by

(n)

(s) =

(2` + 1)

dΩ

4π

)

−n

W (s

)W (s + s

(

s ·

d) . (2.6)

A derivation of eq. (2.5) is included in Appendix C.2. The constraints imposed by FFTs

usually enforce the end-point LOS deﬁnition

d =

where the line of sight is chosen along

the distance vector of one of the galaxies (see [2] for details).

Eq. (2.5) can be calculated using one 1D Fourier transforms (FT) for each k bin with

the complexity O(N

× N

log N

), where N

and N

are the number of bins in k and

, respectively. Additionally we need to calculate the Q

(k) using a 3D FT, which has the

complexity O(N log N), where N is the number of grid cells in which the random galaxies are

binned (see appendix E.1 of [2]). We then obtain Q

(s) through Hankel transforms. In this

– 4 –

Figure 1: The window function multipoles of the low redshift bin 0.2 < z < 0.5 of BOSS

DR12, NGC, calculated following eq. (2.5) at zero order in the wide-angle expansion (n = 0).

The corresponding window function multipoles at ﬁrst order in the wide-angle expansion

(n = 1) are shown in ﬁgure 2. The three columns correspond to the contributions of the three

even multipoles to the ﬁve non-zero multipoles, including the dipole and octopole. Each

window function is plotted multiple times as a function of k

with ﬁxed values of k (vertical

dashed lines). This window function does not account for any bin averaging, but is evaluated

at speciﬁc values k with 16 384 values of k

(here we focus on 0 < k

< 0.12 h Mpc

−1

, while

in practice we calculate the full range of 0 < k

< 0.4 h Mpc

−1

). For visual purposes, each

of the sub-panels is re-normalized by the factor written in the right hand corner. The ﬁnal

window function matrix used for the actual analysis requires bin averaging in both k

and

(see eq. 2.16 for more details).

paper we will use eq. (2.5) to calculate the window function using N

= 16 384 and N

= 400

(from 0 < k < 0.4 h Mpc

−1

in bins of ∆k = 0.001 h Mpc

−1

), which typically takes < 1 minute

on a single state of the art laptop.

Figure 1 shows the window function multipoles W

(n)

(k, k

) for the low redshift bin of

BOSS DR12 in the North Galactic Cap (NGC). This ﬁgure only shows the window function

multipoles at zero order in the wide-angle expansion (n = 0 in eq. 2.5). At this order,

only even multipoles `

are generated, meaning we have a contribution from the monopole

(left column), quadrupole (middle column) and hexadecapole (right column). Each window

function is plotted several times with ﬁxed k (dashed lines). These plots agree with the plots

shown in Figure 7 of [1], where these multipoles have ﬁrst been investigated, but they now

include contributions to the odd multipoles.

Figure 2 shows the corresponding window function multipoles at ﬁrst order in the wide-

angle expansion (n = 1 in eq. 2.5), in which case we only have odd contributions from

the dipole (left column) and octopole (right column). Also note that any window function

multipole W

(n)

in which the combination `+`

is a odd number, results in a complex quantity,

– 5 –

Figure 2: The window function multipoles of the low redshift bin 0.2 < z < 0.5 of BOSS

DR12, NGC, calculated following eq. (2.5) at ﬁrst order in the wide-angle expansion (n = 1).

The corresponding window function multipoles at zero order in the wide-angle expansion

(n = 0) are shown in ﬁgure 1. The two columns correspond to the contributions of the two

odd multipoles to the ﬁve non-zero multipoles. Each window function is plotted multiple

times as a function of k

with ﬁxed values of k (vertical dashed lines). For visual purposes,

each of the sub-panels is re-normalized by the factor written in the right hand corner.

to account for the fact that the estimated odd power spectrum multipoles are complex [2, 28].

Based on the discussion above we can extend eq. (2.1) in section 2, which describes

the convolution of the power spectrum multipoles with the window function multipoles by

including wide-angle eﬀects:

conv

(k) =

02−n

(n)

(k, k

(n),true

)

−

(0)

(k)

(0)

02−n

(n)

(0, k

(n),true

) .

(2.7)

We now deﬁne a new multipole expansion including wide-angle eﬀects as

P (k, µ) =

`,n

(kd)

−n

(n)

(k)L

(

k ·

d) , (2.8)

where d is the LOS vector and d = |

d|. Based on this deﬁnition the power spectrum multi-

poles, P

(n)

(k), at diﬀerent order in the wide-angle expansions n are given by [2, 13]

(n)

(k) = 4π(−i)

ds s

(ks)

(n)

(s)j

(ks) , (2.9)

where ξ

(n)

are the conﬁguration space multipoles at wide-angle order n.

– 6 –

For the remainder of this paper we will use the end-point LOS deﬁnition, meaning the

LOS for each galaxy pair is oriented along the distance vector of one of the galaxies (d = s

In this case the zero order term (n = 0) is equivalent to the commonly used power spectrum

multipoles and at this order only the even multipoles are present. At n = 1 only the odd

multipoles are present and can be obtained as linear combinations of the even multipoles in

conﬁguration space

(1)

= −

(0)

(s) , (2.10)

(1)

(0)

(s) −

(0)

(s) . (2.11)

The second order terms (n = 2) are given in eq. (2.16) - (2.18) of [2]. The analysis in

this paper will be limited to n ≤ 1, even though including higher order terms (n > 1) is

straightforward. Note that the dipole and octopole power spectra can be calculated directly

in Fourier space without any need for a Hankel transform (see also eq. 3.56 of [29]):

(1)

(k) = −i

(0)

(k) + k∂

(0)

(k)

, (2.12)

(1)

(k) = −i





(0)

(k) − k∂

(0)

(k)





(0)

(k) + k∂

(0)

(k)





. (2.13)

We included a derivation of these equations in appendix D. Using linear theory we obtain

(1)

(k) = −if





[3P

(k) + k∂

(k)] , (2.14)

(1)

(k) = −i4f







(2P

(k) − k∂

(k)) +

f (5P

(k) + k∂

(k))



. (2.15)

2.2 Window function convolution as matrix multiplication

If we assume that we calculate our power spectrum model in bins of ∆k

and intend to

compare this model with power spectrum measurements in bins of ‘observed’ bandpowers

∆k

, the window function needs to be integrated over k and k

(n)

, k

) =

dk k

Θ(k

, k)

02−n

(n)

(k, k

)Θ(k

, k

) , (2.16)

where the step function Θ is deﬁned as

Θ(k

, k) =

(

1 if |k

− k| <

∆k

0 otherwise.

(2.17)

We will test the bin averaging employed in our analysis in the next subsection. Before we get

to that, let us deﬁne our matrix nomenclature, starting with the vectors

true







(0),true

(k)

(1),true

(k)

(0),true

(k)

(1),true

(k)

(0),true

(k)







, P

conv







(0),conv

(k)

(1),conv

(k)

(0),conv

(k)

(1),conv

(k)

(0),conv

(k)







, Q =







(0)

(k)

(1)

(k)

(0)

(k)

(1)

(k)

(0)

(k)







, (2.18)

– 7 –

Figure 3: The window function matrix (W

, see section 2.2) and wide-angle matrix (M

section 3) of BOSS DR12 NGC in the low redshift bin. The wide-angle transformation matrix

maps the theoretical power spectrum multipoles with three even multipoles to a vector with

ﬁve multipoles (only accounting for wide-angle eﬀects at order n = 1). The window function

maps the ﬁve (theoretical) power spectrum multipoles (in bins of ∆k

) to the observed/con-

volved power spectrum multipoles (in bins of ∆k

), which can be compared to the data. For

plotting purposes here we use ∆k

= ∆k

= 0.01 h Mpc

−1

, while the analysis in this paper

uses ∆k

= 0.01 h Mpc

−1

and ∆k

= 0.001 h Mpc

−1

. The window function matrix is not

symmetric, since the contribution of the monopole to the dipole is diﬀerent to the contribu-

tion of the dipole to the monopole. The diagonal elements are not 1, since they correspond

to the bin integral of the window function within the bin ∆k

(see eq. 2.16), which is only 1

for ∆k  k

as shown in ﬁgure 6. The z-axis is logarithmic for better contrast.

where P

true

is a vector of model power spectrum multipoles and the convolution matrix is

given by

W =







(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)

(1)

(0)







, (2.19)

where each sub-matrix W

(n)

is of size N

×N

. Based on the deﬁnitions above, we can write

the convolution of the power spectrum multipoles with the window function multipoles as a

matrix multiplication

conv

= WP

true

−

(0)

(0, k

true

) , (2.20)

– 8 –

where

(0)

(0, k

) is a matrix representing the integral constraint correction. For the

remainder of this paper we will absorb the integral constraint into W meaning we re-deﬁne

W ≡ W −

(0)

(0, k

) . (2.21)

This works exactly because the integral constraint eﬀect is linear in the power spectrum just

like the standard window function. In this paper the window function matrix W is generally

deﬁned in the k-range 0.0 < k < 0.4 h Mpc

−1

, while the recommended k-range for any ﬁt is

0.01 < k < 0.3 h Mpc

−1

. The increased k-range accounts for the redistribution of power due

to the window function, which connects modes inside the ﬁtting range with modes beyond

that range. The equations above include wide-angle eﬀects at order n = 0 and n = 1, even

though this formalism could easily be extended to n > 1. Figure 3 shows the matrix W after

performing the integral in eq. (2.16) (and including the integral constraint correction).

2.3 Bin averaging the window function

In eq. (2.16) we wrote down how to take the window function (as derived in eq. 2.5 and

plotted in ﬁgure 1 and 2) an05d account for the bin averaging to match the layout of the

data vector. The default choices for our window functions are ∆k

= 0.001 h Mpc

−1

and

∆k

= 0.01 h Mpc

−1

. Here we will demonstrate how we implemented the bin averaging of the

window function and we will test certain assumptions made in our formalism.

2.3.1 The observational binning ∆k

All power spectrum multipole measurements provided with this paper use k-bins of ∆k

0.001 h Mpc

−1

. However, for practical purposes

we use 10 times larger bins of ∆k

0.01 h Mpc

−1

. The measured power spectra can easily be re-binned into larger bins using

P(k

) =

bin j within ∆k

modes

P(k

) , (2.22)

where ∆k

∆k

= 0.001 h Mpc

−1

and N

modes

is the number of observed k modes in bin

j with

modes

bin j within ∆k

modes

. (2.23)

One crucial point to highlight here is that the fundamental mode for all galaxy datasets

included in this paper is smaller than our choice for ∆k

. This means we need to bin average

our theoretical model in accordance with the average of eq. (2.22). For that reason we measure

the window function in bins of ∆k

= 0.001 h Mpc

−1

and average as

W(k

, k

) =

dk k

W(k, k

)Θ(k

, k) (2.24)

≈

bin j within ∆k

modes

W(k

, k

) . (2.25)

Figure 4 shows the window function contributions to the monopole, W

, with ∆k

0.001 h Mpc

−1

(yellow line) and ∆k

= 0.01 h Mpc

−1

(red line) for the low redshift (z1)

sample of BOSS DR12 NGC.

Mainly to reduce the size of the associated covariance matrix.

– 9 –

Figure 4: Comparison of the window function terms W

for diﬀerent binning schemes at

= 0.055 h Mpc

−1

for BOSS DR12 NGC in the low redshift bin (z1). The yellow line shows

the window with bins of ∆k

= ∆k

= 0.001 h Mpc

−1

. The red line shows the window with

larger observational bins of ∆k

= 0.01 h Mpc

−1

, by averaging 10 bins of the yellow line. The

green line is averaged over both k

and k

. The black dashed lines show the edges of the bin

at k

min

= 0.05 h Mpc

−1

and k

max

= 0.06 h Mpc

−1

. The relative heights of the three windows

are scaled to make them comparable. The red line corresponds to our default choice.

2.3.2 The theoretical binning ∆k

The multiplication of the power spectrum model with the window function matrix eﬀectively

calculates a convolution. The question we want to address here is, what would be the required

resolution for this convolution, meaning how sensitive is the convolved power spectrum to

the choice of ∆k

? In ﬁgure 5 we show convolved power spectrum multipoles using our

default binning of ∆k

= 0.01 h Mpc

−1

but diﬀerent ∆k

. The reference model is based on

the current standard method, where the power spectrum model is (1) Hankel transformed

into conﬁguration-space, (2) multiplied by the conﬁguration-space window and (3) Hankel

transformed back into Fourier space [14, 30]. The result is than bin-averaged to ∆k

0.01 h Mpc

−1

. The diﬀerences between our default choice of ∆k

= 0.001 h Mpc

−1

and the

reference model are < 1% of the measurement uncertainties for the monopole on most scales

and even smaller for all other multipoles.

Finally we note that the above binning tests for the power spectrum and window function

are based on a ΛCDM model (with the default cosmology given at the end of the introduction).

Our default binning choices might not be optimal for non-standard analysis that searches for

variations in the power spectrum smaller than the chosen bin width. Examples for such cases

are primordial features [31] or primordial non-Gaussianity [32]. However, our binning choice

– 10 –

Figure 5: Comparison of convolved power spectrum multipole models based on diﬀerent

window function binning choices of ∆k

using BOSS DR12 NGC in the low redshift bin

(z1). The window function transforms the theoretical binning (∆k

) into the binning used

for the data (∆k

). The reference model is based on the current standard method [14, 30],

which Hankel transforms the model into conﬁguration-space, multiplies with the real-space

window and Hankel transforms back into Fourier space. The window function uses ∆k

0.01 h Mpc

−1

in all cases. The window functions provided with this paper uses a default

binning of ∆k

= 0.001 h Mpc

−1

, except of the case of deconvolution, where we use ∆k

0.01 h Mpc

−1

should be perfectly suitable for the standard RSD and BAO analysis.

3 Wide-angle eﬀects as matrix multiplication

The odd power spectrum multipoles sourced by wide-angle eﬀects are linearly related to

the even multipoles through eq. (2.14) and (2.15). This implies that we can deﬁne a linear

transformation between ‘ﬂat-sky’ statistics (no wide-angle eﬀects) and ‘curved-sky’ statistics

(including wide-angle eﬀects) as

true,ﬂat-sky

= P

true

, (3.1)

where P

true,ﬂat-sky

= (P

, P

) is a vector of the predicted even multipoles and M is a

×3N

matrix, which transforms this vector into a new vector P

true

, which contains 5

– 11 –

multipoles, including the dipole and octopole. We can deﬁne

M =







I 0 0

0 K

2→1

0 I 0

0 K

2→3

4→3

0 0 I







, (3.2)

where I is the identity matrix of size N

× N

and

2→1

= −i



Θ(k

, k

) + ∂

Θ(k

, k

)



= −i



Θ(k

, k

) +

Θ(k

, k

− ∆k) −Θ(k

, k

+ ∆k)

2∆k



, (3.3)

2→3

= −i



Θ(k

, k

) −

Θ(k

, k

− ∆k) −Θ(k

, k

+ ∆k)

2∆k



, (3.4)

4→3

= −i



Θ(k

, k

) +

Θ(k

, k

− ∆k) −Θ(k

, k

+ ∆k)

2∆k



, (3.5)

with Θ deﬁned in eq. (2.17). As shown in these equations, we can implement derivatives within

a matrix multiplication by including oﬀ-diagonal terms. Here we use two-sided derivatives

except for the ﬁrst and last bin in the data vector, where we use forward and backwards

derivatives instead. The transformation matrix M does depend on the order of wide-angle

eﬀects n, since at n = 2 there are new even multipoles sourced by wide-angle eﬀects. Following

the rest of this paper, we only include terms at n ≤ 1. The plot on the right in ﬁgure 3 shows

the matrix M for the low redshift bin of BOSS DR12 NGC. The only survey speciﬁc parameter

in this matrix is the amplitude of the LOS vector d. The code to calculate the matrix M is

publicly available

4 Deconvolution as matrix multiplication

Usually the aim of a power spectrum analysis is to obtain the likelihood

L ∝ exp



−

conv

− WP

true

)

−1

conv

− WP

true

)



, (4.1)

where C

−1

conv

is the inverse covariance matrix of the (convolved) power spectrum multipoles,

conv

represents the measured power spectrum and P

true

is the unconvolved power spectrum

model.

Using eq. (4.1) together with the maximum likelihood condition ∂χ

/∂P

true

= 0 im-

plies that the power spectrum before convolution with the window function is related to the

convolved power spectrum by

−1

conv

−1

conv

≡ P

true

, (4.2)

where

−1

conv

−1

conv

= W

−1

(4.3)

https://github.com/fbeutler/pk_tools/blob/master/wide_angle_tools.py

– 12 –

Figure 6: Left: Diagonal of the window function W

(n)

= k

) for the low redshift bin

of BOSS DR12 NGC (solid lines) and SGC (dashed lines). The diﬀerently colored lines

correspond to the diﬀerent multipoles. All lines in the plot on the left use ∆k

= ∆k

0.01 h Mpc

−1

. The SGC has smaller values for W

(n)

= k

) because its fundamental mode

is larger. Right: The window function W

(n)

= k

= 0.1 h Mpc

−1

) as a function of the

size of the bandpower bin width ∆k. The quantity plotted on the y-axis of these two ﬁgures

scales the bandpower uncertainty for the deconvolved power spectrum multipoles as shown

in eq. (4.5).

Figure 7: Left: Correlation matrix for the 5 power spectrum multipoles of BOSS DR12

NGC in the low redshift bin. These matrices are based on the 2048 BOSS DR12 MultiDark-

Patchy (MD-Patchy) mock catalogs with a binning of ∆k

= 0.01 h Mpc

−1

. Right: The

same correlation matrix using the deconvolved power spectrum multipoles. The wide-angle

expansion used here includes all terms with n < 2.

if W is a square matrix and non-singular. Note that the deconvolution process implied in

eq. (4.2) cannot recover the true underlying power spectrum (which can be calculated in

theory), since the window function did erase all information below the fundamental mode.

– 13 –

Figure 8: Slice through the correlation matrices shown in ﬁgure 7 at k

= 0.105 h Mpc

−1

for

the convolved (solid lines) and deconvolved (dashed lines) cases, respectively. We focus on

the diagonal components of the monopole (yellow), dipole (pink) and quadrupole (green).

The covariance matrix of the deconvolved power spectrum is given by

−1

deconv

= W

−1

conv

W . (4.4)

The correlation matrix, R = C

, for BOSS DR12 NGC in the low redshift bin is

plotted in ﬁgure 7 before (left) and after (right) deconvolution. Deconvolution often leads to

anti-correlated bins [33], which can be intuitively explained by imagining a power spectrum

model with very ﬁne k binning (∆k  fundamental mode k

). If one moves the power

spectrum in one bin up, but compensates by moving the power spectrum in the adjacent

bin down, the convolved version of this power spectrum will look very similar to the original

power spectrum, since the window function averages neighbouring bins. Hence such a mode

of variation is not well constrained in a deconvolved estimate, leading to anti-correlation.

Generally however, there is a decrease in the correlation between bandpowers as clearly

visible in ﬁgure 8, which implies an increase in the variance of modes within that bin, given

deconv

(k) ∝

(n)

(k, k)

. (4.5)

Figure 6 shows the behaviour of W

(n)

(k, k) as a function of k for diﬀerent ∆k in the low

redshift bin of BOSS DR12. This clearly shows that deconvolution can signiﬁcantly increase

the uncertainties in the power spectrum multipoles if ∆k is small. However, this increase in

the uncertainties does not reﬂect any loss of information, but only shows that for smaller ∆k

the window function has a larger impact. The larger uncertainty is compensated by having

less correlation (or anti-correlation) between the bandpowers. The information content of

true

and P

conv

in terms of the likelihood is identical, since we derived P

true

in eq. (4.2) using

the likelihood in eq. (4.1) (assuming P

true

is estimated in at least as many bands as P

conv

It should be noted, however, that our deconvolution procedure is based on a Gaussian

likelihood (as given in eq. 4.1), which might not be true on the largest scales of the survey

(see e.g. [34]). This could bias large-scale signals like primordial non-Gaussianity. Alternative

– 14 –

0.00 0.05 0.10 0.15 0.20 0.25 0.30

max

odd-poles [hMpc

−1

]

−1

/ν

BOSS DR12 NGC, z1

BOSS DR12 SGC, z1

BOSS DR12 NGC, z3

BOSS DR12 SGC, z3

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

k [hMpc

−1

]

0.5

0.6

0.7

0.8

0.9

1.0

(k) without odd poles

(k) with odd poles

BOSS NGC z3

` = 0

` = 2

` = 4

Figure 9: Left: The reduced χ

for ﬁts of band-parameterized even poles (P

true,ﬂat-sky

) to all

the measured even poles, and odd poles up to the maximum k on the horizontal axis. Right:

Diagonal covariance for the inferred ﬂat sky even poles (see eq. 4.7), including the constraint

from k < 0.1 h Mpc

−1

odd poles, divided by the same covariance with no odd pole constraint.

to our procedure one could derive the deconvolved bandpowers using an MCMC approach

without the need to assume a Gaussian likelihood or include a Gaussianisation step as e.g.

suggested in [35].

4.1 Wide-angle compression

In eq. (4.2) we can replace W with

W ≡ WM to compress our vector of 5 multipoles to a

vector of 3 even multipoles, i.e.,

true,ﬂat-sky

= (

−1

conv

−1

conv

, (4.6)

while the corresponding covariance matrix is

−1

true,ﬂat-sky

−1

conv

W . (4.7)

This is equivalent to a minimum χ

ﬁt of the band-parameterized model for P

true,ﬂat-sky

to all

multipoles. This is an over-constrained ﬁt with the degrees of freedom equal to the number of

bins in the odd multipoles

. Because the prediction of the odd multipoles from P

true,ﬂat-sky

is model-independent, the reduced χ

(χ

/ν) for this ﬁt is a model-independent test of the

various quantities we report (C, M, etc.). Unfortunately, we ﬁnd that when we try to ﬁt all

multipoles on all scales this way, χ

is unacceptably bad, i.e., the measured points are not

consistent with any model, given our M, W, and C. As shown in ﬁgure 9 (left), this problem

is driven by the high k bins in the odd multipoles.

Figure 9 shows that χ

/ν is near unity if we drop the odd multipoles with k & 0.1 h Mpc

−1

but quickly grows when including higher k. This is not a straightforward thing to understand,

or even see, by looking at the measured points, covariance matrix, etc. It appears to be driven

by a few special linear combinations of even and odd multipoles which have very low variance

in the mocks used to compute the covariance matrix. These are then not well-predicted by

Assuming the theory is band-parameterized using the same set of k bins as the measurement, i.e., the

number of free parameters is equal to the number of bins in the even power spectrum multipoles so that they

cancel in the calculation of the degrees of freedom.

– 15 –

Survey/Sample N

gal

min

max

] z

eﬀ

(50%) k

(95%) N

A/A

old

[Gpc

] [hMpc

−1

] [hMpc

−1

]

6dFGS DR3 75 117 [0.01, 0.2] 0.096 0.12 0.00514 0.00142 600 1.0168

BOSS DR12 NGC z1 429 182 [0.2, 0.5] 0.38 2.6 0.00222 0.00062 2048 0.9032

BOSS DR12 SGC z1 174 820 [0.2, 0.5] 0.38 1.0 0.00310 0.00086 2048 0.8896

BOSS DR12 NGC z3 435 742 [0.5, 0.75] 0.61 2.8 0.00154 0.00042 2048 0.9104

BOSS DR12 SGC z3 158 262 [0.5, 0.75] 0.61 1.0 0.00226 0.00062 2048 0.9016

eBOSS DR16 QSO NGC 218 209 [0.8, 2.2] 1.52 0.35 0.00094 0.00026 1000 0.9302

eBOSS DR16 QSO SGC 125 499 [0.8, 2.2] 1.52 0.18 0.00114 0.00034 1000 0.8974

Table 1: Properties of the diﬀerent galaxy samples studied in this paper. The eﬀective

redshift is deﬁned as z

eﬀ

, where w

includes any completeness weight as

well as the FKP weight [36]. To calculate the FKP weight as well as the eﬀective volume,

we assume P

= 10 000h

−3

Mpc

for BOSS and 6dFGS and P

= 6 000h

−3

Mpc

for eBOSS.

The eﬀective volume in this table might diﬀer from other references due to (1) a diﬀerent

ﬁducial cosmology and (2) only the raw (unweighted) number density of galaxies is used in

our calculation. k

represents the fundamental mode, which is given by the wavenumber at

which the monopole window Q

(0)

(k) has 50% of the height of Q

(0)

(k → 0) (column 6) or 95%

(column 7). N

represents the number of mock realizations available for each sample, and

the last column gives the ratio between our A deﬁnition, enforcing Q

(s → 0) = 1, and the

normalization used in many previous papers (see section 2). For BOSS we also have catalogs

after applying density ﬁeld reconstruction. However, only 1000 of the 2048 BOSS mock

catalogs have been processed in this way

. More information about the diﬀerent samples can

be found in the relevant data release papers of 6dFGS [37], BOSS [38] and eBOSS [39].

the theory as multiplied by M and W. E.g., for BOSS DR12 NGC, z3, k < 0.2 h Mpc

−1

, we

ﬁnd χ

= 1400, of which 1127 is contributed by the single worst eigenvector of the covariance

matrix. We were unable to ﬁnd any ﬂaw in the calculations that would ﬁx this. Since we

expect the wide-angle eﬀects to be primarily important at low-k, our solution is to simply

drop the odd multipoles at k > 0.1 h Mpc

−1

from ﬁts. The right panel of ﬁgure 9 shows

an example of the impact of the lower k odd multipoles on the inferred deconvolved even

multipole variance. As expected the impact of the odd multipoles is primarily at very low k.

5 Datasets

Here we will introduce the three galaxy redshift survey datasets we analyze in this paper,

namely the 6dFGS DR3 sample, the BOSS DR12 sample and the eBOSS DR16 quasar sample.

5.1 6dFGS DR3

The 6-degree Field Galaxy Survey (6dFGS [37]) is a K-band selected, magnitude limited

(K ≤ 12.9) galaxy survey, based on the 2MASS Extended Source Catalog (2MASS XSC; [40]).

6dFGS covers nearly the entire southern sky and is the lowest redshift sample included in this

paper. The survey made use of the Six-Degree Field (6dF) multi-ﬁbre instrument on the UK

We actually have 997 mock catalogs for the NGC in the low and high redshift bins and 1000 mock catalogs

for the SGC in both redshift bins. Some NGC mock catalogs have been excluded due to issues while processing

the catalogs.

– 16 –

Figure 10: Comparison of the monopole window function for 6dFGS DR3, BOSS DR12 and

eBOSS DR16. The dashed lines in the plot on the left indicate the fundamental mode deﬁned

as the 50% height of the monopole. The Figure on the right shows the monopole weighted

by k

, which is the quantity which goes in the window function in eq. (2.16) at n = 0. The

double peaked shape of the eBOSS DR16 SGC window is probably caused by the narrow

band geometry and/or the split of the SGC into two separate regions (see ﬁgure 2 in [39]).

The curves in the plot on the right hand side are normalized so that the maximum of the

curve is at unity (which allows comparison between the diﬀerent surveys).

Schmidt telescope at the Siding Spring Observatory. The three data release papers [37, 41, 42]

describe 6dFGS in full detail, including comparisons between 6dFGS, 2dFGRS and SDSS.

Here we use the ﬁnal K-band selected sample, which contains 75 117 galaxies and has been

used in several galaxy clustering studies [43–48] (see Table 1 for more details).

Mock catalogs: The 6dFGS mock catalogs are based on COLA simulations [49] with 1728

particles in boxes with 1.2h

−1

Gpc on each side. The simulations use 20 time steps down to

z = 0 and a mass resolution of 2.8 ×10

−1



. A friends-of-friends (FoF) ﬁnder is used to

locate halos with a minimum of 32 dark matter particles per halo. The cosmology used in

these simulations is Ω

= 0.3, Ω

= 0.0478, h = 0.68, σ

= 0.82 and n

= 0.96. The derived

halo catalogs are populated with galaxies using an HOD measured on the dataset itself [43]

(for more details see section 3 of [48]).

The 6dFGS sample has by far the smallest volume of all the datasets included in this

analysis (see table 1 and ﬁgure 10). However, it does occupy a unique redshift range, which

is not covered by BOSS or eBOSS.

5.2 BOSS DR12

The Baryon Oscillation Spectroscopic Survey (BOSS) was part of SDSS-III [7, 50] and mea-

sured spectroscopic redshifts of 1 198 006 million galaxies [38]. The survey covers 10 252 deg

– 17 –

divided in two patches on the sky, the North Galactic Cap (NGC) and the South Galactic Cap

(SGC), over a redshift range of 0.2 - 0.75. Here we split this redshift range into two redshift

bins deﬁned by 0.2 < z < 0.5 and 0.5 < z < 0.75 with the eﬀective redshifts z

eﬀ

= 0.38 and

0.61, respectively. We also include the diﬀerent incompleteness weights as

= (w

+ w

− 1)w

sys

, (5.1)

which account for redshift failures (w

), ﬁbre collisions (w

) and photometric systematics

related to the observational seeing conditions and correlations with stellar density (w

sys

) [51,

52].

Mock catalogs: The BOSS collaboration provided mock catalogs for the ﬁnal BOSS DR12

dataset (MD-Patchy mock catalogs [53]). These catalogs have been produced using approxi-

mate gravity solvers and analytical-statistical biasing models calibrated to a reference sample

from the BigMultiDark simulations [54]. The BigMultiDark simulation is based on gadget-

2 [55] with 3840

particles in a volume of (2.5h

−1

Gpc)

assuming a ΛCDM cosmology with

Ω

= 0.307115, Ω

= 0.048206, σ

= 0.8288, n

= 0.9611 and h = 0.6777. The mock

catalogs use halo abundance matching to reproduce the observed BOSS two- and three-point

clustering measurements [56]. This technique is applied as a function of redshift to reproduce

the BOSS DR12 redshift evolution.

We already saw the BOSS DR12 NGC window function W

(n)

of the low redshift bin

(z1) in ﬁgure 1 and 2. We also include the monopole window function, Q

(0)

(k), in ﬁgure 10.

The BOSS DR12 sample represents the largest galaxy redshift dataset (in terms of eﬀective

volume) [3, 15, 30, 52, 57–59].

5.3 eBOSS DR16 QSO

The extended Baryon Oscillation Spectroscopic Survey ([60], eBOSS) is part of SDSS-IV [61]

and relies on the same optical spectrographs [62] as the SDSS-III BOSS survey. In addition to

observing luminous red galaxies (LRGs) and emission line galaxies (ELGs), eBOSS collected

redshifts for ∼ 500 000 quasars. While the Quasar density is comparatively low, this sample

has the distinction of covering the largest cosmic volume, leading to the smallest fundamental

mode of all samples discussed in this paper (see ﬁgure 10). Here we focus on the eBOSS

quasar sample to avoid any overlap with the other samples.

The eBOSS targets [63, 64] are selected from the DR7 [65] and DR8 [66] photometric

catalogs as well as the Wide Field Infrared Survey Explorer (WISE, [67]), as described in [68].

Just like BOSS, the eBOSS quasars are split in two angular regions, the North Galactic Cap

(NGC) and South Galactic Cap (SGC). The eﬀective areas of these regions are 2924 deg

and

1884 deg

, respectively (see Table 1 for more details). Each eBOSS object has a complete-

ness weight including corrections for ﬁbre collisions, redshift failures [69] and photometric

systematic eﬀects [70]:

= w

npz

sys

. (5.2)

We refer to section 5.6 of [71] for details about these weights. The cosmology results for the

DR16 quasar sample were presented in [4, 72]

, which reported a BAO distance measurement

in the range 0.8 < z < 2.2.

https://www.sdss.org/dr16/

– 18 –

Mock datasets: We make use of a set of 1000 mock catalogs to estimate the covari-

ance matrix of the eBOSS DR16 quasar power spectrum. The mocks are based on the

Extended Zel’dovich (EZ) approximate N-body simulation scheme [73]. These mocks rely

on the Zel’dovich approximation to generate a density ﬁeld, while including nonlinear and

halo biasing eﬀects through the use of free parameters. These free parameters are tuned to

produce two-point and three-point clustering of a desired data set. The EZ mock catalogs

account for the redshift evolution of the eBOSS quasars by constructing a light-cone out of

7 redshift shells, generated from periodic boxes of side length L = 5000h

−1

Mpc at diﬀerent

redshifts. These mock catalogs also mimic the ﬁbre collisions and redshift failures, so that

each object has an associated w

foc

and w

. The cosmology of the EZmocks is ΛCDM with

Ω

= 0.307115, Ω

= 0.048206, h = 0.6777, σ

= 0.8255, and n

= 0.9611 (same cosmology

as the BOSS DR12 MD-Patchy mocks discussed above).

Figure 10 shows the window function monopole for the diﬀerent samples where the NGC

sample of eBOSS DR16 indicates the smallest fundamental mode of all samples discussed

in this paper. The fact that the eBOSS quasar sample probes the largest scales currently

accessible with galaxy redshift surveys, makes it a valuable tool to test primordial non-

Gaussianity [32]. However, the high shot noise level does mean that the eﬀective volume of

the eBOSS DR16 QSO sample at most wavenumbers is below BOSS DR12.

6 Data analysis

Here we will apply the deconvolution formalism developed in section 4 to the three datasets

introduced in the previous section.

6.1 Setup of power spectrum measurements

All power spectra discussed in this section are measured using the estimator of [74] and

[75] including the odd multipoles as discussed in appendix E of [2]. For all power spectrum

measurements we use a cubic grid with L = 2000, 3500 h

−1

Mpc and N = 600, 700 for

6dFGS and BOSS, respectively. For the eBOSS QSO sample we use L = 5400, 6600 h

−1

Mpc

with N = 1040, 1270 for the SGC and NGC, respectively. This setup ensures that for all

three surveys the Nyquist frequency is > 0.6 h Mpc

−1

. The galaxies are assigned to the grid

using the triangular shape cloud procedure and we correct for the associated pixel window

function [76]. To further reduce aliasing eﬀects we included the interlacing procedure of [77].

As deﬁned in Section 2, we use a diﬀerent normalization for the power spectra than other

recent papers, which brings the convolved power closer to the true/deconvolved power. For

comparison with past work, Table 1 gives the ratio, of our deﬁnition of the normalisation to

the one that we would compute using eq. (13) of [15].

6.2 Deconvolution

Figure 11 and 12 show a comparison of the convolved and deconvolved power spectrum

multipoles of BOSS DR12 NGC in the low redshift bin. For these plots we combined 10

measured bandpowers of ∆k

= 0.001 h Mpc

−1

into larger bins of ∆k

= 0.01 h Mpc

−1

. Since

deconvolution requires a square matrix in eq. (4.2), we also use ∆k

= 0.01 h Mpc

−1

. In

ﬁgure 11 we can see the increase in the variance caused by deconvolution (comparison of

the red and black shaded regions). At the same time deconvolution reduces the correlation

between bandpowers. The odd multipoles in ﬁgure 12 also include a best ﬁtting model

– 19 –

Figure 11: Comparison of the convolved and deconvolved even power spectrum multipoles

for the BOSS DR12 low redshift bin (z1) in the NGC. The red line and shaded area shows

the mean of the measured MD-Patchy DR12 power spectrum multipoles together with the

standard deviation. The black line and shaded area shows the equivalent power spectra after

deconvolution. The data points show the actual BOSS DR12 data. The lower panel shows

the diﬀerence between the red and black lines in the upper panel, relative to the uncertainties

post-deconvolution, which highlights the impact of the window function.

(green dashed lines) as well as the intrinsic dipole and octopole (blue dashed lines) as given

by eq. (2.14) and eq. (2.15). These models are based on a ﬁt to the even multipoles as

discussed in section 3.6 of [2]. One can clearly see that deconvolving the dipole removes the

window function contributions, which dominate the dipole on most scales and recovers the

intrinsic dipole expected due to wide-angle eﬀects. We included the corresponding results for

6dFGS and eBOSS in appendix E.

Figure 14 in appendix E shows the convolved and deconvolved power spectrum multi-

poles of 6dFGS. From these plots we can see that the monopole power spectrum of the mock

catalogs does not perfectly match the data power spectrum amplitude, while higher order

multipoles agree well [78]. Even though 6dFGS is the lowest redshift sample, the wide-angle

eﬀects seem far less important compared to BOSS DR12, and the dipole is (by eye) consis-

tent with zero. This agrees with [45] where wide-angle eﬀects are discussed in appendix C.

Window function eﬀects also seem to be much smaller than the statistical noise, which might

be caused by the very compact geometry of the 6dFGS survey (see [37] for details about the

angular and redshift distribution of 6dFGS).

Figure 16 in appendix E compares the power spectrum multipoles of the eBOSS DR16

quasar sample with the corresponding mock results before and after deconvolution. While

the dipole and octopole moments are consistent with zero for each bandpower estimate, there

is a clear systematic dipole signal

– 20 –

Figure 12: Comparison of the convolved and deconvolved odd power spectrum multipoles

for the BOSS DR12 low redshift bin (z1) in the NGC. The red line and shaded area shows

the mean of the measured MD-Patchy DR12 power spectrum multipoles together with the

standard deviation. The black line and shaded area shows the equivalent power spectra after

deconvolution. The data points show the actual BOSS DR12 data. The green dashed line

shows the best ﬁt as discussed in section 3.6 of [2]. The dashed blue line shows the intrinsic

odd multipoles introduced by wide-angle eﬀects based on the same model as the green dashed

line. The lower panel shows the diﬀerence between the red and black lines in the upper panel,

relative to the uncertainties post-deconvolution, which highlights the impact of the window

function.

6.3 Deconvolution and BAO

The convolution of the measured power spectrum with the survey window function could

smear out a signal which exists at the same scale as the fundamental mode (or the k

band

width, whichever is larger). The BAO signal has a wavelength of ∼ 0.06 h Mpc

−1

, which is

much larger than the fundamental mode of the surveys discussed in this paper (see table 1).

Nevertheless, deconvolving the power spectrum does increase the BAO signature as

clearly visible in ﬁgure 13. This ﬁgure shows the BAO signal in the low and high redshift

bins of the BOSS DR12 MD-Patchy mock catalogs together with the best ﬁtting models. The

data points represent the mean and variance of 1000 post-reconstruction MD-Patchy mock

catalogs.

It is important to note that even though the BAO signal appears to be enhanced post-

deconvolution, our deconvolution procedure is based on eq. (4.1) and hence the convolved and

Even though the Nyquist frequency is k

= 0.6 h Mpc

−1

, the eBOSS power spectrum dipole below

k < 0.3 h Mpc

−1

does seem to be aﬀected by aliasing, causing the rise in the dipole at high k. This should be

taken into account when analyzing the eBOSS dataset.

– 21 –

Figure 13: Comparison of the convolved and deconvolved BAO signal in the power spectrum

of the low redshift bin (left) and high redshift bin (right) of the BOSS DR12 MD-Patchy mock

catalogs (we are plotting the mean of the 1000 mock catalogs together with the variance).

The red data points correspond to the convolved power spectrum with the best ﬁtting model

shown as the solid red line. The deconvolved measurements are shown by the blue data points

together with the best ﬁtting model (blue solid line). The data points are the weighted mean

of the NGC and SGC. The errorbars correspond to the square root of the diagonal terms of

the covariance matrix.

deconvolved analysis should lead to the same likelihood and the same model parameters, if the

same cuts and theory binning are used. To demonstrate this point we perform an isotropic

BAO analysis based on the mean of 1000 BOSS DR12 MD-Patchy mock power spectrum

monopoles (post-reconstruction). For the pre-deconvolution ﬁt we build the monopole model

following the current standard analysis pipeline described in [30]. We also provide higher

order multipoles for the quadrupole and hexadecapole based on the simple Kaiser model.

We use the window function and wide-angle matrices (W and M) and limit the likelihood

evaluation to the monopole only (see the second example in appendix A). Our ﬁtting range

is 0.01 < k < 0.3 h Mpc

−1

and we are jointly ﬁtting the NGC and SGC. For the post-

deconvolution ﬁt we only ﬁt the monopole without any window function or wide-angle matrix,

again using the model of [30]. Our ﬁtting procedure is using the Python-based MCMC sampler

zeus [79, 80]

In the low redshift bin of BOSS DR12 we ﬁnd α = 0.999 ± 0.013 before deconvolution

and an identical value after deconvolution. Figure 13 (left) compares the best ﬁtting model

and the mean of the MD-Patchy mocks (the plot shows the weighted average of the NGC

and SGC). The equivalent values for the high redshift bin shown in ﬁgure 13 (right) are

α = 1.003 ± 0.012 before deconvolution and α = 1.002 ± 0.012 after deconvolution. Any

observed diﬀerences are consistent with noise in the MCMC chain, the slightly diﬀerent cuts

implied by using the monopole within 0.01 < k < 0.3 h Mpc

−1

in convolved vs. deconvolved

space, and coarsening of the theory side of the window function used in the deconvolution.

The window matrix used for deconvolution has to be a square matrix with relatively large

k-bins (∆k = 0.01 h Mpc

−1

in our case). Such large k-bins cannot capture small-scale features

in the theory power spectrum (see ﬁgure 4). While these eﬀects should not introduce any

issues within a smooth ΛCDM power spectrum (as demonstrated here for the BAO case), it

https://zeus-mcmc.readthedocs.io/en/latest/

– 22 –

could become relevant for non-ΛCDM models especially if small scale features are present.

7 Conclusion

When analysing galaxy redshift surveys in Fourier space, one needs to account for the survey

window function as well as wide-angle eﬀects. In this paper we leverage recent new develop-

ments dealing with the survey window function and wide-angle eﬀects, to lay out a simple

power spectrum analysis framework based on matrix multiplications. The main results of this

paper are:

(1) We derive a matrix to account for wide-angle eﬀects in the power spectrum multipoles.

We use a new analytic approach rather than the commonly used Hankel transforms.

(2) We expand the window function matrix approach presented in [27] by including wide-

angle eﬀects.

(3) We use this matrix-based analysis framework for the power spectrum multipoles to

demonstrate two possible analysis pipelines, one using the standard path of convolving

the model vector and one based on the deconvolution of the data vector.

(4) We apply the deconvolution procedure to a set of existing galaxy redshift surveys,

namely 6dFGS DR3, BOSS DR12 and eBOSS DR16. Using a BAO analysis we demon-

strate that our deconvolution analysis framework leads to the same likelihood as the

standard analysis.

(5) We provide the power spectrum multipoles as well as the window function matrices,

wide-angle matrices and covariance matrices for 6dFGS DR3, BOSS DR12 and eBOSS

DR16. In the appendix we also provide Python-based examples and a general user guide

for a clustering analysis. These easy to use components hopefully simplify the analysis

of these datasets and make them more accessible for the wider cosmology community.

The deconvolution framework outlined in this paper does not suﬀer from limitations inher-

ent to other methods presented in the literature, such as the assumption of a global plane

parallel approximation. Nevertheless, the inversion of the window function does require a

square window matrix, which enforces large ∆k

bins, a limitation which is not present when

convolving the model vector.

Our analysis focuses on the key science targets of galaxy redshift surveys such as RSD

and BAO. Other observables such as primordial non-Gaussianity through the scale-dependent

bias in the power spectrum, naturally requires to focus on the largest scales of the survey.

The products provided with this paper are sub-optimal for such an observable. However, the

formalism presented in this paper can easily be adapted to suit such an observable.

A similar matrix-based analysis approach could also be developed for higher order statis-

tics, like the bispectrum. The extension of our analysis framework to higher order statistics

will be addressed in future work.

Acknowledgments

The authors would like to thank Antonio Cuesta for help with the reconstructed BOSS cat-

alogs and Richard Neveux, Arnaud De-Mattia and Hector Gil-Marin for helpful discussions

– 23 –

regarding the window function normalisation. This project has received funding from the

European Research Council (ERC) under the European Union’s Horizon 2020 research and

innovation programme (grant agreement 853291). FB is a Royal Society University Research

Fellow. PM was supported by the U.S. Department of Energy, Oﬃce of Science, Oﬃce of

High Energy Physics, under Contract no. DE-AC02-05CH11231.

This work has beneﬁted from a variety of Python packages including numpy [81], scipy [82],

matplotlib [83], zeus [79, 80] and hankl [84].

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algorithm for cosmology, Arxiv:2106.06331v1.

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– 29 –

A User guide

Together with this publication we provide:

1. The window function W, in the form of 5N

× 5N

= 200 × 2000 and 5N

× 5N

200 × 200) matrices.

2. The wide-angle transformation matrix M (5N

ﬂat-sky

× 3N

= 2000 × 1200).

3. The covariance matrix C

conv

derived from the mock catalogs (5N

×5N

= 200 ×200).

4. The power spectrum multipole measurements for the mock catalogs and data in ∆k

0.001 h Mpc

−1

(note that to use them with the products above you need to re-bin to

∆k

= 0.01 h Mpc

−1

, see combine_bins parameter in pk_tools.read_power()). For

BOSS DR12 we also provide the post-reconstruction power spectrum measurements.

All quantities assume 5 multipoles in the k-range 0 < k < 0.4 h Mpc

−1

in bins of ∆k

0.01 h Mpc

−1

and ∆k

= 0.001 h Mpc

−1

. The only exception is the square window function

(200 ×200) meant to be used for deconvolution, which assumes ∆k

= ∆k

= 0.01 h Mpc

−1

All products listed above are available at https://fbeutler.github.io/hub/deconv_paper.

html and Python-based modules to read these quantities are available at https://github.

com/fbeutler/pk_tools.

We note that due to the Nyquist frequency in the measured power spectra, any analysis

should be limited to k

max

< 0.3 h Mpc

−1

. The purpose for the addition of the k-range

0.3 < k < 0.4 h Mpc

−1

is mainly to allow a sensible window function contribution from outside

the ﬁtting range. We generally advice to exclude the ﬁrst k-bin (k

min

> 0.01 h Mpc

−1

), which

could suﬀer from large scale systematics not studied in detail in this analysis.

Here is an example of a likelihood analysis using Python:

1 import numpy as np

2 import pk_tools # see above

3 # Read data power spectrum (Delta k_o =0.01 h/ Mpc ) as dictionary

4 pk_data_dict = pk_tools . read_power(pkfile , combine_bins =10)

5 # Turn dictionary into vector as [P_0 ,P_1 , P_2 ,P_3 ,P_4]

6 kbins , pk_data_vector = pk_tools . dict_to_vec ( pk_data_dict )

7 # Read covariance matrix

8 C = pk_tools . read_matrix ( covfile )

9 Cinv = np. linalg.inv (C)

10 # Read window matrix ( section 2, eq .~2.14)

11 W = pk_tools . read_matrix ( Wfile )

12 # Read expansion matrix ( section 3, eq .~3.2)

13 M = pk_tools . read_matrix ( Mfile )

14 # Get your favourite power spectrum model as dictionary

15 pk_model_dict = your_favourite_model ()

16 # Turn dictionary into vector as [P_0 ,P_2 , P_4 ]

17 kbins_long , pk_model_vector = pk_tools . dict_to_vec ( pk_model_dict , use_ell =[0 , 2 ,4])

18 # Expand pk model (true ,flat−sky) −> ( true) see eq .~3.1

19 expanded_model = np. matmul (M, pk_model_vector )

20 # Convolve with window (true ) −> (conv ) see eq .~2.18

21 convolved_model = np. matmul(W , expanded_model )

22 # Calculate chi2

23 diff = pk_data_vector − convolved_model

24 chi2 = np. dot (diff ,np. dot (Cinv ,diff ))

– 30 –

Note that you can speed up these calculations by multiplying the matrices W and M be-

forehand, which reduces the required matrix multiplications in the likelihood evaluation to

one.

How can I limit the k-range? The power spectrum model should always have a k-range

of 0 < k

< 0.4 h Mpc

−1

in bins of ∆k

= 0.001 h Mpc

−1

. If your model cannot predict

the power spectrum up to k

max

= 0.4 h Mpc

−1

, you should provide some “sensible” estimate,

so that the window function contributions from those scales can be included. The likelihood

analysis should be limit to k

max

< 0.3 h Mpc

−1

, since the power spectra outside this k-range

suﬀer from aliasing given that k

≈ 0.6 h Mpc

−1

for all measured power spectra. One can

specify the k-range as

1 kmin = 0.01

2 kmax = 0.3

3 # Assuming 40 bins in k_o with Delta k_o =0.01 h/ Mpc

4 krange = np . linspace (0.005 , 0.395 , num =40)

5 fit_selection = np. logical_and ( kmin <krange ,krange < kmax)

6 # Select fitting−range for data power spectrum and inverse C

7 fit_pk_data_vector = pk_data_vector [ fit_selection ]

8 fit_Cinv = np. linalg . inv (C[np. ix_ ( fit_selection , fit_selection )])

9 # Select fitting−range for the convolved model power spectrum

10 fit_model = convolved_model [ fit_selection ]

11 # ... and proceed to calculate $\ chi ^2 $

How can I limit my analysis to the monopole? Since the window function couples

the diﬀerent multipoles you have to provide a model for all multipoles, so that the window

function contributions can be calculated. For the likelihood itself one can limit the analysis

to the monopole by

1 # Select multipoles to be included [P_0 , P_1 , P_2 , P_3 , P_4]

2 pole_selection = [ True , False , False , False , False ]

3 fit_selection = np. repeat(pole_selection , 40)

4 # Adjust data power spectrum and inverse C

5 fit_pk_data_vector = pk_data_vector [ fit_selection ]

6 fit_Cinv = np. linalg . inv (C[np. ix_ ( fit_selection , fit_selection )])

7 # Adjust convolved model power spectrum

8 fit_model = convolved_model [ fit_selection ]

9 # ... and proceed to calculate $\ chi ^2 $

Speeding up the likelihood evaluation: Often it is possible to achieve a speedup in

the likelihood analysis if the convolution of the power spectrum model does not have to be

performed in every model-data comparison. We can re-write the likelihood as [27]

L ∼ exp



−

conv

− WMP

true,ﬂat-sky

)

−1

conv

− WMP

true,ﬂat-sky

)



(A.1)

= exp



−

conv,T

−1

conv

+ P

true,ﬂat-sky,T

−1

conv

−

true,ﬂat-sky,T

−1

conv

WMP

true,ﬂat-sky



(A.2)

= exp



−

conv,T

−1

conv

+ P

true,ﬂat-sky

−1

conv,W

conv

−

true,ﬂat-sky,T

−1

conv,WW

true,ﬂat-sky



(A.3)

– 31 –

with

−1

conv,WW

= M

−1

conv

WM (A.4)

−1

conv,W

= M

−1

conv

. (A.5)

An implementation of these equations could look like this:

1 # Select multipoles to be included [P_0 , P_1 , P_2 , P_3 , P_4]

2 pole_selection = [ True , False , False , False , False ]

3 fit_selection = np. repeat(pole_selection , 40)

4 # Select the fitting range

5 Cinv = np. linalg.inv (C[ np .ix_( fit_selection , fit_selection )])

6 pk_data_vector = pk_data_vector [ fit_selection ]

7 W = W[ np.ix_ ( fit_selection , np .ones (2000) < 1.1)]

8 # Pre−calculate the two covariance matrices and the

9 # data−data contribution

10 WCinv = np.matmul(np . transpose (W), Cinv )

11 MWCinv = np .matmul (np. transpose (M), WCinv )

12 MWCinvD = np . matmul (MWCinv , pk_data_vector )

13 MWCinvW = np . matmul (MWCinv , W)

14 MWCinvWM = np. matmul ( MWCinvW , M)

15 data_term = np. dot ( pk_data_vector , np.dot (Cinv , pk_data_vector ))

17 ## The following part needs to be run for every parameter evaluation

18 # Get your favourite power spectrum model as disctionary

19 pk_model_dict = your_favourite_model ()

20 # Turn dictionary into vector as [P_0 ,P_2 , P_4 ]

21 kbins_long , pk_model_vector = pk_tools . dict_to_vec ( pk_model_dict , use_ell =[0 , 2 ,4])

22 # Calculate chi2

23 chi2 = data_term

24 chi2 −= 2.∗np.dot ( pk_model_vector , MWCinvD )

25 chi2 += np. dot (pk_model_vector , np.dot ( MWCinvWM , pk_model_vector ))

This implementation is about a factor of two times faster than the brute-force implementation

shown in the ﬁrst code example. Of course that is only signiﬁcant if the window function

convolution is dominating the likelihood evaluation. In many cases most of the time will be

spend in your_favourite_model().

How can I account for uncertainties in the covariance matrix? When deriving the

covariance matrix from a ﬁnite set of mock realisations, the resulting likelihood is no longer

Gaussian, but follows a t-distribution. If assuming a Gaussian likelihood the parameter

inference will be biased and this bias depends on the ratio of bins in the data vector and the

number of mock realisations [85]. We can account for this by scaling the likelihood as

ln L ∝ −χ

− N

− 2

2(N

− 1)

, (A.6)

where N

is the number of mock realisations (given in table 1) and N

is the size of the data

vector. Alternatively one can directly account for the non-Gaussian likelihood as proposed in

[86]. For all cases discussed in this paper this approach agrees very well with eq. (A.6).

Assuming your MCMC sampler expects log L as a return value you could implement

this equation as

1 # Hartlap et al. (2007)

2 H = Nmocks − len ( pk_data_vector ) − 2

3 H /= ( Nmocks − 1)

4 return −chi2∗H /2.

– 32 –

If the ﬁnal parameter uncertainty is derived from the likelihood itself, you need to account

for a bias caused by the mock based covariance estimate. We can do that with a re-scaling

of the parameter errors [87, 88] by the square root of

1 + B(N

− N

)

1 + A + B(N

+ 1)

(A.7)

with

A =

− N

− 1)(N

− N

− 4)

, (A.8)

B =

− N

− 2

− N

− 1)(N

− N

− 4)

(A.9)

(we ﬁnd that the approach of [86] does not take care of this factor).

How can I deconvolve a power spectrum measurement? To perform a deconvolution,

all you have to do is to follow eq. (4.2). The diﬃculty here is that this equation only holds for

square matrices W. The window functions using ∆k

= ∆k

= 0.01 h Mpc

−1

are available.

1 # Read data power spectrum (Delta k_o =0.01 h/ Mpc ) as disctionary

2 pk_data_dict = pk_tools . read_power(pkfile , combine_bins =10)

3 # Turn dictionary into vector as [P_0 ,P_1 , P_2 ,P_3 ,P_4]

4 kbins , pk_data_vector = pk_tools . dict_to_vec ( pk_data_dict )

5 # Read window matrix (needs to be a square matrix )

6 W = pk_tools . read_matrix ( Wfile )

7 # Invert window function

8 Winv = np. linalg.inv (W)

9 # Deconvolution

10 pk_data_vector_deconvolved = np. matmul(Winv , pk_data_vector )

B Window function pre-factors

Focusing on the ﬁrst 5 multipoles (0 ≤ L ≤ 4) of the window function, including wide-angle

terms up to second order (n < 3), the weights for the individual contributions in eq. (2.5) are

given by

(0)

0`L

= C

(2)

0`L







1 0 0 0 0

0 0 0 0 0

0 0

0 0 0 0 0

0 0 0 0







(1)

0`L







0 0 0 0 0

0 0 0

0 0 0 0 0

0 0 0

0 0 0 0 0







(0)

1`L

= C

(2)

1`L







0 1 0 0 0

0 0 0 0 0

0 0 0







(1)

1`L







0 0 0 0 0

1 0

0 0

0 0 0 0 0

0 0

0 0 0 0 0







(0)

2`L

= C

(2)

2`L







0 0 1 0 0

0 0 0 0 0

1 0

0 0 0 0 0

0 0

100

693







(1)

2`L







0 0 0 0 0







(B.1)

– 33 –

and

(0)

3`L

= C

(2)

3`L







0 0 0 1 0

0 0 0 0 0







(1)

3`L







0 0 0 0 0

0 0

0 0 0 0 0

1 0

0 0 0 0 0







(0)

4`L

= C

(2)

4`L







0 0 0 0 1

0 0 0 0 0

0 0

0 0 0 0 0

1 0

162

1001







(1)

4`L







0 0 0 0 0

0 0 0

0 0 0 0 0







(B.2)

where the wide-angle correction terms at n = 0 and n = 2 have the same shape.

C Derivation of the 2D window function

In this section we derive the equation for the 2D window function W

(n)

(k, k

). We ﬁrst derive

this equation for n = 0 showing consistency with [1], where this equation ﬁrst appeared. We

than include the wide-angle correction terms following [2], which leads to our eq. (2.5).

C.1 Excluding wide-angle terms and consistency with [1]

Here we show the relation between eq. (33) of [1] and eq. (2.2). The convolution of the power

spectrum can be written as (using a LOS of

d =

and following eq. B.1 of [1])

conv

(k) =

2` + 1

dµ

dφ

2π

(2π)

true

)|W (k −k

(

) (C.1)

2` + 1

dµ

dφ

2π

dµ

dφ

4π

2π

true

, µ

)×

ran

i,j,i=j

FKP

ik·s

−ik

·s

(

k ·

) .

(C.2)

Using multipole expansion as well as

iksµ

(2L + 1)j

(ks)L

(µ) , (C.3)

(

s)δ

2` + 1

dµ

dφ

2π

(

k ·

s)L

(

k ·

) , (C.4)

we get

conv

(k) = (−i)

(2` + 1)

2π

true

s)j

(ks)×

ran

i,j,i=j

FKP

(

s)L

(

s) .

(C.5)

Note that our Fourier transform convention is P (k) ≡

dx e

−ik·x

ξ(x).

– 34 –

Now using

ran

i,j,i=j

FKP

) =

W (s

)W (s

)

ds s

dΩ

W (s

)W (s + s

)

(C.6)

and the deﬁnition (see eq. 2.21 of [2])

(s) = (2L + 1)

dΩ

4π

W (s

)W (s + s

(

s ·

) (C.7)

as well as

(

s)L

(

s) =

`+L

p=|`−L|

(2p + 1)



` L p

0 0 0



(

s) , (C.8)

results in

conv

(k) = 4π(−i)

2π

true

)×

ds s

s)j

(ks)(2` + 1)

`+L

p=|`−L|



` L p

0 0 0



(s)

(C.9)

(k, k

true

) , (C.10)

where

(k, k

) = (−i)

ds s

s)j

(ks)A

(s) (C.11)

with

(s) = (2` + 1)

`+L

p=|`−L|



` L p

0 0 0



(s) (C.12)

`+L

p=|`−L|

`Lp

(s) (C.13)

and the factors C

`Lp

= C

(0)

`Lp

are given in appendix B.

C.2 Including wide-angle terms and consistency with [2]

The convolution of the power spectrum multipoles including the wide-angle correction terms

has ﬁrst been derived in [2] and is given by

conv

(k) = (−i)

(2` + 1)

4π

`+L

p=|`−L|



L p `

0 0 0



n+2

(ks)ξ

(n)

(s)Q

(n)

(s) . (C.14)

We added a factor of 1/4π to account for the diﬀerence in the window function deﬁnition (see eq. 2.21

in [2] and eq. 2.6 in this paper.)

– 35 –

Now using

(n)

= i

dk k

2π

(ks)

−n

(n)

(k)j

(ks) (C.15)

we get

conv

(k) = (−i)

(n)

(s)

n+2

(ks)×

−n

(n),true

(C.16)

02−n

L,n

(n)

(k, k

(n),true

) , (C.17)

where

(n)

(k, k

) = (−i)

ds s

s)j

(ks)A

(n)

(s) (C.18)

and

(n)

(s) = (2` + 1)

`+L

p=|`−L|



L p `

0 0 0



(n)

(s) (C.19)

`+L

p=|`−L|

(n)

`Lp

(n)

(s) . (C.20)

The factors C

(n)

`Lp

are given in appendix B. The equation above is consistent with eq. (C.12)

since



0 0 0





0 0 0



. (C.21)

D Analytic calculation of the odd power spectrum multipoles

Here we derive eq. (2.14) and eq. (2.15) used in section 2.1. The dipole power spectrum is

given by

(1)

(k) = −ik

5π

(0)

)

ds j

s)j

(ks) . (D.1)

Following eq. (F.1), (F.6) and (F.10) of [12] we can write

ds j

s)j

(ks) = −k

∂



− k)



, (D.2)

where δ

is the Dirac delta function. Using

11−1

(k, k

) =

− k) (D.3)

to replace the integral with a derivative, results in

(1)

(k) = ik

(0)

)∂



− k)



(D.4)

= ik

(0)

)



−

− k) +

∂

− k)



. (D.5)

– 36 –

Now we can use

dxf(x)∂

(x − a) = −∂

f(a) , (D.6)

yielding

(1)

(k) = −i

(0)

(k) + k∂

(0)

(k)

(D.7)

= −if





[3P

(k) + k∂

(k)] , (D.8)

where the second line assumes linear theory. This ﬁnal equation is numerically much easier

to evaluate compared to the equation we started with.

Equivalently we can derive the octopole

(1)

(k) = −i





(0)

(k) − k∂

(0)

(k)





(0)

(k) + k∂

(0)

(k)





(D.9)

= −i4f







(2P

(k) − k∂

(k)) +

f (5P

(k) + k∂

(k))



(D.10)

E Deconvolution results for all datasets

Figure 14, 15 and 16 show the comparisons for the convolved and deconvolved power spectra

for 6dFGS DR3, BOSS DR12 and the eBOSS DR16 QSO samples. Similar plots for the low

redshift bin of BOSS DR12 NGC are included in the main text (see ﬁgure 11 and 12). The

remaining plots are included here rather than the main text, to not interrupt the ﬂow of the

paper.

Figure 14: Comparison between the power spectrum multipoles of 6dFGS DR3 measured

in the mock catalogs (gray and red shaded area) and in the data (data points). The results

before deconvolution are shown as the red shaded area and solid red line (mocks) and the

green data points. The deconvolved results are shown as the gray shaded area and solid black

line (mocks) and the blue data points. The residuals in the lower panel show that wide-angle

and window function eﬀects are sub-dominant in 6dFGS.

– 37 –

Figure 15: Comparison between the power spectrum multipoles of BOSS DR12 measured

in the mock catalogs (gray and red shaded area) and in the data (data points). The results

before deconvolution are shown as the red shaded area and solid red line (mocks) and the

green data points. The deconvolved results are shown as the gray shaded area and solid black

line (mocks) and the blue data points. The equivalent BOSS DR12 NGC results for the low

redshift bin are included in the main text (see ﬁgure 11 and 12).

– 38 –

Figure 16: Comparison between the power spectrum multipoles of the eBOSS DR16 QSO

sample measured in the mock catalogs (gray and red shaded area) and in the data (data

points). The results before deconvolution are shown as the red shaded area and solid red line

(mocks) and the green data points. The deconvolved results are shown as the gray shaded

area and solid black line (mocks) and the blue data points. The increasing in the dipole power

spectrum at high k seems to be an aliasing eﬀect, even though the Nyquist frequency is twice

as high as the scale range plotted here (k

> 0.6 h Mpc

−1

– 39 –