# The RSD Sorting Hat: Unmixing Radial Scales in Projection

Peter L. Taylor<EMAIL_ADDRESS>Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109,
USA  Katarina Markovič  Jet Propulsion Laboratory, California Institute of
Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  Alkistis Pourtsidou
School of Physics and Astronomy, Queen Mary University of London, Mile End
Road, London E1 4NS, UK  Department of Physics and Astronomy, University of
the Western Cape, Cape Town 7535, South Africa  Eric Huff  Jet Propulsion
Laboratory, California Institute of Technology, 4800 Oak Grove Drive,
Pasadena, CA 91109, USA

###### Abstract

Future data sets will enable cross-correlations between redshift space
distortions (RSD) and weak lensing (WL). While photometric lensing and
clustering cross-correlations have provided some of the tightest cosmological
constraints to date, it is not well understood how to optimally perform
similar RSD/WL joint analyses in a lossless way. RSD is typically measured in
$3D$ redshift space, but WL is inherently a projected signal, making angular
statistics a natural choice for the combined analysis. Thus, we determine the
amount of RSD information that can be extracted using projected statistics.
Specifically we perform a Fisher analysis to forecast constraints and model
bias comparing two different Fingers-of-God (FoG) models using both, the $3D$
power spectrum, $P(k,\mu)$, and tomographic $C(\ell)$. We find that because
naïve tomographic projection mixes large scales with poorly modelled nonlinear
radial modes, it does not provide competitive constraints to the $3D$ RSD
power spectrum without the model bias becoming unacceptably large. This is
true even in the limit of narrow tomographic bins. In light of this we propose
a new radial weighting scheme which unmixes radial RSD scales in projection
yielding competitive constraints to the $3D$ RSD power spectrum, while keeping
the model bias small. This work lays the groundwork for optimal joint analyses
of RSD and cosmic shear.

††© 2021. All rights reserved.

##  I Introduction

Analyses of the RSD power spectrum from spectroscopic surveys have yielded
some of the most precise cosmological constraints to date Reid  _et al._
(2012, 2012); Macaulay  _et al._ (2013); Beutler  _et al._ (2014); Gil-Marín
_et al._ (2016); Simpson  _et al._ (2016); Blake  _et al._ (2011); Macaulay
_et al._ (2013); Alam  _et al._ (2021); Guzzo  _et al._ (2008). These will
further improve as data from Stage IV experiments including
Euclid111http://euclid-ec.org Blanchard  _et al._ (2020); Laureijs  _et al._
(2011), the Nancy Grace Roman Space Telescope222https://www.nasa.gov/roman
Spergel  _et al._ (2015) and the Dark Energy Spectroscopic Instrument
(DESI)333https://www.desi.lbl.gov/ Aghamousa  _et al._ (2016) become
available.

As a standalone probe RSD provides extremely tight constraints on gravity on
cosmological scales and specifically the logarithmic growth function, $f$
Lahav  _et al._ (1991); Linder (2005). Measurements of this parameter can
distinguish between theories of modified gravity which are indistinguishable
from background expansion probes alone Huterer  _et al._ (2015).

Yet, small scale radial modes (high $k_{\parallel}$) are poorly understood and
there are large discrepancies between models Hamilton (1997); Taruya  _et al._
(2010); Bernardeau  _et al._ (2002); Chen  _et al._ (2020); Scoccimarro
(2004); Shirasaki  _et al._ (2021). Thus, when performing parameter inference
one typically assumes a nonlinear RSD model and marginalizes over a set of
nuisance parameters which govern the behaviour on intermediate nonlinear
scales, while removing (or severely down-weighting) scales deep into the
nonlinear regime Markovic  _et al._ (2019). This is the primary source of
modelling uncertainty limiting the constraining power of RSD surveys.

Meanwhile, analyses of photometric surveys, using a combination of weak
lensing Troxel  _et al._ (2018); Hikage  _et al._ (2019); Asgari  _et al._
(2021), galaxy clustering Elvin-Poole  _et al._ (2018) and the cross-
correlation between the two Abbott  _et al._ (2018), provide complimentary
information. Over the next decade, data from Euclid, the Vera Rubin
Observatory Alonso  _et al._ (2018) and Roman will enable these measurements
to achieve unprecedented precision.

By breaking the degeneracy between cosmological and nuisance parameters, the
full ‘$3\times 2$ point’ constraints including galaxy-galaxy lensing are
dramatically more precise than those from photometric cosmic shear or
clustering alone. This has been shown with existing data Abbott  _et al._
(2018); Heymans  _et al._ (2021a) and it is predicted Tutusaus  _et al._
(2020) that for Euclid, the inclusion of the cross-correlation between the two
signals improves the dark energy $w_{0}-w_{a}$ figure of merit by a factor of
five.

By constraining highly uncertain nonlinear RSD models, increasing the signal-
to-noise and acting as an important consistency check – a combined RSD/WL
analysis could provide many of the same advantages as a $3\times 2$ point
analysis and yield some of the most competitive constraints on dark energy and
modified gravity. The velocity and position of foreground galaxies is
correlated with lensing structure, so the WL and RSD signals should be highly
covariant provided that 1) the spectroscopic RSD sample lies at a similar
redshift to the peak of the lensing kernel and 2) the WL and RSD samples cover
overlapping regions of the sky. This will be the case for the DESI luminous
red galaxies (LRGs), emission line galaxies (ELGs), the bright galaxy sample
(BGS) and Euclid weakly lensed galaxies, and for the spectroscopic /
photometric overlap in Roman Eifler  _et al._ (2020). The Euclid spectroscopic
sample meanwhile is likely at too high a redshift to cross-correlate with WL.

Several studies have previously sought to exploit this covariance. The
analysis of Lee  _et al._ (2021) used weak lensing data to constrain the
galaxy bias of the RSD sample. However as the RSD and WL samples were in non-
overlapping fields, the data vector in this study contained no RSD/WL cross-
correlation terms. Instead a photometric sample was constructed to match the
properties of the RSD sample and the resulting galaxy-galaxy lensing
measurement was used to constrain the galaxy bias. Other analyses have also
sought to exploit the cross-correlation between photometric cosmic shear and
spectroscopic galaxy clustering by performing $3\times 2$ point analysis
Heymans  _et al._ (2021b); Joudaki  _et al._ (2018). Since these studies use
photometric redshifts for the lensed galaxies and the tomographic bin widths
were taken to be much larger than the nonlinear RSD scale, one should not
expect this type of analysis to provide additional constraints on nonlinear
RSD i.e the high $k_{\parallel}$-modes. In fact binning the spectroscopic
clustering sample into broad redshift bins leads to a loss of information. But
if one had spectral information for the lensed galaxies from cross-matching
the RSD and WL catalogs, it is possible that by providing additional
measurements of the small scale radial modes, the WL/RSD cross-signal could
help constrain nonlinear RSD.

Given the large overlap between Euclid and DESI and the Roman photometric and
spectroscopic survey Eifler  _et al._ (2020), we anticipate a sample of
approximately $30$ million galaxies where we have both lensing and
spectroscopic information within the next decade.444There are also proposals
to perform kinematic weak lensing studies which will provide spectra for
lensed galaxies Huff  _et al._ (2013). A first detection of this signal was
recently made in Gurri  _et al._ (2020). To take advantage of this cross-probe
signal, we first need an appropriate two-point estimator.

While RSD studies typically use the 3D anisotropic power spectrum555We will
use $P(k,\mu)$ and $P(k_{\parallel},k_{\perp})$ interchangeably through the
remainder of the text using the relations $k_{\parallel}=k\mu$ and
$k=\sqrt{k_{\parallel}^{2}+k_{\perp}^{2}}$., $P(k,\mu)$, the anisotropic two-
point correlation function, $\xi(s_{\perp},s_{\parallel})$, or Legendre
multipoles $P_{\ell}(k)$, weak lensing is a projected observable, so cosmic
shear studies use angular statistics, in particular the tomographic cosmic
shear power spectrum, $C(\ell)$, or the two-point correlation functions,
$\xi_{\pm}(\theta)$. Projection is not a completely invertible operation, so
we should seek to perform the joint analysis in projected space.

This raises the question of whether it is even possible to capture all the
information in the anisotropic power spectrum, $P(k,\mu)$ using projected
statistics. One may naïvely wish to use classical projected tomographic power
spectra Jalilvand  _et al._ (2020); Grasshorn Gebhardt and Jeong (2020);
Camera  _et al._ (2018); Asorey  _et al._ (2012); Padmanabhan  _et al._
(2007); Loureiro  _et al._ (2019), $C_{ij}(\ell)$, binning galaxies in
sufficiently narrow redshift bins (with bin numbers $i$ and $j$) to capture
the $k_{\parallel}$-modes down to sufficiently small scales. As shown in
Jalilvand  _et al._ (2020), this mixes large and small $k_{\parallel}$-modes
for each fixed perpendicular scale, $k_{\perp}$. This has two problems:

  * •

Cosmological constraining power is lost because each ${\bf
k}=(k_{\parallel},\left|{\bf k_{\perp}}\right|)$ contains independent
information.

  * •

As we are mixing $k_{\parallel}$-scales for fixed $\ell$, we must cut data
points to avoid model bias from nonlinear RSD scales, even when these data
points also contain useful information about linear $k_{\parallel}$-scales.

To sidestep these issues, we will show how to construct a radial weighting
function (basis) which sorts $k_{\parallel}$-scales in projection. For radial
weights $w(\eta_{a},z)$ and $w(\eta_{b},z)$ in this new basis, we write the
angular power spectrum as $C^{\eta_{a}\eta_{b}}(\ell)$. We will demonstrate
there are tight correspondences between $\eta-k_{\parallel}$ and $\ell-
k_{\perp}$. This can be thought of as the RSD analogue of $k$-cut and $x$-cut
cosmic shear which sorts lensing 2-point statistics by sensitivity to
difference structure scales Taylor  _et al._ (2021, 2018a, 2020); Bernardeau
_et al._ (2014).

We will show that one must still use the RSD power spectrum, $P(k,\mu)$, to
optimally extract information from large linear radial scales before
discussing how to use the two estimators, $C^{\eta_{b}\eta_{b}}(\ell)$ and
$P(k,\mu)$, in a joint analysis. This is somewhat similar to the hybrid
estimator developed in Wang  _et al._ (2020) except we work in projected space
on small scales while Wang  _et al._ (2020) worked in spherical-Bessel
projected space on large scales to handle wide-angle effects. Although this
means in the future we will not be able to extract information from large
scale radial modes in the WL/RSD cross-signal, this is acceptable as we are
primarily interested in the cross-correlations on smaller radial scales to
constrain nonlinear RSD.

The structure of this paper is as follows. In Section II we summarize the
formalism of projected RSD before presenting our new basis. In Section III we
perform a Fisher analysis. Using two phenomological FoG models, we compare the
precision and model bias when using the RSD power spectrum, $P(k,\mu)$,
standard tomography, $C_{ij}(\ell)$ and our new basis,
$C^{\eta_{a}\eta_{b}}(\ell)$, before concluding in Section V. Finally we
discuss issues related to performing a likelihood analysis with real data in
the Appendix. Throughout the remainder of the paper we take
$\Omega_{m}=0.315$, $\Omega_{b}=0.04$, $h_{0}=0.67$, $n_{s}=0.96$ and
$\sigma_{8}=0.8$.

##  II RSD Angular Power Spectra

###  II.1 A Brief Review of the RSD Anisotropic Power Spectrum

Before discussing projected RSD, we first review the formalism of the
anisotropic RSD power spectrum, $P(k,\mu)$. We decompose the RSD spectrum into
isotropic and anistropic parts Grasshorn Gebhardt and Jeong (2020)

$P(k,\mu)=\widetilde{A}^{2}_{\mathrm{RSD}}(\mu,k\mu)P(k),$ (1)

where $\mu=cos(\theta)$, $\theta$ is the angle between the wave-vector and the
line of sight, $P(k)$ is the matter power spectrum and $\widetilde{A}$ is the
RSD operator which takes the power spectrum into redshift space and accounts
for galaxy bias. As in Grasshorn Gebhardt and Jeong (2020), we decompose it as

$\widetilde{A}_{\mathrm{RSD}}(\mu,k\mu)=b_{g}\left(1+\beta\mu^{2}\right)\widetilde{A}_{\mathrm{nl}}(\mu,k\mu),$
(2)

where $b_{g}\left(1+\beta\mu^{2}\right)$ is the Kaiser term Kaiser (1987)
acting on linear scales, $b_{g}$ is the linear galaxy bias, $\beta=f/b_{g}$
and $\widetilde{A}_{\mathrm{nl}}$ is the operator that accounts for the
nonlinear RSD model. In this paper we consider two simple phenomenological
nonlinear models which account for the FoG. For a Gaussian FoG, the nonlinear
RSD operator is Hamilton (1997)

$\widetilde{A}_{\rm
Gauss}(k\mu)=\exp\left[-f^{2}\sigma_{v}^{2}k^{2}\mu^{2}\right],$ (3)

while for the Lorentzian model it is given by

$\widetilde{A}_{\rm
Lorentz}(k\mu)=\frac{1}{1+f^{2}\sigma_{v}^{2}k^{2}\mu^{2}}.$ (4)

Throughout the remainder of this work we use the nonlinear power spectrum from
Halofit Takahashi  _et al._ (2012) to generate the isotropic part, $P(k)$ and
take the Lorentzian FoG as the fiducial model666In this work we are primarily
interested in the difference between the two models as proxy for nonlinear RSD
model uncertainty. unless explicitly stated otherwise. This phenomenological
model is as accurate as more physically motivated models Jalilvand  _et al._
(2020). We choose to ignore the impact of spectroscopic redshift uncertainty
as this is subdominant to the FoG.

###  II.2 Survey Selection Functions, Windows and Tomographic Bins

In this section we aim to clarify some definitions which will be used through
the remainder of the text. Given a spectroscopic survey, the observed radial
distribution of galaxies will have some distribution in redshift, $z$, which
we will refer to as the survey radial selection function and which we write as
$N(z)$.

We divide the radial selection function into several coarse redshift bins
which we will refer to as radial windows or simply as windows. We use the
notation $W(z)$ to denote the galaxy redshift distribution of a window. It is
standard practice in RSD anisotropic power spectrum studies to sub-divide the
survey selection function into coarse redshift bins, assume there is no
redshift evolution of the power spectrum in the window and analyse each
redshift bin separately. For the remainder of this work we compute distances
and the power spectrum at the average redshift of the window $z_{w}=\int{\rm
d}z\ zW(z)/\int{\rm d}z\ W(z)$, writing $r_{w}$ for the comoving distance at
$z_{w}$. The impact of choosing an effective redshift are studied in Obuljen
and Percival (2021). We will also restrict our attention to a single window
for the remainder of the paper. Finally when we consider projected tomographic
power spectra we will further sub-divide the window into tomographic bins in
redshift. In this case we denote the $i^{\rm th}$ tomographic bin window as
$W_{i}(z)$.

###  II.3 Angular Tomographic Power Spectra in the Flat-Sky Approximation

Since the RSD power spectrum $P(k,\mu)$ is anisotropic, care must be taken
when projecting to angular statistics. For the remainder of this work we will
assume the plane-parallel approximation Grasshorn Gebhardt and Jeong (2020);
Jalilvand  _et al._ (2020); Matthewson and Durrer (2021) which is an extremely
good approximation for $\ell>100$ assuming that the radial separation of
tomographic bins is small Matthewson and Durrer (2021). Then for tomographic
bin numbers $i$ and $j$, following the notation of Grasshorn Gebhardt and
Jeong (2020), we can write the projected tomographic angular power
spectrum777Formally since the RSD field is anisotropic
$C_{\ell\ell^{\prime}}\neq 0$ for $\ell\neq\ell^{\prime}$. We choose to ignore
this effect since the coupling is weak and contained to a few adjacent
$\ell$-modes Grasshorn Gebhardt and Jeong (2018). In practice with real data,
one must also use wide bandpowers in $\ell$ to deconvolve the survey mask,
mitigating the impact of coupling. as

$C_{ij}(\ell)=\frac{1}{r_{i}r_{j}}C\left(k_{\perp}=\frac{\ell+1/2}{r_{0}};r_{i},r_{j}\right),$
(5)

where $r_{i}$ and $r_{j}$ give the comoving radial distance to tomographic
bins $i$ and $j$, $r_{0}=(r_{i}+r_{j})/2$ and

$\displaystyle
C\left(k_{\perp};r_{i},r_{j}\right)=\frac{1}{\pi}\int_{0}^{\infty}{\rm
d}k_{\parallel}\ P(k,\mu)\widetilde{K}_{ij}(k_{\parallel}),$ (6)

where $k=\sqrt{k_{\perp}^{2}+k_{\parallel}^{2}}$ and $\mu=k_{\parallel}/k$,
and we define the radial-mode efficiency kernel,
$\widetilde{K}_{ij}(k_{\parallel})$, as

$\widetilde{K}_{ij}(k_{\parallel})=\widetilde{W}_{i}^{*}(k_{\parallel})\widetilde{W}_{j}(k_{\parallel}).$
(7)

Here $\widetilde{W}_{i}(k_{\parallel})$ is the Fourier transform of the radial
window for tomographic bin $i$, so that,

$\widetilde{W}_{i}(k_{\parallel})=\int_{r[z_{\rm min}]}^{r[z_{\rm max}]}{\rm
d}r\ W_{i}\left(r[z]\right)\exp\left(-ik_{\parallel}r[z]\right)$ (8)

with the normalization

$\int_{r[z_{\rm min}]}^{r[z_{\rm max}]}{\rm d}r\ \ W_{i}\left(r[z]\right)=1.$
(9)

The radial-mode efficiency kernel, $\widetilde{K}_{ij}(k_{\parallel})$,
defined in Eqn. 7, is an important quantity. We will refer to it at many
points in the remainder of this work as it defines how sensitive the projected
spectra are to different $k_{\parallel}$-modes. It is important to note that
it depends on the comoving distance, $r$, and hence is a function of the
background cosmology.

In the absence of a survey mask, the expected observed power spectrum is a sum
of the signal in Eqn. 5 and the shot-noise. The shot-noise is strictly a
function of the number of galaxies in the bin, $N_{i,{\rm gals}}$, so we can
use the expression from the isotropic case Blanchard  _et al._ (2020)

$N_{ij}(\ell)=\frac{\delta_{ij}}{N_{i,{\rm gals}}},$ (10)

where $\delta_{ij}$ is the Kronecker delta.

###  II.4 Generalized Angular Power Spectra in the Flat-Sky Approximation

To extract information we weight every galaxy by some radial weight function,
$w(\eta,z)$. Then inside each window the cosmological information is contained
in the generalized power spectra $C^{\eta_{a},\eta_{b}}(\ell)$. Power spectra
with generalized weights have previously been proposed for weak lensing Taylor
_et al._ (2018b, c). It should be noted that standard tomography is just a
special case of the generalized power spectra where one takes
$\\{w(\eta,z)\\}$ to be a set of narrow non-overlapping top-hat functions.

After applying the weighting, $w(\eta,z)$, the radial window is

$\mathcal{W}(\eta,r[z])=w(\eta,z)W(r[z]).$ (11)

Generalizing the expressions in the previous section, it follows that

$C^{\eta_{a}\eta_{b}}(\ell)=\frac{1}{r_{w}^{2}}C^{\eta_{a}\eta_{b}}\left(k_{\perp}=\frac{\ell+1/2}{r_{w}};r_{w}\right)$
(12)

where

$\displaystyle C^{\eta_{a}\eta_{b}}$
$\displaystyle\left(k_{\perp};r_{w}\right)=\frac{1}{\pi}\int_{0}^{\infty}{\rm
d}k_{\parallel}\ P(k,\mu)\widetilde{K}(k_{\parallel};\eta_{a},\eta_{b}),$ (13)

and the generalized radial-mode efficiency kernel,
$K(k_{\parallel};\eta_{a},\eta_{b})$, is given by

$\widetilde{K}(k_{\parallel};\eta_{a},\eta_{b})=\widetilde{W}^{*}(\eta_{a},k_{\parallel})\widetilde{W}(\eta_{b},k_{\parallel}).$
(14)

The Fourier transform of the radial windows in the new basis are

$\widetilde{W}(\eta,k_{\parallel})=\int_{r[z_{\rm min}]}^{r[z_{\rm max}]}{\rm
d}r\ \mathcal{W}\left(\eta,r[z]\right)\exp\left(-ik_{\parallel}r[z]\right).$
(15)

Finally the shot-noise is given by

$N^{\eta_{a}\eta_{b}}(\ell)=\frac{1}{N_{{\rm gals}}}\int_{z_{\rm min}}^{z_{\rm
max}}{d}z\ W(z)w(\eta_{a},z)w(\eta_{b},z),$ (16)

where in this case we assume the normalization

$\int_{z_{\rm min}}^{z_{\rm max}}{d}z\ W(z)=1.$ (17)

We refer the reader to Heavens (2003) for the derivation of the shot-noise
with a generic weighting function. In practice the shot-noise can be estimated
from random realizations.

###  II.5 The Tomographic Radial-Mode Efficiency Kernel for Top-hat Window
Functions

It is in general difficult to compute the radial-mode efficiency kernels at
each point in cosmological parameter space rapidly enough for cosmological
parameter inference if the window (and hence tomographic bin windows) are not
simple analytic functions. We discuss a fast numerical approach888In practice
all results in this paper are computed using this fast numerical approaches
given in Appendices A-B. in Appendix A for more realistic windows. For the
time being, we restrict our attention to top-hat tomographic bin windows,
$W_{i}(z)$, since it is possible to proceed analytically. This is a good
starting point since in the limit of narrow tomographic bins, the tomographic
bin windows are well approximated by top-hat functions.

Assuming the tomographic bin window is centered at $r_{i}$, with bin width
$\Delta r_{i}$, noting that the Fourier transform of a top-hat function is a
sinc function, we find Grasshorn Gebhardt and Jeong (2020); Jalilvand  _et
al._ (2020)

$\widetilde{K}_{ii}(k_{\parallel})={\rm
sinc}^{2}\left(\frac{k_{\parallel}\Delta r_{i}}{2}\right),$ (18)

where ${\rm sinc}(x)=\sin(x)/x$.

Figure 1: The radial mode-efficiency kernels,
$\widetilde{K}_{ij}(k_{\parallel})$, for top-hat tomographic windows with
redshift widths $\Delta z$, centered around $z=0.5$. The radial mode-
efficiency kernel defines the sensitivity of the tomographic $C(\ell)$ to
$k_{\parallel}$-modes in the RSD power spectrum (see Eqn. 6). Smaller bins
widths lead to more sensitivity to smaller scale parallel modes. Since the
kernels are broad in $k_{\parallel}$, even in the limit of small bin widths,
information is still lost when using tomographic $C(\ell)$ as we mix
independent $k_{\parallel}$-modes across a broad range of scales.

Figure 2: The derivative of the RSD power spectrum
$P(k_{\parallel},k_{\perp})$ with respect to $b_{g}$, $f$ and $\sigma_{v}$.
Each parameter affects the power spectrum in different regions of
$k_{\parallel}-k_{\perp}$ space. This information is partially lost when
integrating over $k_{\parallel}$ to compute tomographic $C(\ell)s$ (see Eqn.
6). We quantify this statement in Section III.

In Fig. 1 we plot the radial-mode efficiency kernels for tomographic top-hat
windows of different widths, $\Delta z$, at a redshift $z=0.5$. Taking the bin
width to be $\Delta z=0.1$, we see that almost all nonlinear RSD information
is lost as the radial mode-efficiency kernel is effectively zero for
$k_{\parallel}\gtrsim 0.03\ h{\rm Mpc}^{-1}$. Decreasing the bin width to
$\Delta z=0.01$, we start to probe the mildly nonlinear regime
$k_{\parallel}\lesssim 0.2\ h{\rm Mpc}^{-1}$. But if we decrease the bin width
to $\Delta z=0.001$ the tomographic power spectra will remain sensitive to
radial modes extremely deep into the nonlinear regime, up to
$k_{\parallel}\lesssim 3\ h{\rm Mpc}^{-1}$. Hence we see that decreasing the
bin width yields sensitivity to larger $k_{\parallel}$-modes. We will restrict
our attention to $\Delta z>0.01$ to avoid sensitivity to the deeply nonlinear
regime for the remainder of this work.

It is worth emphasizing two key points. First, tomographic RSD angular power
spectra probe nonlinear radial modes even in the limit of low-$\ell$ (see Eqns
5-6). Unlike in photometric galaxy clustering studies, taking $\ell$-cuts can
not be used to completely mitigate bias from poorly understood nonlinear
scales. Second, the mixing of scales in the radial direction is not a
consequence of the plane parallel approximation. This can be seen by taking
the high-$\ell$ limit where the plane parallel power spectra are equivalent to
the full non-Limber spherical power spectra Grasshorn Gebhardt and Jeong
(2020). Instead the mixing of scales is a direct consequence of the anisotropy
of the RSD field.

Let us suppose that we have a model of nonlinear RSD which is unbiased for all
$k_{\parallel}<k_{\parallel}^{\rm max}$, for some $k_{\parallel}^{\rm max}$.
This suggests that we should keep decreasing $\Delta z$ as long as
$\widetilde{K}_{ij}(k_{\parallel})\sim 0$ for all $k>k_{\parallel}^{\rm max}$.
This ensures we capture information from all modes with
$k_{\parallel}<k_{\parallel}^{\rm max}$, while removing sensitivity to all
modes with $k_{\parallel}>k_{\parallel}^{\rm max}$.

With this choice of bin width, we still do not retain all the information
available in $P(k,\mu)$. To see this note that each $k_{\parallel}$-mode
yields independent information. In fact, many of the parameters of interest
such as the galaxy bias, growth function and FoG velocity dispersion all
impact $P(k,\mu)$ in different regions of $k_{\parallel}-k_{\perp}$ space.
This is shown in Fig. 2, where we plot the RSD power spectrum derivatives with
respect to these parameters, normalized against the RSD power spectrum. Yet
from Eqn. 6 we see that the tomographic $C(\ell)$ are given as an integral
over $k_{\parallel}$-modes weighted by the kernel
$\widetilde{K}_{ij}(k_{\parallel})$. These kernels are broad in
$k_{\parallel}$ leading to a mixing of modes across a wide range of radial
scales so that in projection over $k_{\parallel}$, detailed information about
$P(k_{\perp},k_{\parallel})$ and hence cosmological parameters, is lost. This
is also manifested in additional degeneracies between cosmological parameters
in projection. We quantify these statement in Section III.

###  II.6 Angular Radial-Harmonic Power Spectra

As we have just argued, the primary problem with tomographic $C(\ell)$ is that
the radial-mode efficiency kernels, $\widetilde{K}_{ij}(k_{\parallel})$ are
broad in $k_{\parallel}$. In this section, we will use the generalized power
spectra formalism and find a set of weight functions, $\\{w(\eta,z)\\}$ which
ensure that $\widetilde{K}(k_{\parallel};\eta_{a},\eta_{b})$ are narrow in
$k_{\parallel}$ and form a spanning set over $k_{\parallel}$.

For the sake of argument, let us suppose the window $W(r[z])$ is flat in
redshift. It follows from Eqns. 11, 14 and 15 that to ensure the radial-mode
efficiency kernels are narrow in $k_{\parallel}$, we should choose a set of
weight functions, $\\{w(\eta,z)\\}$, which are narrow in Fourier space.
Meeting this criteria, the key insight of this paper is to note that the
Fourier transform of a cosine is a delta function, inspiring the choice:

$w(\eta,z)=\cos\left(\frac{2\pi\eta r^{\rm ref}[z]}{\Delta r^{\rm
ref}[z]}\right).$ (19)

Here, $r^{\rm ref}$ and $\Delta r^{\rm ref}$ are the comoving distance to the
window and the width of the window assuming a fiducial reference cosmology. In
a likelihood analysis this weight should be applied to the data only once, and
when computing the theory vector, the weight remains invariant even as the
background cosmology is changed while sampling cosmological parameter space.
The pre-factors are chosen so that the weight function has a period of $\Delta
r^{\rm ref}/\eta$. With this choice of weight function, we refer to the
resulting $C^{\eta_{a}\eta_{b}}(\ell)$ as the radial-harmonic power spectra.

###  II.7 $k_{\parallel}$-mode Separation in the Radial-Harmonic Basis and the
$P(k,\mu)-C^{\eta_{a}\eta_{b}}(\ell)$ Hybrid Estimator

In Figure 3, we plot the radial-mode efficiency kernels for a Stage IV-like
survey centered on $z=0.5$. The radial window, $W(r[z])$ is shown in the top
panel. This is not flat in redshift to demonstrate that the weight function in
Eqn. 19 works as intended in a more general setting.

Figure 3: Top: The radial window function of a Stage IV-like survey. Middle:
The radial-mode efficiency kernel normalized against its maximal value. With
the hybrid estimator we use $P(k_{\parallel},k_{\perp})$ to probe scales to
the left of the black dashed line in grey and $C^{\eta_{a}\eta_{b}}(\ell)$ to
probe scales to the right of the line. In this regime the kernels are narrow
so there is a nearly one-to-one correspondence between $\eta$ and
$k_{\parallel}$, implying that in contrast to the tomographic case (see Figure
1), radial $k_{\parallel}$ scales are unmixed in projection. Bottom: The
radial-mode efficiency kernels. For $\eta_{a}\neq\eta_{b}$ the magnitude of
the radial-mode efficiency kernel is much smaller than for the auto-
correlations between $\eta$-modes. Hence for fixed $\ell$,
$C^{\eta_{a}\eta_{b}}(\ell)$ is nearly diagonal.

The radial-mode efficiency kernels normalized against their maximal values are
shown in the middle row. As the value of $\eta$ is increased, the kernels
become narrower and peak at higher values of $k_{\parallel}$. This means that
in contrast to the tomographic power spectrum which mixes
$k_{\parallel}$-scales, for $\eta\gtrsim 3$ the radially harmonic power
spectrum, $C^{\eta\eta}(\ell)$, is only sensitive to $k_{\parallel}$-modes
inside a narrow range and we probe different $k_{\parallel}$ scales for
different values of $\eta$. Very roughly the scaling goes as

$k_{\parallel}\sim\frac{2\pi\eta}{\Delta r}.$ (20)

In fact, for $\eta\gtrsim 3$ ($k_{\parallel}\gtrsim 0.05\ h{\rm Mpc}^{-1}$)
there is a nearly a one-to-one correspondence between $k_{\parallel}$ and
$\eta$. From Eqn. 12 there is also a one-to-one correspondence between $\ell$
and $k_{\perp}$. Thus nearly all the cosmological information that exists in
the RSD power spectrum $P(k_{\parallel},k_{\perp})$ is retained after
projection to $C^{\eta_{a}\eta_{b}}(\ell)$ – at least in the regime where
$k_{\parallel}\gtrsim 0.05\ h{\rm Mpc}^{-1}$.

As $\eta\rightarrow 0$, we approach the tomographic case as the wavelengths of
the radial weighting become much larger than the window. The fact that the
kernels are broad and mix $k_{\parallel}$ scales for low-$\eta$
(low-$k_{\parallel}$) motivates the choice of a new
$P(k,\mu)-C^{\eta_{a}\eta_{b}}(\ell)$ hybrid estimator. We shall refer to this
as the radial-harmonic hybrid estimator. In the high-$k_{\parallel}$
(high-$\eta$) regime we will use the projected radial-harmonic power spectra
$C^{\eta_{a}\eta_{b}}(\ell)$, enabling a cross-correlation with lensing at
high-$k_{\parallel}$ to constrain nonlinear RSD. Meanwhile on large radial
scales, we will use the RSD power spectrum $P(k_{\parallel},k_{\perp})$.

We combine the two estimators as follows:

  * •

Choose a minimum $\eta$-value, $\eta^{\rm min}$. This choice is not formally
defined, but it should be as small a value as possible while ensuring that the
radial-mode efficiency kernel, $\widetilde{K}(k_{\parallel};\eta,\eta)$ is
narrow.

  * •

Using the correspondence between $\eta$ and $k_{\parallel}$, define a scale
$k_{\parallel}^{\rm max}(\eta=\eta^{\rm min})$ which is the largest value of
$k_{\parallel}$ such that $\widetilde{K}(k_{\parallel};\eta^{\rm
min},\eta^{\rm min})$ is ‘small’. This ensures that
$P(k_{\parallel},k_{\perp})$ and $C^{\eta_{a}\eta_{b}}(\ell)$ probe different
$k_{\parallel}$ scales so that there is negligible covariance between the two.
For the remainder of this paper we will take $k_{\parallel}^{\rm
max}(\eta^{\rm min})$ as the maximum $k_{\parallel}$ such that
$K(k_{\parallel};\eta^{\rm min},\eta^{\rm min})$ is less than $20\%$ of its
maximal value over $k_{\parallel}$, corresponding to the horizontal dashed
line in the middle row of Fig. 3.

Following this procedure, if we choose $\eta^{\rm min}=3$, then
$k_{\parallel}^{\rm max}(\eta^{\rm min})=0.04$ which is indicated by the
dotted black line in the middle panel of Figure 3. The grey shaded region to
the left of this line indicates the $k_{\parallel}$ modes which are probed
with $P(k_{\parallel},k_{\perp})$ and the unshaded region denoted scales which
are probed with $C^{\eta_{a}\eta_{b}}(\ell)$.

The bottom panel of Figure 3 shows the radial-mode efficiency kernels for
$(\eta_{a},\eta_{b})=(1.,1.),(5.,5.),(1.,5.)$. We see that for
$\eta_{a}\neq\eta_{b}$, the magnitude of the radial-mode efficiency kernel is
much smaller than for the auto-correlations between $\eta$-modes so that for
fixed $\ell$, $C^{\eta_{a}\eta_{b}}(\ell)$ is nearly diagonal.

###  II.8 Comparison of Radial-harmonic and Tomographic Power Spectra

We start by validating our tomographic code against Cosmosis Zuntz  _et al._
(2015). For this test we assume no anisotropic RSD contributions, i.e we set
$\widetilde{A}^{2}_{\mathrm{RSD}}=1$ in Eqn. 1 and assume a top-hat window in
the range $z\in[0.45,0.55]$. All data points agree within $5\%$ with average
agreement within $1\%$ for $\ell\in[100,1000]$. Cosmosis makes the Limber
approximation LoVerde and Afshordi (2008) and assumes a temporally evolving
power spectrum while we assume that the power spectrum is fixed inside the
tomographic bin and assume the plane parallel approximation Jalilvand  _et
al._ (2020). In both cases we use Camb Lewis  _et al._ (2000) to generate the
linear power spectrum and Halofit Takahashi  _et al._ (2012) to generate the
nonlinear power spectrum, which we call from PyCamb in our code.

In Fig. 4 we plot the tomographic $C_{ii}(\ell)$ that cover the top hat window
over the range $z\in[0.5,0.75]$ for $1$, $2$ and $10$ subdivisions
corresponding to tomographic bin-widths, $\Delta z$, of $0.25$, $0.125$ and
$0.025$ respectively.

Comparing the three panels of Fig. 4, we notice that as the tomographic bin-
width is decreased the magnitude of the $C_{ii}(\ell)$ increases. This is
precisely as expected because on comparison with Fig. 1, it is clear that
narrow tomographic bins are sensitive to high-$k_{\parallel}$ modes as well as
the low-$k_{\parallel}$ modes which are probed with broader tomographic bins.
It is important to note that for narrow tomographic bins, even though we must
use all tomographic bin pairs to maximize the signal-to-noise, they are all
sensitive to the same poorly modelled nonlinear $k_{\parallel}$-modes, even
for low-$\ell$.

Figure 4: Tomographic $C(\ell)$ in the redshift range $z\in[0.5,0.75]$. Top,
middle and bottom panels correspond to $1$, $2$ and $10$ tomographic
subdivisions, corresponding to tomographic bin-width, $\Delta z$, of $0.25$,
$0.125$ and $0.025$ respectively. The $C_{ii}(\ell)$ of narrow bins have more
power as these functions probe high-$k_{\parallel}$ modes in addition to
low-$k_{\parallel}$ modes (see Fig. 1). For a given tomographic bin width, the
$C(\ell)$ of each tomographic bin probe approximately the same scales so that
$C_{ii}(\ell)\approx C_{jj}(\ell)$ for $i\neq j$. This implies that if
$C_{ii}(\ell)$ is biased for some $\ell$-mode due to modelling uncertainty,
then so is $C_{jj}(\ell)$ for $i\neq j$.

In Fig. 5 we show the radial-harmonic power for different values of $\eta$.
Unlike in the tomographic case, the projected spectra probe different
$k_{\parallel}$-scales at each $\eta$-value, resulting in differences in the
power spectra. One can easily remove sensitivity to poorly modelled
$k_{\parallel}$-modes by taking an $\eta$-cut to the data vector. For some
fixed $\ell$-mode, this preserves useful information at low-$\eta$ while
removing data points which are biased due to nonlinear RSD modelling
uncertainty at high-$\eta$.

Figure 5: Radial-harmonic power spectra inside the redshift range
$z=[0.5,0.75]$. Different values of $\eta$ probe different
$k_{\parallel}$-modes resulting in differences between spectra in contrast to
the tomographic case (see Fig. 4).

For parameters $f$, $b_{g}$ and $\sigma_{v}$, in Fig. 6 we plot the absolute
value of the derivatives of the tomographic power spectra normalized against
the power spectrum. This is a measure of how sensitive the tomographic power
spectra are to these parameters (see Sec. III.1 for more details). As before
we consider tomographic bins widths, $\Delta z$, of $0.25$, $0.125$ and
$0.025$. In each case we only consider the lowest redshift tomographic bin,
since the differences between each bin’s power spectra are small (see Fig. 4).

The normalized derivative in the top panel of Fig. 6 is largest for narrow
tomographic bins implying that the sensitivity to the high
$k_{\parallel}$-modes yields additional constraining power on $f$. From the
bottom plot in Fig. 6 we find that the power spectra of narrower bins are
significantly more sensitive to, $\sigma_{v}$, as these power spectra probe
high $k_{\parallel}$-modes (see Fig. 2). More generally data points that are
sensitive to $\sigma_{v}$ are those which are most prone to nonlinear RSD
model bias. This plot shows that although narrow bins would yield the tightest
constraints on $f$, this comes at the cost of model bias.

Figure 6: The absolute value of the derivatives of the tomographic power
spectra normalized against the power spectrum. This is a measure of how
sensitive the tomographic power spectra are to different parameters (see Sec.
III.1). The top row indicates that narrow tomographic bins are the most
sensitive to $f$. However, from the bottom row, narrow tomographic bins are
the most sensitive to $\sigma_{v}$, and more generally nonlinear RSD modelling
uncertainties, so this binning strategy is the most prone to model bias.

Finally the normalized derivatives of the radial-harmonic power spectra are
shown in Fig. 7. In this case we find that at low-$\ell$, the lower value
$\eta$-modes are sensitive to $f$. Crucially these data points are extremely
insensitive to $\sigma_{v}$ and nonlinear RSD scales in general, so the model
bias is small. Meanwhile from the bottom panel of Fig. 7 it is evident that we
are most sensitive to $\sigma_{v}$, and nonlinear RSD in general, for
large-$\eta$. These data points can easily be removed from the data vector to
mitigate model bias, while constraining $f$ with the data points at
low-$\eta$.

Figure 7: Same as Fig. 6, but for the radial-harmonic power spectra. From the
top row all power spectra are sensitive to $f$. Crucially from the bottom row,
only the high-$\eta$ power spectra are sensitive to $\sigma_{v}$ and more
generally nonlinear RSD modelling uncertainties. These data points can be
easily removed from the data vector, while preserving data points at
low-$\eta$ which constrain the growth function, $f$. This is in contrast to
the tomographic case, where we choose a tomographic bin width at the outset so
that all data points are affected by modelling uncertainties.

##  III Fisher Results

###  III.1 Fisher Formalism

To compare the constraining power and model bias of the anisotropic power
spectrum, tomography and the radial-harmonic hybrid estimator, we will perform
a Fisher analysis. For this analysis we will assume a Gaussian covariance
neglecting both the Super-Sample Covariance (SSC) Takada and Hu (2013); Lacasa
and Grain (2019); Barreira  _et al._ (2018) and Non-Gaussian (NG) Barreira
_et al._ (2018); Sato and Nishimichi (2013) terms.

Given a set of model parameters, $\\{\theta_{\alpha}\\}$, if we assume the
data vector follows a Gaussian likelihood and that it is linear in the model
parameters, then a good estimate of the marginal error on $\theta_{\alpha}$,
is

$\sigma(\theta_{\alpha})=\sqrt{(F^{-1})_{\alpha\alpha}},$ (21)

where $F$ is the Fisher matrix. For tomographic $C(\ell)$ the Fisher matrix is
Blanchard  _et al._ (2020)

$\displaystyle F^{\rm
tomo}_{\alpha\beta}=\sum_{\ell=\ell_{\min}}^{\ell_{\max}}\sum_{ij,mn}\frac{\partial
C_{ij}(\ell)}{\partial\theta_{\alpha}}\left[\Delta C^{-1}(\ell)\right]_{jm}$
(22) $\displaystyle\times\frac{\partial
C_{mn}(\ell)}{\partial\theta_{\beta}}\left[\Delta C^{-1}(\ell)\right]_{ni},$

where $\alpha$ and $\beta$ label the cosmological parameters, $i,j,m$ and $n$
label tomographic bins and,

$\Delta
C_{ij}(\ell)=\sqrt{\frac{2}{(2\ell+1)f_{\mathrm{sky}}\Delta\ell}}C_{ij}(\ell).$
(23)

Here $\Delta\ell$ is the multipole bandwidth and $f_{\mathrm{sky}}$ is the
fraction of the sky covered by the survey.

Let us suppose that we have two different nonlinear RSD models for the
tomographic power spectra $C_{ij}(\ell)$ and we write the difference between
the two models, $C^{m_{1}}_{ij}(\ell)$ and $C^{m_{2}}_{ij}(\ell)$, by defining

$\delta C_{ij}(\ell)=C^{m_{1}}_{ij}(\ell)-C^{m_{2}}_{ij}(\ell).$ (24)

Then for parameter $\theta_{\alpha}$, the expected bias between models,
$b_{\alpha}$, is given by Amara and Refregier (2008)

$b^{\rm tomo}_{\alpha}=\sum_{\beta}\left[(F^{\rm
tomo})^{-1}\right]_{\alpha\beta}B^{\rm tomo}_{\beta},$ (25)

where

$\displaystyle B^{\rm
tomo}_{\beta}=\sum_{\ell=\ell_{\min}}^{\ell_{\max}}\sum_{ij,mn}\delta
C_{ij}(\ell)\left[\Delta C^{-1}(\ell)\right]_{jm}$ (26)
$\displaystyle\times\frac{\partial
C_{mn}(\ell)}{\partial\theta_{\beta}}\left[\Delta C^{-1}(\ell)\right]_{ni}.$

In terms of the RSD power spectrum the Fisher matrix is given by Tegmark
(1997)

$\displaystyle F^{3D}_{\alpha\beta}=\frac{V_{s}}{8\pi^{2}}\sum_{k=k_{\rm
min}}^{k_{\rm max}}\sum^{1}_{\mu=-1}k^{2}\frac{\partial
P(k,\mu)}{\partial\theta_{\beta}}\ \frac{\partial
P(k,\mu)}{\partial\theta_{\beta}}$ (27) $\displaystyle\times{\rm
Cov}\left[P(k,\mu),P(k^{\prime},\mu^{\prime})\right],$

where $V_{s}$ is the survey volume and the RSD power spectrum covariance is

$\displaystyle{\rm Cov}\left[P(k,\mu),P(k^{\prime},\mu^{\prime})\right]$
$\displaystyle=\frac{2}{N_{k}}\left[P(k,\mu)+\frac{1}{\bar{n}_{g}}\right]^{2}$
(28) $\displaystyle\times\delta(k-k^{\prime})\delta(\mu-\mu^{\prime}),$

and the number of modes in the survey volume, $N_{k}$, is

$N_{k}=\frac{k^{2}\Delta k\Delta\mu}{4\pi^{2}}V_{s}.$ (29)

Assuming that we have two different nonlinear RSD models with power spectra
$P^{m_{1}}(k,\mu)$ and $P^{m_{2}}(k,\mu)$, and we write

$\delta P(k,\mu)=P^{m_{1}}(k,\mu)-P^{m_{2}}(k,\mu),$ (30)

and

$\displaystyle B^{3D}_{\beta}=\frac{V_{s}}{8\pi^{2}}\sum_{k=k_{\rm
min}}^{k_{\rm max}}\sum^{1}_{\mu=-1}k^{2}\delta P(k,\mu)\ \frac{\partial
P(k,\mu)}{\partial\theta_{\beta}}$ (31) $\displaystyle\times{\rm
Cov}\left[P(k,\mu),P(k^{\prime},\mu^{\prime})\right],$

the bias between models, for cosmological parameter $\alpha$, is

$b^{3D}_{\alpha}=\sum_{\beta}\left[(F^{3D})^{-1}\right]_{\alpha\beta}B^{3D}_{\beta}.$
(32)

This result can be derived by following the derivation given in the Appendix
of Taylor  _et al._ (2007), which computes the linear bias for any estimator
with a Gaussian likelihood.

For the angular radial-harmonic power spectra, $C^{\eta_{a}\eta_{b}}(\ell)$,
we define $F_{\alpha\beta}^{\rm harm}$, $B_{\beta}^{\rm harm}$ and
$b_{\beta}^{\rm harm}$ exactly as in the tomographic case, but now $i$, $j$,
$m$ and $n$ label the $\eta$-modes rather than the tomographic bins.

In the hybrid case we cut scales as described in Sec. II.7 and compute $F^{\rm
harm}_{\alpha\beta}$, $B^{\rm harm}_{\beta}$, $F^{\rm 3D}_{\alpha\beta}$ and
$B^{\rm 3D}_{\beta}$ as described above using the new scale cuts. Using the
fact that the two estimators used in the hybrid approach probe different
scales, the hybrid Fisher matrix is

$F^{\rm hybrid}_{\alpha\beta}=F^{\rm 3D}_{\alpha\beta}+F^{\rm
harm}_{\alpha\beta},$ (33)

and the bias

$b^{\rm hybrid}_{\alpha}=\sum_{\beta}\left[(F^{\rm
hybrid})^{-1}\right]_{\alpha\beta}B^{\rm hybrid}_{\beta},$ (34)

where

$B^{\rm hybrid}_{\beta}=B^{\rm 3D}_{\beta}+B^{\rm harm}_{\beta}.$ (35)

###  III.2 Fisher Results $z\in[0.5,0.75]$

We use the Fisher formalism described in the previous section to compute the
errors on $f$, $\sigma_{v}$ and $b_{g}$ for different $k_{\parallel}$-cuts,
while simultaneously computing the linear model bias between the Gaussian and
Lorentzian FoGs. The primary aim of this analysis is to determine how tightly
the three estimators can constrain cosmology, and in particular the growth
function, while keeping the nonlinear RSD model bias in check. Since the
Gaussian and Lorentzian FoG models disagree at high-$k_{\parallel}$ (see Eqns.
3 and 4), we use the linear bias between the two models as a proxy for RSD
nonlinear model uncertainty.

Figure 8: The Fisher $68\%$ and $95\%$ confidence regions for tomography, the
anisotropic power spectrum and the radial-harmonic hybrid estimator. The axes
do no match for the three cases. For a direct comparison see Fig. 10. The
contours are shifted by the linear model bias between the Lorentzian and
Gaussian FoG simulating the impact of an incorrect nonlinear RSD model. The
model bias increases with more optimistic scale cuts and as the precision of
the constraints improves. This is particularly true for $\sigma_{v}$ which is
the most sensitive to nonlinear scales.

Figure 9: Top: The linear bias normalized against the Fisher error,
$|b|/\sigma$, as a function of cut scale. Bottom: The error on the growth
function, $\sigma(f)$. The dashed line indicates the maximum cut scale where
the growth function constraints are unbiased in the sense that normalized bias
is less than $10\%$, i.e. $|b|/\sigma<0.1$. This delineates our fiducial scale
cuts. For this choice $\sigma(f)=0.013$ for the anisotropic power spectrum,
$\sigma(f)=0.013$ for the radial-harmonic hybrid estimator and
$\sigma(f)=0.055$ for tomography. This amounts to a factor of $4$ degradation
in constraining power when using tomography compared to the other two
estimators. The full Fisher constraints are shown in Fig. 10. Figure 10: Same
as Fig. 8, but now all three estimators are shown together to compare errors.
We take the fiducial choice of scale cut, which maximizes precision while
ensuring the bias on $f$ is closest to $10\%$ (see Fig. 10 for more details).
The radial-harmonic and anisotropic power spectrum constraints are nearly
identical, while the tomographic errors are much larger due to the loss of
information from mixing $k_{\parallel}$-scales in projection.

For this analysis we assume that $f_{\rm sky}=0.3$ and that the radial window
contains $7.5$ million galaxies uniformly distributed in the range
$z\in[0.5,0.75]$. This is chosen to match the redshift range of the
highest-$z$ window in Beutler  _et al._ (2017), so that we can use the
measured $\sigma_{v}$ from this study. For comparison the DESI Aghamousa  _et
al._ (2016) catalogue will cover approximately a third of the sky and comprise
of roughly $20$ million objects in the redshift range $z\in[0.5,1.5]$.999An
additional bright galaxy sample BGS will comprise of an extra 10 million
objects with a median redshift of $z\sim 0.2$. We cut all perpendicular scales
with $k_{\perp}>0.25$, corresponding to $\ell>350$ in the projected case. We
take $\eta^{\rm min}=5$, ensuring the radial-mode efficiency kernels are
narrow.

The Fisher results as a function of the radial-mode scale cut are displayed in
Fig. 8. In the anisotropic power spectrum case, the scale cut is defined by a
simple cut in $k_{\parallel}$, while for the radial-harmonic power spectrum,
the cut corresponds to an $\eta$-cut which is converted to an effective cut-
scale101010For fixed $\eta$, we define the cut-scale, $k_{\parallel}^{\rm
eff,cut}$, as the maximum $k_{\parallel}$ such that the radial-mode efficiency
kernel is within $50\%$ of it’s maximal value. Since the kernels are narrow in
$k_{\parallel}$-space (see Fig. 3) the precise way the cut is defined will
have a negligible effect., $k^{\rm eff,cut}_{\parallel}$. Meanwhile the scale
for tomography is given by the bin width. Smaller bin widths correspond to
more optimistic cuts in $k_{\parallel}$.

From Fig. 8 we see that as expected, for each of the three estimators, the
model bias increases with more optimistic scale cuts as the precision of the
constraints improve. This is particularly true for $\sigma_{v}$ which is the
most sensitive to nonlinear scales.

A comparison of the three estimators is displayed in Fig. 9. Each column
corresponds to a different estimator. The grey shaded region in the hybrid
case corresponds to the scales probed using the anisotropic power spectrum
while the white regions indicate scale probed by the radial-harmonic power
spectrum as in Fig. 3. In the first row of Fig. 9 we display the linear model
bias normalized against the Fisher error, $|b|/\sigma$. The error on the
growth function after marginalising over $\sigma_{v}$ and $b_{g}$ is shown in
the bottom row. While $|b|$ and $\sigma$ should respectively increase and
decrease monotonically with more optimistic scale cuts, $|b|/\sigma$ is not
necessarily strictly monotonic and it is not in the tomographic case.

We say that a parameter is unbiased if the normalized bias, $|b|/\sigma<0.1$,
i.e. the bias is less than $10\%$. This is similar to the criterion used in
Massey  _et al._ (2013) and corresponds to approximately a $95\%$ overlap in
the marginalized posterior probability distribution function Massey  _et al._
(2013). The scale cut at which the growth function, $f$, becomes biased is
delineated by dashed line in Fig. 9.

Since we are primarily concerned with the growth function, the parameter which
distinguishes between gravitational models Huterer  _et al._ (2015), we take
the point at which $f$ becomes biased as our fiducial choice of scale cut. In
the anisotropic power spectrum case this corresponds to $k_{\parallel}^{\rm
cut}\sim 0.1\ h{\rm Mpc}^{-1}$, in the tomographic case this corresponds to a
tomographic bin width of $\Delta z=0.05$ or $5$ tomographic bins covering the
range $z\in[0.5,0.75]$, and in the radial-harmonic case this corresponds to an
effective cut at $k_{\parallel}^{\rm eff,cut}\sim 0.13\ h{\rm Mpc}^{-1}$. We
do not expect the $k^{\rm cut}_{\parallel}$ in the anisotropic power spectrum
case and $k^{\rm eff,cut}_{\parallel}$ in the radial-harmonic hybrid case to
be identical. This is because the $k_{\parallel}$-$\eta$ correspondence is not
absolute since the radial-mode, although narrow in $k_{\parallel}$-space by
construction, still has some width. We have also been intentionally
conservative when transitioning from $P(k_{\parallel},k_{\perp})$ to
$C^{\eta_{a}\eta_{b}}(\ell)$ to ensure $k_{\parallel}$-modes are not double
counted. This results in a small loss of information (see II.7 for more
details).

In the second row of Fig. 9 we plot the error on the growth function,
$\sigma(f)$, as a function of the cut scale. For our fiducial choice of scale
(i.e $|b|\ \sigma<0.1$), $\sigma(f)=0.013$ for the anisotropic power spectrum,
$\sigma(f)=0.013$ for the radial-harmonic hybrid estimator and
$\sigma(f)=0.055$ for tomography. This amounts to a factor of four degradation
in constraining power when using tomography compared to the anisotropic power
spectrum, caused by mixing of $k_{\parallel}$-scales in projection. Meanwhile
the radial-harmonic power spectrum constraints are almost identical the
anisotropic case demonstrating the harmonic weight works as intended.

We plot the Fisher contours and bias for the three estimators in Fig. 10
assuming the fiducial scale cuts. The dotted line indicates the input
cosmology, while the contours are shifted by the linear model bias between the
Lorentzian and Gaussian FoG simulating the impact of using an incorrect
nonlinear RSD model. Tomography is by far the least constraining for all
parameters. The anisotropic power spectrum yield slightly tighter constrain on
$\sigma_{v}$, but the constraints are otherwise indistinguishable.

##  IV Fisher Results for $z\in[1,1.25]$

Figure 11: Same as Fig. 11, except the redshift range is now $z\in[1.,1.25]$.
The dashed line indicates the maximum scale cut where the growth function
constraints are unbiased in the sense that normalized bias is less than
$10\%$, i.e. $|b|/\sigma<0.1$. For this choice of scale cut, $\sigma(f)=0.010$
for the radial-harmonic hybrid estimator, $\sigma(f)=0.052$ for tomography and
$\sigma(f)=0.011$ for the anisotropic power spectrum.

To ensure that our method works over a broad range in redshift, we repeat our
analysis for the redshift bin $z\in[1,1.25]$. Assuming the fiducial cosmology,
at this redshift $f=0.89$. As before we assume $f_{\rm sky}=0.3$ and that the
radial window contains 7.5 million galaxies. Assuming $\sigma_{v}(z)$ scales
with the growth factor $D(z)$ Blanchard  _et al._ (2020) and using our
previous fiducial value $\sigma_{v}(z=0.675)=5.1$ implies
$\sigma_{v}(z=1.125)=4.2$. We cut perpendicular scales $k_{\perp}>0.25$, which
corresponds to $\ell>519$ in the projected case. We take $\eta^{\rm min}=3.4$.
This is intended to correspond to the same $k_{\parallel}$-mode as in the
previous section, but we have rescaled $\eta^{\rm min}$ by the fractional
change in the window width following the scaling relation in Eqn. 20.

We plot the normalized bias and the error on the growth function in Fig. 11.
As in the previous section, we take the most optimistic scale cut while
ensuring that the normalized bias on $f$ does not exceed $10\%$. For this
choice of scale cut, $\sigma(f)=0.010$ for the radial-harmonic hybrid
estimator, $\sigma(f)=0.011$ for the anisotropic power spectrum and
$\sigma(f)=0.052$ for tomography. Thus the radial-harmonic hybrid estimator
again yields an almost fourfold improvement in constraining power compared to
tomography.

##  V Conclusion

We have presented the formalism for projected anisotropic RSD spectra for an
arbitrary radial galaxy weighting in the flat-sky approximation. Writing the
projected spectra as a projection of the anisotropic power spectrum over
$k_{\parallel}$-modes weighted by a radial-mode efficiency kernel has provided
insights into the efficacy of different radial weighting schemes.

Although commonly applied to photometric survey data, we have argued that the
standard projected tomographic power spectra estimator (i.e. top-hat
weighting) has several drawbacks when applied to spectroscopic data sets in
redshift space. Narrow tomographic binning leads to a mixing of poorly modeled
small-scale radial modes with well modelled large-scale radial modes, inducing
model bias. Although this can be avoided by choosing wider tomographic bins,
even then detailed information about the structure of the anisotropic power
spectrum in $(k_{\parallel},k_{\perp})$-space is lost. This is because for
standard tomography the radial efficiency kernels are broad in
$k_{\parallel}$.

To avoid mixing between $k_{\parallel}$-scales, one should choose a weighting
which gives narrow radial-mode efficiency kernels in $k_{\parallel}$-space.
One can then also cleanly cut the parts of the data vector which are sensitive
to poorly modelled $k_{\parallel}$-modes. Inspired by the fact that the
Fourier transform of a cosine is a delta function, we have argued that a
harmonic weighting, labeled by a wavemode-$\eta$, is a natural choice. We
refer to the resulting power spectra as the radial-harmonic power spectra.

A limiting factor is that for low-$\eta$, the wavelength of the weight
function becomes significantly larger than the window. In this regime the
harmonic weighting is virtually equivalent to tomography resulting in broad
efficiency kernels in $k_{\parallel}$ and mixing of scales. Hence we advocate
using the anisotropic power spectrum to probe large parallel wavemodes and the
radial-harmonic $C(\ell)$ to probe small wavemodes, while carefully choosing
the scale cuts to ensure the covariance between the two estimators is
negligible.

We have performed a Fisher analysis for the three estimators. In the
tomographic and radial-harmonic case, scale cuts are defined by the
tomographic bin width and $\eta$, respectively. Using the Fisher bias between
a Gaussian and Lorentzian FoG as a proxy for model bias due to nonlinear RSD
modelling uncertainties we established two key results:

  * •

When we choose the most optimistic $k_{\parallel}$-scale cut for each method,
while also ensuring the model bias is small, the radial-harmonic hybrid and
the anisotropic power spectrum yield significantly tighter constraints on the
growth function, $f$, than tomography.

  * •

The radial-harmonic hybrid estimator and anisotropic power spectrum estimator
yield almost identical constraints showing that it is possible to extract
nearly all the information in projection with a suitable radial weighting.

In an upcoming paper we will extend the radial-harmonic formalism presented in
this work to include cross-correlations between weak lensing and RSD and
explore the prospects for this type of analysis using current and future data
sets.

##  VI Acknowledgements

The authors are grateful for useful conversations with Ruth Durrer, Henry
Grasshorn Gebhardt, Mona Jalivand, Tom Kitching and Jason Rhodes and to Anurag
Deshpande for providing a Fisher contour plotting routine. AP is a UK Research
and Innovation Future Leaders Fellow, grant MR/S016066/1. PLT acknowledges
support for this work from a NASA Postdoctoral Program Fellowship. Part of the
research was carried out at the Jet Propulsion Laboratory, California
Institute of Technology, under a contract with the National Aeronautics and
Space Administration. We acknowledge use of the open source software Virtanen
_et al._ (2020); Harris  _et al._ (2020); Hunter (2007).

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##  Appendix A Numerical Evaluation of the Radial-Mode Efficiency Kernels in
the Tomographic Case

From Eqn. 6 we see that to evaluate the tomographic power spectra,
$C_{ij}(\ell)$, we must integrate the product of the anisotropic power
spectrum, $P(k,\mu)$, and the efficiency kernel
$\widetilde{K}_{ij}(k_{\parallel})$. Since the power spectrum is naturally
expressed in logarithmic space, this implies that to evaluate this integral
numerically, we should evaluate $\widetilde{K}_{ij}(k_{\parallel})$ on
logarithmic intervals in $k_{\parallel}$. From Eqn. 8 this reduces to finding
the Fourier transform of the tomographic bin window, $W_{i}(r[z])$, in log-
space. We can not use the Fast Fourier Transform (FFT) as it works on a
regular grid.

Instead we approximate $W_{i}(r[z])$ as piecewise function in $z$, and
integrate each piece analytically. This is the strategy employed in Bloomfield
_et al._ (2017) to evaluate integrals over Bessel functions, and may have
further applications in cosmological settings where one must rapidly evaluate
oscillatory integrals.

We start by approximating $W_{i}(r[z])$ as a sum of $N$ non-overlapping top-
hat functions in $z$. Suppose that $W_{i}(r[z])$ is only non-zero in the range
$[z_{\rm min},z_{\rm max}]$. Defining $\Delta z=(z_{\rm max}-z_{\rm min})/N$,
$\alpha_{n}=W_{i}(r_{n})$ where $r_{n}=r[z_{\rm min}+(n+1/2)\Delta z]$, we can
write

$W_{i}\left(r[z]\right)\approx\sum_{n=0}^{N-1}\alpha_{n}T_{n}(r[z]),$ (36)

where $T_{n}(r[z])$ is a top-hat function

$T_{n}(r[z])=\begin{cases}1,&\text{if}\ r_{n}^{-}<r[z]<r_{n}^{+}\\\
0,&\text{otherwise}.\end{cases}$ (37)

Then

$\displaystyle\widetilde{W}_{i}(k_{\parallel})=\sum_{n=0}^{N-1}\alpha_{n}\int_{r[z_{\rm
min}]}^{r[z_{\rm max}]}{\rm d}r\ T_{n}\left(r[z]\right)$ (38)
$\displaystyle\times\exp\left(-ik_{\parallel}r[z]\right).$

If we then define $\Delta r_{n}=r_{n}^{+}-r_{n}^{-}$ where $r_{n}=r[z_{\rm
min}+(n+1/2)\Delta z]$, $r^{+}_{n}=r[z_{\rm min}+(n+1)\Delta z]$ and
$r^{-}_{n}=r[z_{\rm min}+n\Delta z]$ we can integrate Eqn. 38 analytically
Grasshorn Gebhardt and Jeong (2020), so that

$\widetilde{W}_{i}(k_{\parallel})=\sum_{n=0}^{N-1}\alpha_{n}\Delta
r_{n}e^{-ik_{\parallel}r_{n}}{\rm sinc}\left(\frac{k_{\parallel}\Delta
r_{n}}{2}\right).$ (39)

Hence,

$\displaystyle\widetilde{K}_{ij}(k_{\parallel})=\sum_{n,n^{\prime}}\alpha_{n}\alpha_{n^{\prime}}\Delta
r_{n}\Delta r_{n^{\prime}}\cos(k_{\parallel}[r_{n}-r_{n^{\prime}}])$ (40)
$\displaystyle\times{\rm sinc}\left(\frac{k_{\parallel}\Delta
r_{n}}{2}\right){\rm sinc}\left(\frac{k_{\parallel}\Delta
r_{n^{\prime}}}{2}\right).$

##  Appendix B Numerical Evaluation of the Radial-Mode Efficiency Kernels in
the Radial-Harmonic Case

To evaluate the efficiency kernels in the radial-harmonic case we use the same
strategy as in the tomographic case. As in Eqn. 36 we write

$W(\eta,r[z])\approx\sum_{n=0}^{N-1}\alpha_{n}T_{n}(r[z])\cos\left(\frac{2\pi\eta
r^{\rm ref}[z]}{\Delta r^{\rm ref}[z]}\right),$ (41)

Hence,

$\displaystyle\widetilde{W}(\eta,k_{\parallel})=\sum_{n=0}^{N-1}\alpha_{n}\int_{r[z_{\rm
min}]}^{r[z_{\rm max}]}{\rm d}r\ T_{n}\left(r[z]\right)$ (42)
$\displaystyle\times\cos\left(\frac{2\pi\eta r^{\rm ref}[z]}{\Delta r^{\rm
ref}[z]}\right)\exp\left(-ik_{\parallel}r[z]\right).$

Now we must write the weight, which is written as a function of the comoving
distance in the reference cosmology, as a function of the comoving distance in
which we are trying to evaluate the efficiency kernels. Assuming, $\Delta z$,
is small we can assume that the comoving distances in the two cosmologies are
linearly related inside each piecewise step, $n$, so that

$r^{\rm ref}[z]=a_{n}r[z]+b_{n},$ (43)

To solve for $a_{n}$ and $b_{n}$ we first assume than inside each piecewise
step the comoving distance is a linear function of redshift

$\displaystyle r[z]$ $\displaystyle=\nu_{n}z+\beta_{n},$ (44) $\displaystyle
r^{\rm ref}[z]$ $\displaystyle=\nu^{\rm ref}_{n}z+\beta_{n}^{\rm ref}.$

A good choice is to take

$\displaystyle\nu_{n}$ $\displaystyle=\frac{r_{n}^{+}-r_{n}^{-}}{\Delta z},$
(45) $\displaystyle\nu_{n}^{\rm ref}$ $\displaystyle=\frac{r_{n}^{+,{\rm
ref}}-r_{n}^{-,{\rm ref}}}{\Delta z}$

and

$\displaystyle\beta_{n}$ $\displaystyle=r_{n},$ (46)
$\displaystyle\beta_{n}^{\rm ref}$ $\displaystyle=r_{n}^{\rm ref},$

where we define $r_{n}$ etc. as in Section A. From Eqns. 43-46 it follows that

$\displaystyle a_{n}$ $\displaystyle=\frac{\nu_{n}^{\rm ref}}{\nu_{n}}$ (47)
$\displaystyle b_{n}$ $\displaystyle=\beta_{n}^{\rm
ref}-\beta_{n}\left(\frac{\nu_{n}^{\rm ref}}{\nu_{n}}\right).$

Hence we can write

$\cos\left(\frac{2\pi mr^{\rm ref}[z]}{\Delta r^{\rm
ref}[z]}\right)=\cos(c_{nm}r[z]+d_{nm})$ (48)

where

$c_{nm}=\frac{2\pi m}{\Delta r},\ d_{nm}=\frac{2\pi mb_{n}}{a_{n}\Delta r}.$
(49)

Then using the fact

$\displaystyle\int^{b}_{a}{\rm d}x\
\cos(cx+d)e^{-ikx}=R[k;a,b,c,d]+iI[k;a,b,c,d]$ (50)

where

$\displaystyle R[k;a,b,c,d]$ $\displaystyle=R2[k;a,c,d]-R1[k;a,c,d]$ (51)
$\displaystyle+R1[k;b,c,d]-R2[k;b,c,d]$ $\displaystyle I[k;a,b,c,d]$
$\displaystyle=I1[k;a,c,d]+I2[k;a,c,d]$
$\displaystyle-I1[k;b,c,d]-I2[k;b,c,d]$

and

$\displaystyle R1[k;r,c,d]$
$\displaystyle=\frac{c\cos(rk)\sin(rc+d)}{c^{2}-k^{2}},$ (52) $\displaystyle
R2[k;r,c,d]$ $\displaystyle=\frac{k\sin(rk)\cos(rc+d)}{c^{2}-k^{2}},$
$\displaystyle I1[k;r,c,d]$
$\displaystyle=\frac{c\sin(rk)\sin(rc+d)}{c^{2}-k^{2}},$ $\displaystyle
I2[k;r,c,d]$ $\displaystyle=\frac{k\cos(rk)\cos(rc+d)}{c^{2}-k^{2}}$

we can evaluate each piece of Eqn. 42 analytically as in the tomographic case.
We find

$\displaystyle\widetilde{K}(k_{\parallel};\eta_{a},\eta_{b})=\sum_{n,n^{\prime}}\alpha_{n}\alpha_{n^{\prime}}$
(53)
$\displaystyle\bigg{(}R[k_{\parallel};r^{-}_{n},r^{+}_{n},c_{n\eta_{a}},d_{n\eta_{a}}]R[k_{\parallel};r^{-}_{n^{\prime}},r^{+}_{n^{\prime}},c_{n^{\prime}\eta_{b}},d_{n^{\prime}\eta_{b}}]$
$\displaystyle+I[k_{\parallel};r^{-}_{n},r^{+}_{n}c_{n\eta_{a}},d_{n\eta_{a}}]I[k_{\parallel};r^{-}_{n^{\prime}},r^{+}_{n^{\prime}}c_{n^{\prime}\eta_{b}},d_{n^{\prime}\eta_{b}}]\bigg{)}.$

##  Appendix C Power Spectrum Estimators in the Radial-Harmonic Basis

The radial-harmonic $C(\ell)$ can be computed from a spectroscopic galaxy
catalogue as follows:

  * •

For each $\eta$, compute a pixelized map $m^{\eta}(\boldsymbol{\theta}_{P})$.
For some pixel, $P$, at an angle $\boldsymbol{\theta}_{P}$ on the sky, the map
is defined as a weighted sum of all galaxies, $g$, inside the pixel. That is

$m^{\eta}(\boldsymbol{\theta}_{P})=\sum_{g\in\boldsymbol{\theta_{P}}}w(\eta,z_{g}),$
(54)

where $z_{g}$ is the redshift of galaxy $g$ and $w(\eta,z_{g})$ is the
harmonic weight defined in Eqn. 19 assuming a fiducial cosmology to convert
redshifts to distances.

  * •

Use the map to compute the spherical harmonic coefficients $a_{\ell m}(\eta)$
using e.g. Healpix Gorski  _et al._ (2005).

  * •

A good estimate of the pseudo-spectrum (including shot-noise contributions) is
then

$\widetilde{C}^{\eta_{a}\eta_{b}}(\ell)=\frac{w_{\ell}^{-2}}{2\ell+1}\sum_{\ell=-m}^{m}a_{\ell
m}(\eta_{a})a^{*}_{\ell m}(\eta_{b}),$ (55)

where $w_{\ell}$ is the pixel window function which accounts for the
suppression of power on small scales due to pixelization Leistedt  _et al._
(2013).

  * •

A key feature of the harmonic weight presented in this work is that it is not
coupled to angular scales, so that we can use the pseudo-$C(\ell)$ method
Wandelt  _et al._ (2001); Alonso  _et al._ (2019) to deconvolve the angular
survey mask and estimate $C^{\eta_{a}\eta_{b}}(\ell)$.

##  Appendix D Configuration Space Radial-Harmonic Formalism and Estimators

Since the radial-harmonic weighting is independent of the angular wavemode, it
is possible to compute the radial-harmonic two-point correlations function in
configuration space. This sidesteps the need to deconvolve the survey mask
which can lead to biases and information loss. If we assume the coupling
between $\ell$-modes due to the RSD anisotropy is weak111111This approximation
must be tested in a follow-up study., we can write the correlation function as
a function of the angular power spectrum as in the isotropic case. Then in the
flat-sky approximation, the radial-harmonic correlation function,
$w^{\eta_{a}\eta_{b}}(\theta)$, is given by Abbott  _et al._ (2018)

$w^{\eta_{a}\eta_{b}}(\theta)=\frac{1}{2\pi}\int{\rm d}\ell\ \ell
J_{0}(\ell\theta)C^{\eta_{a}\eta_{b}}(\ell),$ (56)

where $\theta$ is the angular separation between pairs of galaxies and $J_{0}$
is the zeroth-order Bessel function. After applying the radial-harmonic weight
to the galaxy catalogue, the data vector can be computed using a public code
such as TreeCorr121212https://github.com/rmjarvis/TreeCorr Jarvis  _et al._
(2004).

##  Appendix E The Likelihood of the Radial-Harmonic Power Spectrum and Hybrid
Estimators

In order to perform a radial-harmonic likelihood analysis, we must know the
functional form of the likelihood. In the absence of a mask and assuming that
the coupling of $m$-modes due to the RSD anisotropy is weak, each spherical-
harmonic coefficient, $a_{\ell m}(\eta_{a})$, in the radial-harmonic basis is
statistically independent. From Eqn. 55, the radial-harmonic power spectrum is
a sum over independent, identically distributed random variables, so from the
central limit theorem, we expect the likelihood to be Gaussian. For the same
reason, binning data vector into bandpowers further Gaussianizes the
likelihood.

This Gaussian likelihood approximation must be explicitly checked in a follow-
up study. This can be done by performing a mock likelihood analysis using
simulated data and comparing to a likelihood-free (e.g. Alsing  _et al._
(2019); Akeret  _et al._ (2015)) approach as in Taylor  _et al._ (2019).

Meanwhile the likelihood of the anisotropic power spectrum
$P(k_{\parallel},k_{\perp})$ is effectively Gaussian except on extremely large
scales where the non-Gaussianity can be handled with existing methods Wang
_et al._ (2019).