# A new analysis of galaxy 2-point functions in the BOSS survey, including
full-shape information and post-reconstruction BAO

Shi-Fan Chen  Zvonimir Vlah  Martin White

###### Abstract

We present a new method for consistent, joint analysis of the pre- and post-
reconstruction galaxy two-point functions of the BOSS survey. The post-
reconstruction correlation function is used to accurately measure the
distance-redshift relation and expansion history, while the pre-reconstruction
power spectrum multipoles constrain the broad-band shape and the rate-of-
growth of large-scale structure. Our technique uses Lagrangian perturbation
theory to self-consistently work at the level of two-point functions, i.e.
directly with the measured data, without approximating the constraints with
summary statistics normalized by the drag scale. Combining galaxies across the
full redshift range and both hemispheres we constrain $\Omega_{m}=0.303\pm
0.0082$, $H_{0}=69.23\pm 0.77$ and $\sigma_{8}=0.733\pm 0.047$ within the
context of $\Lambda$CDM. These constraints are consistent both with the Planck
primary CMB anisotropy data and recent cosmic shear surveys.

##  1 Introduction

The large-scale structure of the Universe, as traced by galaxies, provides
fundamental physics information through its connection to both initial
conditions in the primordial universe and general relativity through the
gravitational formation of structures on the largest observable scales [1, 2].
A well-established measure of this structure is the redshift-space two-point
function as measured by galaxy redshift surveys [3, 4], which encodes both the
power spectrum shape of fluctuations in the early universe and, through the
quirk that line-of-sight distances in such surveys are inferred from their
redshifts, cosmological velocities in the form of redshift-space distortions
(RSD).

A particularly interesting and relevant interplay of the initial conditions
and gravitational dynamics occurs in the galaxy baryon acoustic oscillations
(BAO) signal. The BAO signal is the imprint of early-universe acoustic waves
on the observed clustering of galaxies, manifesting as a localized peak in the
galaxy correlation function at separations around the characteristic size of
these waves [5]. The linear physics underlying the size and shape of the BAO
feature is well-understood [6, 7], making it a robust cosmological signal of
both the early universe and the redshift-distance relation. However, the
process of nonlinear structure formation leads to a slight wrinkle in this
picture: nonlinearities, particular due to bulk displacements of galaxies on
large scales, tend to dampen and shift the BAO signal [8, 9]. To better
extract the BAO signal, ref. [10] proposed a now-standard method known as
“reconstruction” to cancel a large portion of these effects by estimating
large-scale displacements and subtracting them from the observed positions of
galaxies.

In recent years there have been significant advances in the modeling and
theoretical understanding of both redshift-space distortions and the nonlinear
damping of BAO. For the former, the recasting of cosmological perturbation
theory in the language of effective field theories has provided a systematic
way in which to write down the possible contributions to galaxy clustering on
quasilinear scales based on fundamental symmetries while clarifying and taming
dependences on small-scale physics (see e.g. refs [11, 12, 13, 14, 15, 16, 17,
18]). For the latter, improved understanding of the effects of large-scale
displacements through so-called infrared (IR) resummations allows us to
quantitatively describe nonlinear damping of the BAO peak, both for the raw
and reconstructed power spectrum, in perturbation theory [19, 20, 21, 22, 23].
These developments allow us to model diverse sets of cosmological observables—
in real and redshift space, power spectrum and correlation function, pre- and
post-reconstruction— within the consistent theoretical framework of
perturbation theory.

Our aim in this paper is to present a consistent analysis of the pre- and
post-reconstruction 2-point correlation function in Fourier and configuration
space using the BOSS survey [24] as an example. We will work within the
framework of Lagrangian perturbation theory [25], which the present authors
have used to develop models for redshift-space distortions and reconstruction
[23, 18]. We will operate directly at the level of two-point correlation
functions, i.e. given a set of cosmological and galaxy bias parameters
$\Theta$, we will use the redshift-space power spectra and correlation
function multipoles, pre- or post-reconstruction, to construct a likelihood:

$\mathcal{L}\propto\exp\Big{\\{}-\frac{1}{2}\big{(}m(\Theta)-\hat{d}\big{)}^{T}C^{-1}\big{(}m(\Theta)-\hat{d}\big{)}\Big{\\}}\quad,\quad\hat{d}=(P_{\ell},\xi^{\rm
recon}_{\ell},\cdots),$ (1.1)

where $m(\Theta)$ is the model, $\hat{d}$ is a data vector composed of two-
point correlation functions measured from data and $C$ is the covariance
matrix.

This work is not the first to analyze galaxy clustering with an effective
theory framework, and follows a number of papers analyzing the BOSS redshift
space power spectrum using effective Eulerian perturbation theory [26, 27] as
well as work by refs. [28, 29] combining these analyses with additional BAO
information through reconstruction. Our goal, rather, is to perform a joint
analysis of pre- and post-reconstruction data without resorting to additional
assumptions or machinery (Fig. 1). In particular, previous work combining pre-
and post-reconstruction measurements (e.g. refs [30, 28, 29]) have typically
sought to distill the content of the latter by fitting a set of BAO parameters
$\tilde{\alpha}_{\parallel,\perp}$ (§ 3.2) which scale the BAO signal in a
template, or fiducial, linear power spectrum to match that of the observed
signal. Then, since the BAO oscillations in the template differ from the
observed by the cosmological dependence of the sound horizon ($r_{d}$) in
addition to simple distance scalings by redshift, these best-fit
$\tilde{\alpha}$’s are fit to the ratio of cosmological distances to $r_{d}$
in conjunction to the pre-reconstruction power spectra, such that the data
vector is instead
$\hat{d}=(P_{\ell},\tilde{\alpha}_{\parallel},\tilde{\alpha}_{\perp})$. The
covariance of the power spectra and $\tilde{\alpha}$’s is then inferred from
measurements of $\tilde{\alpha}$ from the power spectra in mock catalogs. A
schematic comparing our approach in this work to the standard one is shown in
Fig. 1. For a given set of cosmological parameters the information about
$r_{d}$ and cosmological distances is inherent in the perturbation-theory
prediction $\xi^{\rm recon}_{\ell}(\Theta)$, allowing us to bypass the need to
approximate the BAO signal as a scaled version of a fixed power spectrum
template and directly compare cosmology with data. In particular, since the
BAO information in the correlation function is effectively isolated in
configuration space as a sharp peak at large scales, our fiducial setup will
combine the post-reconstruction correlation function around the peak with a
full-shape analysis of the pre-reconstruction power spectrum. Note that while
we have focused our discussion on the post-reconstruction BAO measurement, the
BOSS collaboration (but not refs. [28, 29]) similarly also distilled the RSD
signal pre-reconstruction into a best-fit $f\sigma_{8}$, such that the final
data vector fit to cosmological parameters was
$\hat{d}=(f\sigma_{8},\tilde{\alpha}_{\parallel},\tilde{\alpha}_{\perp}$).

The simplified approach advocated for in this work has a number of advantages
for obtaining cosmological constraints from surveys like BOSS. For a given set
of theory parameters (including cosmology and galaxy bias) there is a unique
‘forward’ mapping — shown in the top row of Fig. 1 — from these parameters to
the (model-independent) observational data. This implies that constraining
information in the data that is implied by the theory model is captured
without loss. By comparison, in the standard approach one assumes e.g. that
any BAO information post-reconstruction can be distilled into two BAO scaling
parameters which, by themselves, do not uniquely map into the space of post-
reconstruction observables, potentially leading to information loss (§3).
Furthermore, while the standard approach requires measuring the covariance of
these summary statistics from approximate mock catalogs, computing the
likelihood at the level of model-independent data in principle allows us to
straightforwardly use the statistical uncertainties implied by the theory
model itself, e.g. by computing covariance matrices analytically within
perturbation theory [31]111See, however, ref. [28] for an idealized
calculation using Fisher matrices.. Future theory calculations of noise-free
data covariances have the potential to ease numerical difficulties from
estimating covariances of statistics beyond the pre-reconstruction power
spectrum with only a finite number of mocks.

Figure 1: Flowchart comparing our method with the standard approach to
combining full-shape RSD and BAO analyses. Arrows denote the unique mapping
from theory parameters to observed data. Our approach allows for a direct
translation from cosmological parameters into measured 2-point correlation
functions (black dashed box) via a theory of structure formation (LPT) and
does not rely on power spectrum templates or model-dependent BAO parameters
derived therefrom. Squares highlighted in red indicate the actual data vector
fit in the likelihoods of each approach, related to cosmological parameters
through a model $m(\Theta)$ (red arrows), and each row to the right of
“cosmology” indicates separate fits which must be combined using simulated
mocks. Our approach features a single fit to the observed data while the
standard approach separately fits theory-dependent BAO parameters that depend
highly non-linearly on the data and the pre-reconstruction clustering.

The outline of the paper is as follows. In §2 we present the galaxy samples
that we analyze, which are all drawn from the BOSS survey [24]. The models we
fit to these data, all based on cosmological perturbation theory, are
described in §3. Our fiducial analysis setup, including scale cuts, parameter
choices and priors are presented in §4. Our final cosmoloical constraints are
given in §5, where we also discuss constraints from different subsamples of
the BOSS data and compare to constraints from other groups and experiments. We
conclude in §6. Some technical details are relegated to a series of
Appendices.

##  2 Data

Figure 2: The pre-reconstruction power spectrum (top) and pre- and post-
reconstruction correlation functions (bottom) of the BOSS DR12 galaxies. The
monopole and quadrupole of each are both shown in blue and orange,
respectively. For the power spectrum, the separate measurements for the NGC
and SGC samples are shown as filled and open circles. Pre- and post-
reconstruction (“Raw” and “Rec”) correlation function measurements are shown
with round and crossed markers. The correlation function is measured jointly
across both galactic caps. Error bars represent the diagonals of the
covariance matrix computed using 1000 Patchy mocks.

We analyze the clustering of galaxies drawn from the BOSS galaxy redshift
survey [24], part of the Sloan Digital Sky Survey III [32]. Galaxies in BOSS
were targeted with two independent selection criteria: one (LOWZ) targeted
luminous red galaxies up to $z=0.4$ while another (CMASS) targeted massive
galaxies with $0.4<z<0.7$; however, due to an incorrect application of the
LOWZ criteria in the first nine months of the survey, two additional samples
(LOWZE2, LOWZE3) had to be separated out. All of these samples are described
in more detail in ref. [33]. Ref. [34] combined these samples into three
redshift bins with $0.2<z<0.5$, $0.4<z<0.6$ and $0.5<z<0.75$ (named z1, z2 and
z3, respectively). Since z2 overlaps both z1 or z3 in redshift and thus gives
correlated constraints we choose to analyze z1 and z3 only in this work. Each
redshift bin can be further split into galaxies observed in the Northern (NGC)
and Southern (SGC) galactic caps. Because the imaging in the north and the
south differ slightly, the samples have slightly different properties and
should be analyzed separately. The final BOSS sample covers 1,198,006 galaxies
in total over 10,252 square degrees of sky.

The power spectra and correlation function multipoles of these samples were
measured in refs. [30, 35] and the data are shown in Fig. 2. The BOSS two-
point function measurements were computed assuming a flat $\Lambda$CDM
cosmology with present-day matter density $\Omega_{M,\rm fid}=0.31$. This
implies that the reported redshift-space power spectrum is related to its
value in the coordinates of the true cosmology by

$P^{\rm obs}_{s}(\textbf{k}_{\rm
obs})=\alpha_{\perp}^{-2}\alpha_{\parallel}^{-1}P_{s}(\textbf{k})\quad,\quad
k^{\rm
obs}_{\parallel,\perp}=\alpha_{\parallel,\perp}k_{\parallel,\perp}\quad,$

where the Alcock-Paczynski parameters are defined as [36, 37]

$\alpha_{\parallel}=\frac{H^{\rm
fid}(z)}{H(z)}\quad,\quad\alpha_{\perp}=\frac{D_{A}(z)}{D^{\rm
fid}_{A}(z)}\quad.$ (2.1)

Since all distances are reported in $h^{-1}$Mpc units the above ratios should
be computed assuming a fixed $h$. The equivalent relations for the correlation
function are simply the Fourier transforms of the above equations. The
mismatch between true and fiducial coordinates imprints additional anisotropy
in the galaxy two-point function and serves as a further source of
cosmological information known as the Alcock-Paczynski (AP) effect [36]
(Appendix C).

For the power spectrum we use the updated pre-reconstruction measurements of
both data and mock catalogs presented in ref. [38]. On top of the AP effect
the geometry of the survey itself leaves an imprint in the clustering of the
galaxies, so that the measured power spectra are the convolution of the true
clustering signal with a window function and taking into account wide-angle
effects [39, 40, 41, 42]; to this end ref. [38] provide for each sample a wide
angle matrix M and window function matrix W such that for an input vector of
appropriately-binned theoretical (including AP) power spectrum multipoles P
the observed (binned) power spectra are given by222We use the updated values
of the power spectrum multipoles and window functions from a revised version
of ref. [38]. These measurements fix a mismatch in the normalizations of the
window function and power spectra at the roughly $10\%$ level that afflicted
earlier results and which improves the agreement with correlation function
fits which do not require multiplying by a window function. This normalization
issue is discussed in detail in the published version of ref. [38], with
resulting amplitude corrections to each BOSS sample tabulated in Table 1 of
that work; roughly, there is an overall degeneracy between the power spectrum
and window function amplitudes, such that no results are affected when both
are adjusted simultaneously by a single multiplicative factor–however, this is
premised upon the two being normalized consistently when computed. We thank
Pat McDonald and Florian Beutler for helpful discussions of this issue. Upated
versions of the BOSS power spectra and window functions can be found in
https://fbeutler.github.io/hub/deconv_paper.html.

$\textbf{P}^{\rm
conv}=\textbf{W}\textbf{M}\textbf{P},\quad\textbf{P}=(P_{0},P_{2},P_{4}).$

While the output $\textbf{P}^{\rm conv}$ contains both the hexadecapole as
well as odd multipoles $P_{1,3}$ we will restrict our analysis in this paper
to the monopole and quadrupole only.

For our post-reconstruction BAO analysis we use the post-reconstruction
correlation function multipoles obtained in ref. [35]. Both pre- and post-
reconstruction correlation function multipoles were measured for the combined
NGC and SGC samples at the redshift bins z1 and z3. The reconstruction in the
public BOSS data was performed using the so-called RecIso convention, with
reconstructed displacements solved-for using a finite-difference approach on
the observed galaxy density field smoothed by a Gaussian filter with width
$R=15h^{-1}$ Mpc. We will discuss further details of the procedure in Section
3.2.

Figure 3: The joint correlation matrix of the z1 and z3 pre- and post-
reconstruction two-point function samples, computed using 1000 Patchy mocks.
For clarity of presentation we have restricted the power spectrum to
wavenumbers $0.02\,h\,{\rm Mpc}^{-1}<k<0.20\,h\,{\rm Mpc}^{-1}$ (18 bins) and
the correlation function to $80\,h^{-1}\text{Mpc}<r<130\,h^{-1}\text{Mpc}$ (20
bins). The numbers on the $x$ axis denote bin number.

Finally, to obtain the joint covariances of the power spectrum333For the power
spectrum mocks, see: https://fbeutler.github.io/hub/deconv_paper.html. and
correlation function444We thank Mariana Vargas for providing the correlation
function mock measurements. measurements used in our analysis we used the V6C
BigMultiDark Patchy mocks [43] released with DR12 of the SDSS-III Survey.
These mocks were prepared using approximate gravity solvers with galaxy
biasing calibrated to the BigMultiDark simulation in a redshift-dependent way
to capture the time-depedence of the BOSS galaxy sample [43]. The thus-derived
correlation matrix for the power spectrum and post-reconstruction correlation
function is shown in Fig. 3. Since the correlation-function mock measurements
were only obtained for 1000 of these mocks that is also the number of power
spectrum measurments we use to obtain the joint pre- and post-reconstruction
covariance. The thousand mocks are sufficient for our purposes; concretely,
our most extensive analysis setup will include (for each independent redshift
bin) $18$ $k$-bins per power spectrum multipole per galactic cap and $10$
radial bins per correlation function multipole, yielding a $92$-element data
vector. Applying a multiplicative correction to unbias the precision matrix
[44] would lead to rescalings of the parameter errors of less than $5\%$, with
no change in the matrix structure. As a further test, reducing to $N_{\rm
mocks}=500$ changed the $\chi^{2}$ of the best-fit model found for the z3
sample by $\Delta\chi^{2}=3.5$, i.e. far less than one per d.o.f. for our
fiducial setup (§ 4.1), suggesting that one thousand mocks measurements is
sufficient for our purposes. Finally, while we have not found it necessary in
our analysis, we note that if we were to rebin the power spectrum data into
broader $k$ bins we would reduce the number of degrees of freedom and more
cleanly separate the BAO and broadband shape information between the
correlation function and power spectrum. This may be beneficial in future
analyses.

##  3 Model

Our aim in this work is to jointly model the redshift-space galaxy two-point
function both pre- and post-reconstruction within a consistent theoretical
framework. Specifically, we will operate within Lagrangian perturbation theory
(LPT; [25, 19]), which models nonlinear structure formation through the
evolution of galaxy displacements $\Psi(\textbf{q},\tau)$, relating observed
(Eulerian) positions x at a given time $\tau$ to initial (Lagrangian)
positions q through $\textbf{x}=\textbf{q}+\Psi$. The displacements are
expanded order-by-order in the initial conditions
$\Psi=\Psi^{(1)}+\Psi^{(2)}+\Psi^{(3)}+...$. In this work we will operate
within the EdS approximation wherein the nth order displacement scales as the
nth power of the linear growth factor $D(z)$; this has been shown to be an
excellent approximation for current and upcoming galaxy surveys in [45, 46,
47, 48, 49], with a slight caveat due to scale dependence from massless
neutrinos which we will address in Section 4. The nature of the mapping
between Lagrangian and Eulerian coordinates makes LPT a natural arena in which
to understand both redshift-space distortions and nonlinear BAO damping, which
we discuss in term below. In addition, in order to speed up our calculations
and avoid repeatedly calling Boltzmann codes at each new point in our Markov
chains, in this paper we have chosen to approximate our theory components as a
Taylor series in the cosmological parameters. The grid of PT predictions used
to compute the Taylor-series coefficients was computed using CLASS [50] and
velocileptors [17, 18]. The details of our approximation scheme are discussed
in Appendix A.

###  3.1 One-Loop Redshift Space Power Spectrum

In spectroscopic galaxy surveys like BOSS a galaxy’s position along the line
of sight (LOS) is inferred from its measured redshift. Since a galaxy’s
cosmological redshift and peculiar velocities both contribute to this
redshift, its redshift-space position s is boosted along the LOS by its
peculiar LOS velocity in the appropriate units, that is
$\textbf{s}=\textbf{x}+\textbf{u}$, where
$\textbf{u}=(\hat{n}\cdot\textbf{v})\hat{n}/\mathcal{H}$ and $\mathcal{H}$ is
the conformal Hubble parameter [3]. Within LPT this is equivalent to boosting
the displacement $\Psi$ by the LOS component of its (appropriately normalized)
time derivative, i.e. $\Psi_{s}=\Psi+\dot{\Psi}$ [19, 18]; conveniently, this
boost can be recast as a coordinate transformation

$\Psi^{(n)}_{s,i}=\Psi^{(n)}_{i}+nf\hat{n}_{i}\hat{n}_{j}\Psi^{(n)}_{j}\equiv
R^{(n)}_{ij}\Psi^{(n)}_{j}\quad.$ (3.1)

For the remainder of this paper we will in addition make the plane-parallel
approximation that the LOS vector $\hat{n}$ is a constant independent of
position.

The translation between densities and displacements follows from number
conservation. Assuming galaxies are sampled from the initial conditions
$\delta_{0}$ according to some functional
$\rho_{g}(\textbf{q})=F[\delta_{0}(\textbf{q})]$ this implies that the present
day density satisfies
$\rho_{g}(\textbf{x})d^{3}\textbf{x}=\rho_{g}(\textbf{q})d^{3}\textbf{q}$ or,
in Fourier space, [19, 51, 52, 18]

$1+\delta_{g}(\textbf{k})=\int d^{3}\textbf{q}\
e^{i\textbf{k}\cdot(\textbf{q}+\Psi)}F(\textbf{q})\quad.$ (3.2)

The equivalent equation in redshift space follows by substituting $\Psi$ for
$\Psi_{s}$. The bias functional $F(\textbf{q})$ can be perturbatively expanded
as

$F(\textbf{q})=b_{1}\delta_{0}(\textbf{q})+\frac{1}{2}b_{2}(\delta_{0}(\textbf{q})^{2}-\left\langle\delta_{0}^{2}\right\rangle)+b_{s}(s^{2}_{0}(\textbf{q})-\left\langle
s^{2}\right\rangle)$ (3.3)

where
$s^{2}_{0}=(\partial_{i}\partial_{j}/\partial^{2}-\delta_{ij}/3)\delta_{0}$ is
the shear tensor. For a complete accounting of terms relevant to the one-loop
power spectrum we in principle have to also account for third order bias
operators; however, since we expect them to be small for all but the most
massive halos [53, 26, 17], and highly degenerate with the effective-theory
corrections discussed below, we will set them to zero for the rest of this
work.

The power spectrum in redshift space is then given by [19, 18]

$P_{s}(\textbf{k})=\int d^{3}\textbf{q}\ \left\langle
e^{i\textbf{k}\cdot(\textbf{q}+\Delta_{s})}F(\textbf{q}_{1})F(\textbf{q}_{2})\right\rangle_{\textbf{q}=\textbf{q}_{1}-\textbf{q}_{2}},$
(3.4)

where we have defined the pairwise displacement in redshift space
$\Delta_{s}=\Psi_{s}(\textbf{q}_{1})-\Psi_{s}(\textbf{q}_{2})$. In the case of
matter ($F=1$) within first-order LPT (Zeldovich approximatin) Equation 3.4
can be evaluated exactly to give $P_{\rm Zel}=\int
d^{3}\textbf{q}\exp[ik_{i}q_{i}-k_{i}k_{j}A_{ij}/2]$ where we have defined the
second cumulant of pairwise displacements
$A_{ij}=\left\langle\Delta_{i}\Delta_{j}\right\rangle$. The exponentiation of
the pairwise displacement in Equation 3.4 is highly significant and allows LPT
to capture the nonlinear damping of BAO due to the large-scale (bulk)
displacements [54, 14, 55, 51, 56]. In practice we keep only these long-
wavelength displacements exponentiated and perturbatively expand those above a
certain wavenumber ($k_{\rm IR}=0.2\,h\,{\rm Mpc}^{-1}$) perturbatively; for
further details we refer interested readers to ref. [18], which also provides
a detailed exposition of the various terms implied in Equation 3.4 as well as
relevant numerical methods. Finally, as an effective perturbation theory LPT
requires a number of counterterms to properly tame its sensitivity to small-
scale (UV) physics; in this work we adopt the parametrization of ref. [17] and
write

$P_{s}(\textbf{k})=P_{s}^{\rm
PT}(\textbf{k})+(\alpha_{0}+\alpha_{2}\mu^{2})k^{2}P_{\rm
Zel}(\textbf{k})+R_{h}^{3}(1+\sigma^{2}k^{2}\mu^{2})$ (3.5)

where $P_{\rm Zel}$ is the Zeldovich matter power spectrum. Note that while
the above parametrization is not exhaustive (i.e. we should in principle
include terms like $\alpha_{4}\mu^{4}$), this parameter set has been tested
extensively against simulations (e.g. refs. [23, 17, 57, 18]) and the
neglected terms are extremeley degenerate with those listed when fitting the
monopole and quadrupole only. The above parameter set can also be used to fit
the pre-reconstruction correlation function, which can be obtained directly by
Fourier transforming the theory prediction for the power spectrum.

###  3.2 Nonlinear BAO Damping Post Reconstruction

It is well known that nonlinear structure formation smooths out the BAO peak
in the galaxy two-point function [58, 8, 19, 9, 59, 60], reducing its
prominence and as a result also the signal-to-noise of BAO measurements.
Within LPT this phenomenon can be understood by looking at contributions to
Equation 3.4 such as [57]

$P_{\rm BAO}(\textbf{k})\sim\int d^{3}\textbf{q}\
e^{i\textbf{k}\cdot\textbf{q}-\frac{1}{2}k_{i}k_{j}A_{ij}(\textbf{q})}\
\xi_{\rm BAO}(\textbf{q})\approx e^{-\frac{1}{2}k^{2}\Sigma^{2}}\int
d^{3}\textbf{q}\ e^{i\textbf{k}\cdot\textbf{q}}\ \xi_{\rm BAO}(\textbf{q}).$

Since the BAO peak is well-localized at the sound horizon at the drag epoch
$r_{d}$, it acts to pick out a particular scale at which to evaluate $A_{ij}$,
from which a specific damping scale $\Sigma^{2}=\left\langle
A_{ij}\right\rangle_{|\textbf{q}|=r_{d}}$, leading to Gaussian damping of the
BAO. Within $\Lambda$CDM most of the displacement power comes from relatively
low wavenumbers [8].

The purpose of standard reconstruction [10] is to sharpen the BAO feature by
undoing some of this damping. To do so, one smooths the observed galaxy
density field using a Gaussian filter $S(k)$ on a sufficiently large scale
that the Kaiser formula $\delta_{g}(\textbf{k})=(b+f\mu^{2})\delta_{m}$ is a
good approximation, uses the smoothed field to solve for the (smoothed) linear
Zeldovich displacement $S(k)\Psi_{\rm
Zel}(\textbf{k})=i\textbf{k}/k^{2}S(k)\delta_{m}$, and subtracts these
displacements from the observed galaxy positions. To preserve power on large
scales a random catalog is shifted using the same displacements, with the
difference between the displaced galaxies ($d$) and shifted randoms ($s$)
constituting the full reconstructed density field $\delta_{\rm
rec}=\delta_{d}-\delta_{s}$. The presence of redshift space distortions
presents a slight complication and there is no general settled-upon convention
in the literature: the BOSS data analyzed in this work followed the so-called
RecIso convention wherein the galaxies were displaced with RSD (that is, the
reconstructed displacement multiplied by $R^{(1)}_{ij}$) while the randoms
were shifted without RSD [61].

The residual damping left in the BAO signal after reconstruction can be
modeled in the same way as was the damping pre-reconstruction. Specifically,
we can think of the displaced galaxies ($d$) and shifted randoms ($s$) as two
additional tracers with displacements [59, 60, 62, 23]

$\Psi^{d}_{i}=R^{(1)}_{ij}(1-\mathcal{S})\ast\Psi^{\rm
Zel}_{j}\quad,\quad\Psi^{s}_{i}=-\mathcal{S}\ast\Psi^{\rm Zel}_{i}.$ (3.6)

The two-point statistics of these displacements can then be computed at the
sound horizon $r_{d}$ as in the pre-reconstruction case and used to damp the
BAO feature. Specifically, for a given linear power spectrum $P_{\rm lin}$ we
can decompose it into a “wiggle” component with the BAO feaure and a smooth
component without, i.e. $P_{\rm lin}=P_{w}+P_{nw}$. The prediction for each
(cross) spectrum between $d$, $s$ is then

$P^{ab}(\textbf{k})=K^{ab}(k,\mu)\left[P_{w}(k)e^{-k^{2}\Sigma^{ab}(\mu)/2}+P_{nw}(k)\right]$
(3.7)

where $K^{ab}(k,\mu)$ is a linear-theory function of the growth rate $f(z)$
and linear bias $b_{1}$ taking into account Kaiser infall and the smoothing
filter.

Equation 3.7 captures (resums) the nonperturbative effect of bulk
displacements on linear-theory BAO wiggles; in principle, nonlinear effects
such as mode coupling can induce further modifications to the BAO feature. A
full treatment of these additional effects for galaxies at one-loop order is
complex and beyond the scope of this work (see however ref. [63] for the
calculation in the case of matter only). Calculations taking into account
nonlinear bias within the Zeldovich approximation show that phase shifts due
to mode coupling are substantially reduced post-reconstruction, suggesting
that the in-phase damping of the BAO feature due to IR displacements is the
dominant nonlinear effect [23]. However, in order to further insulate our
full-shape fits from potential systematics of the reconstruction procedure,
including due to residual nonlinear contributions, we set the linear bias and
growth rate in the BAO model to be free parameters called $F$ and $B_{1}$, and
in addition allow for broadband deviations between the theory predictions for
$\xi_{\ell}$ and the data by fitting adding a linear template to the theory
predictions, that is

$\xi_{\ell}(r)=\xi^{\rm th}_{\ell}(r)+a_{\ell,0}+\frac{a_{\ell,1}}{r}$ (3.8)

where $\xi^{\rm th}_{\ell}$ are the multipoles of the Fourier transformed
(with AP) theory predictions of Equation 3.7. Detailed functional forms and
integrals are given in Appendix B.

Let us conclude this section by comparing our approach in BAO fitting with
that in previous work. Traditionally, the BAO have been fit using a template
power spectrum whose BAO component is scaled by

$\tilde{\alpha}_{\parallel,\perp}=\Bigg{(}\frac{r^{\rm
fid}_{d}}{r_{d}}\Bigg{)}\ \alpha_{\parallel,\perp}.$ (3.9)

This takes into account stretching of the BAO signal due both to changing
distance scales ($\alpha_{\parallel,\perp}$) and sound horizon ($r_{d}$).
Typically this template model is used to fit directly for the
$\tilde{\alpha}$’s, whose output likelihoods can then be used in broader
cosmological analyses. Note that this prescription mixes two in-principle
distinct physical effects: the cosmological dependence of the sound horizon
$r_{d}$ and the anisotropy due to the Alcock-Paczynski effect
($\alpha_{\parallel,\perp}$), While the latter would exist even in the absence
of the BAO peak, the conventional method assumes the BAO carries the bulk of
the signal, with other effects from the broadband shape of the power spectrum
discarded, or marginalized away, via polynomial parameters.

In this work we eschew the use of templates for both pre- and post-
reconstruction fitting, choosing instead to extract the BAO wiggles for each
cosmology directly from the transfer-function outputs of Boltzmann codes, with
the perspective that Equation 3.4 and 3.7 are definite predictions of the pre-
and post-reconstruction galaxy two-point function for any given cosmology.
This obviates the need for the $r_{d}$ scaling in $\tilde{\alpha}$, since the
frequency of the BAO wiggles is automatically encoded in the linear power
spectrum, and automatically includes any distance and anisotropy information
present in the data. Modeling the reconstructed BAO at the data instead of the
summary statistic level also has advantages for constraining non-standard
physical effects like beyond $\Lambda$CDM physics. While measurements of the
BAO scale are robust to many such physical effects [64], there are notable and
theoretically well-understood exceptions, for example features around the BAO
scale due to relative perturbations between baryons and dark matter [65, 66].
Beyond robustness, measurements of BAO scaling parameters alone cannot capture
many effects related to oscillatory features in the power spectrum that can be
sharpened by reconstruction, including neutrino or light-relic induced phase
shifts in the BAO [67] or inflationary signatures [68, 69, 57]. Indeed, ref.
[70] detected the neutrino-induced phase shift in reconstructed BOSS data by
expanding the standard template fit with an additional phase-shift parameter.
On the other hand, for a given cosmological model, any such effects are
automatically included when fitting at the data level without modification
and, once included in a theory model, cannot act as a theoretical systematic
by default. For completeness, and to correct some typos in the literature, we
include a detailed description of the standard method in Appendix C.

##  4 Analysis Setup

In this paper we aim to perform a joint analysis combining full-shape
information in pre-reconstruction power spectrum and additional information in
the post-reconstruction correlation function. To this end, our fiducial setup
will combine the former with the latter in a narrow band around the peak where
most of the BAO information is isolated. Our goal in this section is to
outline and explain this setup, and lay out the accompanying cosmological and
effective-theory parameter choices and priors.

###  4.1 Scale Cuts

Figure 4: Best fit models to the NGCz3 pre-reconstruction power spectrum
multipoles (top) and their residuals (bottom), fixing $\Omega_{m}$ and $h$ to
their Planck best-fit values and a scan in $\sigma_{8}$ (different curves; see
legend). The biases and counterterms are varied to find the best fit for each
$\sigma_{8}$ and so differ between curves. The models primarily vary in their
predictions for the low $k$ quadrupole amplitude while making essentially
identical predictions to the high $k$ amplitude across different
$\sigma_{8}$’s. This suggests the constraining power on $\sigma_{8}$ comes
primarily from large scales.

We begin by setting up the Fourier-space side of our analysis. Throughout this
paper we will adopt the Fourier-space scale cuts $k_{\rm min}=0.02\,h\,{\rm
Mpc}^{-1}$ and $k_{\rm max}=0.20\,h\,{\rm Mpc}^{-1}$. The LPT model we use in
this work was shown to yield unbiased cosmological constraints on this range
of scales for BOSS-like samples even at significantly larger volumes [18]. We
drop the lowest two $k$ bins below $0.02\,h\,{\rm Mpc}^{-1}$; including them
increased the best-fit $\chi^{2}$ for the NGCz3 sample when limiting to a
Planck cosmology by $\Delta\chi^{2}=20$, suggesting systematic errors in the
data beyond our theoretical modeling.

It is worth noting that we have chosen to adopt a more conservative scale cut
than some other works in the literature, like Ref. [26] who adopt $k_{\rm
max}=0.25\,h\,{\rm Mpc}^{-1}$, even though we expect our theoretical model to
perform as well as other effective-theory models on the pre-reconstruction
power spectrum. We have made this choice on the reasoning that the information
on the primordial (linear) power spectrum rests primarily on large scales,
such that fitting smaller scales mainly serves to fit the shape of
nonlinearities in the redshift-space power spectrum like fingers of god
(FoGs). In fact, ref. [17] found that $P_{\ell}$ became dominated by
nonlinear, higher-order velocity statistics at around our chosen scale cut for
$\ell>0$. As an illustrative example of these effects, in Figure 4 we show the
best-fit models to the NGCz3 power spectrum multipoles for a series of
cosmologies with $\Omega_{m}$ and $h$ fixed to their Planck best-fit values
and $\sigma_{8}$ scanned from 0.63 to 0.89. For each $\sigma_{8}$ we then vary
the bias parameters and counter terms to fit the NGCz3 multipoles. Notably,
the only change from varying $\sigma_{8}$ is in the quadrupole at $k\lesssim
0.1\,h\,{\rm Mpc}^{-1}$, with the higher $k$ points having essentially the
same residuals compared to the data across a broad range of power spectrum
amplitudes. The best-fitting values of the biases and counter terms that we
find appear reasonable for each of the $\sigma_{8}$ values we tried. We can
understand this effect as follows: since the linear bias is well-constrained
by the monopole, varying $\sigma_{8}$ effectively tunes the linear quadrupole
amplitude, but as effective-theory corrections like nonlinear bias turn on at
higher wavenumbers this variation is erased by all the nonlinear parameters
conspiring to fit a relatively smooth and featureless $P_{2}$. This suggests
that, absent a detailed understanding of the small-scale physics underlying
the bias parameters, there is not signficant information to be gained at
smaller scales. Conversely, the fact that the variations in the quadrupole
amplitude on large scales cannot be fit away by (reasonable) bias parameters
is a demonstration that in effective theories the large scale clustering
signal cannot be polluted by small-scale physics in unphysical ways. This also
suggests that future surveys capable of reducing the errors at low $k$ would
significantly improve the constraints on the amplitude.

For the configuration-space side of our analysis, since we are primarily
interested in fitting the BAO feature, we restrict our fitting of the post-
reconstruction correlation function to $80<s<130\,h^{-1}$ Mpc for our fiducial
setup. This range of scales effectively isolates the BAO peak and most of the
additional distance information coming from reconstruction. It is worth noting
that a fit using the same model in Fourier space would have required fitting
over a wide range of wavenumbers to cover all the BAO wiggles in the power
spectrum and risked numerical issues in the covariance matrix due to
significant correlation with the pre-reconstruction power spectrum; the fact
that the BAO feature is localized at a large scale in the correlation
function, which happens to also be significantly sharpened by reconstruction
at those radii, is thus a useful fact in combining pre- and post-
reconstruction data. We also note that using broader bins in $P_{\ell}(k)$
could further separate the BAO and broad-band information, making the
constraints even less correlated, but we have not needed to take that step for
BOSS. Finally, as a consistency check we will also want to compare fits to the
pre-reconstruction correlation functions for z1 and z3 samples to our fiducial
setup fitting the pre-reconstruction power spectrum; for these fits we will
fit the correlation function over the entire range shown in the bottom panel
of Figure 2, i.e. $r>25\,h^{-1}\,{\rm Mpc}$.

###  4.2 Parameters and Priors

We choose to sample the cosmological parameter space uniformly in
$\Omega_{m}$, $h$ and $\ln(10^{10}A_{s})$. For the purposes of our analysis we
will fix the values of the baryon density $\Omega_{b}h^{2}=0.02242$ and
spectral index $n_{s}=0.9665$ to the best-fit values from Planck [71], since
the BOSS data are not very sensitive to these parameters which are very well-
determined in the CMB and, in the case of the baryon density, also big-bang
nucleosynthesis (BBN) [26]. We also fix the sum of the neutrino masses to be
the minimal allowed $M_{\nu}=0.06$ eV. While a number of recent works (e.g.
ref. [72]) have sought to more exactly integrate the scale-dependent effects
of massive neutrinos into LPT, in this work we will approximate the effect of
neutrinos on galaxy clustering by using the “cb” prescription [73]. This
prescription is motivated by the intuition that galaxies trace the cold dark-
matter and baryon fluid with linear power spectrum $P_{cb}$ and small-scale
growth rate $f_{c}=(1-3f_{\nu}/5)f_{\Lambda\rm CDM}$, where $f_{\nu}$ is the
neutrino mass fraction. This was recently shown to be an excellent
approximation using phase-matched simulations in ref [74].

For our pre-reconstruction power spectrum model we choose to fit each of the
four samples with independent sets of bias parameters. As can be seen in
Figure 2 the NGC and SGC samples at z1 have signficantly different power
spectrum multipoles on all scales shown; indeed, using the best-fit bias
parameters for a Planck cosmology for the NGC sample and convolving it with
the SGCz1 window function yields a noticeably worse fit ($\Delta\chi^{2}=19$
with our fiducial scale cuts). The multipoles at z3 show greater agreement,
and performing the same exercise of convolving the best fit model from one
galactic cap with the window function of the other yields a much better fit.
Nonetheless, we have opted for the more conservative approach and fit all four
(statistically independent) samples pre-reconstruction using separate sets of
bias parameters. Since the correlation function data were computed jointly for
the NGC and SGC samples we are forced to use a unified set of bias parameters
when fitting in configuration space; however, since our primary interest in
fitting the post-reconstruction $\xi_{\ell}$’s are to extract large-scale BAO
information, and since we find that the linear biases between the two galactic
caps are in good agreement in our power-spectrum only fits, we do not expect
this to significantly affect our results.

Our priors on the effective-theory and bias parameters are listed in Table 1.
We adopt broad, uninformative priors for the BAO broadband parameters
$B_{1},F$, with similarly broad priors on the polynomial broadband terms with
widths such that they do not dominate the clustering signal on BAO scales. For
the pre-reconstruction power spectrum we use generally broad priors for the
quadratic biases $b_{2},b_{s}$ on the assumption that they are free effective-
theory coefficients of order unity. Our counterterms $\alpha_{0},\alpha_{2}$
are set to Gaussian priors with the expectation that the galaxy power spectrum
deviate from linear theory by a factor less than unity on perturbative scales.
Finally, while one might expect the isotropic stochastic term $R_{h}^{3}$ to
be a free parameter in the ball-park of the inverse galaxy number density
$\bar{n}^{-1}\approx 3000\,h^{-3}\,{\rm Mpc}^{3}$, we put a relatively tight
Gaussian prior with width $1/(3\bar{n})$ on it given that we find an almost
exact degeneracy between it and $\alpha_{0}$ in our fits that did not
correlate significantly with any cosmological parameters for reasonable values
of $R_{h}^{3}$ and $\alpha_{0}$. We put a physically motivated prior on
$R_{h}^{3}\sigma^{2}$ based on the expectation that characteristic halo
velocities for BOSS LRGs are around $500$ km/s, or about $5\,h^{-1}\,{\rm
Mpc}$. Our pre-reconstruction fits to the pre-reconstruction correlation
function follow the same set of priors as the power spectrum fits.

Parameter | Prior
---|---
$\ln(10^{10}A_{s})$ | $\mathcal{U}(1.61,3.91)$
$\Omega_{m}$ | $\mathcal{U}(0.20,0.40)$
$H_{0}$ [km/s/Mpc] | $\mathcal{U}(60.0,80.0)$
$B_{1}$ | $\mathcal{U}(0,5.0)$
$F$ | $\mathcal{U}(0,5.0)$
$a_{\ell,0}$ | $\mathcal{N}(0,0.05)$
$a_{\ell,1}$ [$h^{-1}$ Mpc] | $\mathcal{N}(0,5)$

Parameter | Prior
---|---
$(1+b_{1})\sigma_{8}$ | $\mathcal{U}(0.5,3.0)$
$b_{2}$ | $\mathcal{N}(0,10)$
$b_{s}$ | $\mathcal{N}(0,5)$
$\alpha_{0}$ [h-2 Mpc2] | $\mathcal{N}(0,100)$
$\alpha_{2}$ [h-2 Mpc2] | $\mathcal{N}(0,100)$
$R_{h}^{3}$ [h-3 Mpc3] | $\mathcal{N}(0,1000)$
$R_{h}^{3}\sigma^{2}$ [h-5 Mpc5] | $\mathcal{N}(0,5\times 10^{4})$

Table 1: Parameter priors for our analysis. Uniform and normal distributions
are indicated by $U(x_{min},x_{max})$ and $N(\mu,\sigma)$, respectively.

###  4.3 Test on Mocks

| $P_{\ell}$ |  $P_{\ell}$ \+ BAO
---|---|---
$\ln(10^{10}A_{s})$ | $3.09\pm 0.18$ | $3.10\pm 0.17$
$\Omega_{m}$ | $0.294\pm 0.017$ | $0.298\pm 0.014$
$H_{0}$ [km/s/Mpc] | $67.9\pm 1.7$ | $67.8\pm 1.3$
$\sigma_{8}$ | $0.797\pm 0.062$ | $0.810\pm 0.063$

Figure 5: (Left) Contours of mock constraints from the mean of 1000 Patchy
mocks for the z3 sample fitting redshift-space power spectrum monopole and
quadrupole between $0.02\,h\,{\rm Mpc}^{-1}<k<0.20\,h\,{\rm Mpc}^{-1}$ with
(red) and without (blue) additional BAO information from the post-
reconstruction correlation function near the BAO peak. Gray lines show the
true cosmology of the Patchy mocks. (Right) A table of the mock constraints
(mean$\pm 1\sigma$). The true cosmology is given by
$\Omega_{m}=0.307115,H_{0}=67.77,\sigma_{8}=0.8288$, well within the $1\sigma$
bounds shown.

As we have discussed, the model described in Section 3, i.e. LPT, has been
tested extensively against simulated galaxy samples both pre- and post-
reconstruction, e.g. in refs. [18, 23]. These tests were performed using
periodic boxes without observational effects like window functions and
realistic survey geometries; while these effects are well understood and not
expected to significantly affect the accuracy of our models over the relevant
scales, it is worth checking that our fiducial setup in this paper (jointly
fitting the redshift-space power spectrum and post-reconstruction correlation
function near the BAO peak) does not yield unexpected biases in cosmological
constraints. To this end, in this subsection we apply the same analysis
pipeline we will use to analyze the BOSS data to obtain mock constraints from
the mean of the 1000 Patchy mocks released by the BOSS collaboration described
in Section 2. In particular, we perform our test on mocks of the z3 sample,
including the NGC and SGC power spectra and combined correlation function
multipoles. While these mocks employ approximate dynamics that may not exactly
match the ab initio predictions of perturbation theory, they are designed to
match the survey geometry and observational systematics of the BOSS survey,
and are thus a reasonable way to test whether these effects or our joint
Fourier and configuration space setup meaningfully affect our results.

The results of this test are shown in Figure 5. The constraints both with and
without BAO information recover the true cosmology
($\Omega_{m}=0.307115,H_{0}=67.77,\sigma_{8}=0.8288$) of the Patchy mocks to
within $1\sigma$, and adding in the post-reconstruction correlation function
tightens but does not lead to significant ($<0.3\sigma$) shifts in the
resulting constraints. We note that our results both with and without
reconstruction show up to $0.5\sigma$ deviations from “truth” in both
$\Omega_{m}$ and $\sigma_{8}$ despite the low statistical scatter from
averaging over 1000 mocks, which could be due to either the approximate nature
of the mocks themselves or parameter projection effects; nonetheless, these
results are satisfactory for our purposes since (1) they demonstrate the main
goal of this subsection, which was to show that adding in the post-
reconstruction correlation function does not bias our results and (2) the
parameter shifts occur both with and without reconstruction, despite our
having tested the latter case in simulation volumes significantly
($100\times$; [75]) larger than the BOSS survey and with correspondingly
tighter constraints and recovered unbiased constraints. Indeed, adding in
post-reconstruction BAO shifts both $\Omega_{m}$ and $\sigma_{8}$ closer to
“truth” while leaving $H_{0}$ firmly centered at the true value, potentially
due to reduced parameter-projection effects coming from tighter constraints.

##  5 Results

###  5.1 $\Lambda$CDM Constraints from BOSS with and without BAO

| $P_{\ell}$ |  $P_{\ell}$ \+ BAO | Planck
---|---|---|---
$\ln(10^{10}A_{s})$ | $2.84\pm 0.13$ | $2.81\pm 0.12$ | $3.044\pm 0.014$
$\Omega_{m}$ | $0.305\pm 0.01$ | $0.303\pm 0.0082$ | $0.3153\pm 0.0073$
$H_{0}$ [km/s/Mpc] | $68.5\pm 1.1$ | $69.23\pm 0.77$ | $67.36\pm 0.54$
$\sigma_{8}$ | $0.738\pm 0.048$ | $0.733\pm 0.047$ | $0.8111\pm 0.0060$
Table 2: Constraints from the full BOSS sample, i.e. NGCz1 SGCz1, NGCz3 and
SGCz3, with and without additional BAO information from the reconstructed
correlation function, summarized as mean $\pm 1\sigma$. The equivalent
constraints from Planck are also tabulated for comparison.

Figure 6: (Left) Constraint contours for fits to the BOSS galaxy power spectra
alone (blue) and with post-reconstruction correlation function multipoles
(BAO) added (red) compared to posteriors from Planck (blue), with which our
constraints are broadly consistent. (Right) Binned best-fit models for the
power spectrum and post-reconstruction correlation function multipoles from
our chains. Here we show only the results for NGCz3 power spectra and z3
correlation functions for brevity; the other samples are similarly well fit,
with total $\chi^{2}/\text{d.o.f}=1.06.$ Gray bands in the correlation
function plot show separations excluded by our fit.

The main results of this paper – constraints on $\Lambda$CDM parameters from
pre-reconstruction power spectra and post-reconstruction correlation function
multipoles for the full BOSS sample, including both galactic caps and redshift
slices – are shown in Figure 6 and listed in Table 2. Fitting to pre-
reconstruction power spectra alone we constrain $\Omega_{m}=0.305\pm 0.01$,
$H_{0}=68.5\pm 1.1$ and $\sigma_{8}=0.738\pm 0.048$; adding in the post-
reconstruction correlation function gives $\Omega_{m}=0.303\pm 0.0082$,
$H_{0}=69.23\pm 0.77$ and $\sigma_{8}=0.733\pm 0.047$. When fit with a shared
set of cosmological parameters, our model provides good fits to all of the
individual statistics included in the likelihood (pre-recon $P_{\ell}$ and
post-recon $\xi_{\ell}$ with $\ell=0$ and 2), with a combined
$\chi^{2}/\text{d.o.f}=1.05.$ As an example, the right panel of Figure 6 shows
the best-fit $P_{\ell}$ and $\xi_{\ell}$ for the NGCz3 and z3 samples,
respectively. The other samples look similar.

Comparing the red and blue contours in Figure 6, we see that the primary
effect of including post-reconstruction correlation functions is to tighten
constraints on the Hubble parameter $H_{0}$ (by around $40\%$) while also
slightly tightening constraints on $\Omega_{m}$ and keeping the $\sigma_{8}$
constraint largely untouched. This is to be expected since (1) the main
purpose of standard reconstruction is to sharpen the BAO peak and (2) we have
fit the post-reconstruction correlation function only near the peak at
$80\,h^{-1}\,{\rm Mpc}<r<130\,h^{-1}\,{\rm Mpc}$. Our motivation to include
the post-reconstruction correlation function in this paper was to include the
information in the linear power spectrum isolated at the BAO peak in
configuration space and not to use it as a further probe of nonlinearities in
structure formation; indeed, the purpose of freeing the Kaiser factors
$B_{1}$, $F$ and linear broadband polynomials in our correlation-function
model was precisely to prevent systematics in reconstruction from biasing our
constraints on the broadband amplitude of the linear power spectrum.

Let us conclude this subsection by comparing our results to those of other
groups and experiments. As can be seen in the left panel of Figure 6, our
constraints from BOSS, both with and without additional information in the
form of the reconstructed correlation function, are also broadly consistent
with cosmic microwave background (CMB) constraints, lensing included, from
Planck [71]. Indeed, our pre-reconstruction fit has the Planck best-fit
$\Omega_{m}$ and $H_{0}$ within its 1$\sigma$ contours, and $\sigma_{8}$ at
about $1.5\sigma$ lower. This is in contrast with earlier papers using
effective theory approaches to fitting the BOSS data, e.g. ref.’s [26, 76,
27], which found $\sigma_{8}$’s significantly ($>2\sigma$) lower than Planck.
These discrepancies are potentially attributable to the data normalization
issue described in Section 2 as our pre-reconstruction $\Omega_{m}$ and
$H_{0}$ constraints are in much better agreement with those earlier works;
specifically, this normalization issue resulted in power spectra that were
roughly $10\%$ too low in amplitude for a fixed window function normalization,
translating to a roughly $5\%$ lower best-fit $\sigma_{8}$. We recover
essentially identical $\sigma_{8}$ constraints to previous works if this
correction is neglected. Indeed, after our paper was first posted to the
arXiv, updated results free of window-function systematics from the authors of
the aforementioned works appeared: specifically, ref. [77] used correlation
functions to obtain $\sigma_{8}=0.7537^{+0.055}_{-0.06}$ and
$0.7559^{+0.052}_{-0.062}$ with and without BAO, in good agreement with
results from their independently measured power spectra, and ref. [78]
obtained $\sigma_{8}=0.729^{+0.040}_{-0.045}$ using a window-function free
power spectrum estimator and $\sigma_{8}=0.737^{+0.040}_{-0.044}$ when using
the same updated BOSS power spectra as the present work, virtually identical
to our pre-reconstruction constraints555For a clean comparison, we have quoted
their constraints fixing the spectral tilt $n_{s}$ and without including the
bispectrum monopole or finger-of-god reduced two-point statistics.. Our
results are also in excellent agreement with constraints from the BOSS data
using an emulator approach [79] and the configuration-space analysis in ref.
[80].

Adding the post-reconstruction correlation function, which sharpens the BAO
peak and tightens the $H_{0}$ constraint, puts the Hubble parameter best-fit
from Planck at about 2$\sigma$. This is in contrast to the BOSS fits in Ref.
[28], who find $H_{0}=67.81^{+0.68}_{-0.69}$666We compare to their results
keeping $n_{s}$ fixed rather than free for a more apples-to-apples comparison.
after adding in fits to $\tilde{\alpha}$’s from post-reconstruction power
spectra despite a similar tightening of constraints, though we note that we
use different post-reconstruction data ($P_{\ell}$ vs. $\xi_{\ell}$) and
covariance matrices and that the mock tests in §4.3 suggest that adding in the
post-reconstruction correlation function does not bias our results.
Nonetheless, our constraints with and without BAO both lie on the surface of
constant $\Omega_{m}h^{3}$ given by the Planck best-fit parameters; this
combination is close to a principle component of the Planck posterior and is
much better constrained than either $H_{0}$ or $\Omega_{m}$ alone.

Finally, our redshift-space constraints offer an independent check to the
measurements of power spectrum amplitude from weak lensing surveys like DES
and KiDS, which primarily measure the hybrid quantity
$S_{8}=\sigma_{8}(\Omega_{m}/0.5)^{0.5}$ and have found it to be significantly
lower than the value implied by Planck ($0.832\pm 0.013$ [71]). Figure 7
summarizes these constraints in the $\Omega_{m}-\sigma_{8}$ plane. Since
$\sigma_{8}$ and $\Omega_{m}$ are positively correlated in our posteriors, our
constraints are not optimized to measure $S_{8}$; nonetheless, we obtain
$S_{8}=0.736\pm 0.051$, slightly less than $2\sigma$ lower than the Planck
result and measurements of $0.775^{+0.026}_{-0.024}$ ($3\times 2$pt only) or
$0.812\pm 0.008$ (+BAO, RSD and SNIA) from DES Y3 [81]777We caution that these
analyses, unlike ours, freed the total neutrino mass $M_{\nu}$. and of
$0.766^{+0.020}_{-0.014}$ from a joint KiDS, BOSS and 2dFLenS analysis [82].
At lower redshifts, once BAO and weak priors are included, the constraint from
CMB lensing measured by Planck is
$\Sigma_{8}\equiv\sigma_{8}(\Omega_{m}/0.3)^{0.25}=0.815\pm 0.016$ [83]. Our
joint analysis finds $\Sigma_{8}=0.735\pm 0.0495$, about $1.5\sigma$ lower.

Figure 7: A comparison of our $\sigma_{8}$-$\Omega_{m}$ constraints with a
selection of other experiments, including Planck (including lensing) [71],
Planck lensing (P-lens) with BAO prior [83], KiDS+BOSS+2dFlens analysis [82],
and unWISE galaxy-CMB lensing cross correlations [84]. Our BOSS constraint
probes different degeneracy directions than these (primarily lensing) surveys,
but is nonetheless consistent with each.

###  5.2 Consistency Checks

Figure 8: (Left) Constraints from the two independent redshift slices z1
(blue) and z3 (orange) compared to the joint constraint (BOSS) from both
samples (gray). The two redshift bins are broadly consistent with each other,
as well as with constraints from Planck (black). (Right) Constraints using
pre-reconstruction power spectra and correlation functions in each of the
redshift slices, fit using the same theory model.

As discussed in Section 3, (Lagrangian) perturbation theory provides a
framework within which we can model large-scale structure observables like the
power spectrum and correlation function with a consistent theoretical model.
This allows us to, for example, jointly model the pre-reconstruction power
spectrum and post-reconstruction BAO feature without resorting to intermediate
statistics like $\alpha_{\rm BAO}$’s measured from fixed-shape templates. In
addition, the fact that we can model all these observables within the same
framework allows us check the consistency of our model assumptions about the
background cosmology and nonlinear structure formation or, alternatively, to
check for systematics in each statistic. In this subsection we will describe
two such consistency tests: between the high and low redshift samples and
between Fourier and configuration space.

The left panel of Figure 8 shows constraints obtained by fitting the two
redshift slices, z1 and z3, independently while including post-reconstruction
BAO information in each. The cosmological posteriors derived from the two
samples are broadly consistent with each other, with significant overlap
between the 1$\sigma$ regions: the $\Omega_{m}$ and $\sigma_{8}$ distributions
both have means consistent to within $1\sigma$. On the other hand, while the
$\bf{z1}$ sample prefers a value of $H_{0}$ very similar to the Planck (black
contours) best fit, $\bf{z3}$ prefers values around 2$\sigma$ above Planck; we
emphasize however that the combined constraints are themselves consistent with
Planck.

The right panel of Figure 8 compares fits to the pre-reconstruction power
spectra (solid) and correlation functions (dashed) in each redshift slice.
Again, the constraints from each $P_{\ell}$, $\xi_{\ell}$ pair are broadly
consistent, with overlapping $1\sigma$ regions for all three cosmological
parameters. However, while both $\sigma_{8}$ and $H_{0}$ constraints vary
consistently across samples, slightly decreasing and increasing respectively
with sample redshift, the $\Omega_{m}$ constraints are slightly lower in
configuration space than Fourier space. Given the different ranges over which
we fit these statistics and the different data subsamples we do not expect
perfect agreement. Nonetheless, all three parameters are broadly consistent,
especially taking into account that (1) the configuration space constraints on
$\Omega_{m}$ and $H_{0}$ are substantially less tight than their Fourier space
counterparts and (2) unlike the power spectra, the correlation functions were
computed assuming that the NGC and SGC subsamples could be meaningfully
combined into one. While we find the large scale (linear) bias to be
compatible between the two galactic caps at both redshifts, this does not
appear to be true for the full set of bias parameters, i.e. taking the
unconvolved best fit power spectrum from fitting the NGC and convolving it
with the SGC window function does not yield a comparably good fit to the SGC
power spectrum, particularly for the z1 samples, leading to potential
systematic differences between the configuration and Fourier space fits. The
differences are not important for our post-reconstruction BAO constraints, and
a full re-measurement of the BOSS correlation function, and its covariance, in
each galactic cap is beyond the scope of this paper.

We can also compare our results to the growth-rate ($f\sigma_{8}$)
measurements made by the BOSS survey: our indepenent chains for each redshift
bin imply $f\sigma_{8}(z_{\rm eff}=0.38)=0.419^{+0.035}_{-0.041}$ and
$f\sigma_{8}(z_{\rm eff}=0.61)=0.422^{+0.035}_{-0.040}$, in mild tension with
the official BOSS survey results. Their consensus results (including full-
shape and BAO, both using template fits) were $0.497\pm 0.046$ and $0.436\pm
0.035$888Here we have combined the statistical and systematic uncertainties
via quadrature. A complete table of the official BOSS results can be found in
Table 7 of ref. [34]., respectively [34]. It is worth noting that the BOSS
consensus results are a weighted combination of various analyses, including
both Fourier-space and configuration-space results. The configuration space
results are not affected by issues like that of window-function normalization
discussed in Section 2. Comparison of results in each of these categories
within the BOSS full-shape analysis shows differences between mean
$f\sigma_{8}$’s greater than one standard deviation, whereas our analysis
using the new window function normalizations yield Fourier and configuration
space means within $1\sigma$; for z1 we get $f\sigma_{8}$ constraints of
$0.434\pm 0.038$ and $0.470\pm 0.054$ and for z3 we get $0.413\pm 0.039$ and
$0.414\pm 0.050$ in Fourier and configuration space, respectively, pre-
reconstruction.

##  6 Conclusions

Galaxy redshift surveys are an important source of cosmological information,
allowing us to constrain properties of the early universe and general
relativity through measurements of redshift-space distortions and baryon
acoustic oscillations in galaxy clustering. Recent developments in
cosmological perturbation theory, particularly in the arena of effective
theories and IR resummation, have further put these data on rigorous and
precise footing, making it possible to consistently fit a range of
measurements from these surveys within a consistent framework. In the next few
years, surveys like DESI [85] and Euclid [86] will probe larger volumes at
higher redshifts, greatly increasing our constraining power on cosmological
parameters while making accurate modeling on quasilinear scales ever more
important.

In this paper we have presented an analysis of the pre- and post-
reconstruction power spectra and correlation functions from the BOSS survey
[24] within the framework of Lagrangian perturbation theory. Unlike previous
works, we do not combine pre- and post-reconstruction data through additional
fitting parameters for the BAO (e.g. $\tilde{\alpha}_{\parallel,\perp}$) whose
covariances are determined from mocks. Rather, for a given set of cosmological
parameters ($\Omega_{m},h,\sigma_{8}$) and galaxy bias coefficients we compute
directly the power spectrum and correlation function as predicted by
perturbation theory and compare them with observations to compute the
likelihood.

For our main result, we jointly fit pre-reconstruction power spectrum and
post-reconstruction correlation function multipoles for the full BOSS sample.
In order to avoid undue correlation with the power spectrum measurements, we
take advantage of the fact that the BAO signal is well-isolated in the
correlation function, particularly after reconstruction, and only fit the
correlation function in a narrow band containing the peak ($80\,h^{-1}\,{\rm
Mpc}<r<130\,h^{-1}\,{\rm Mpc}$). Our results are consistent with constraints
from Planck, as well as with $S_{8}$ measurements from weak lensing surveys.
We have further checked our analyses by considering constraints from each of
the redshift slices ($z_{\rm eff}=0.38$, 0.61) with and without post-recon
BAO, and in the former case by fitting both the power spectrum and correlation
function, finding that constraints from each subsample or observable are
broadly consistent.

Let us conclude by pointing out some possible future directions. A natural
extension of this work is to include further observables in our analysis: LPT
in the context of galaxy-lensing cross correlations has been studied in ref.
[87, 88, 89] and formed the basis of the model applied to the unWISE data in
ref. [84], as well as luminous red galaxies (LRGs) from DESI in ref. [90].
Recent work [91, 92, 93, 94] combining the Lagrangian bias scheme used in this
paper with nonlinear dynamics from N-body simulations should further extend
the reach and applicability of LPT for analyzing lensing cross correlations
within a Lagrangian framework. This would allow for a consistent analysis of
lensing surveys along with RSD and BAO (as shown in this paper) which will
become very powerful in the era of DESI [85], Euclid [86], Rubin [95] and
CMB-S4 [96]. In parallel, our analysis can be extended to include other
physical effects such as relative baryon-dark matter perturbations or more
exotic early-universe physics such as early dark energy light relics or
primordial features in the power spectrum; the modeling of these effects have
been studied within LPT [66, 57, 97] and applied to data within Eulerian
perturbation theory without reconstruction [98, 65, 99, 100]. Many of these
signatures are sharpened by reconstruction. Another potential effect is that
of anisotropic secondary bias due to line-of-sight selection biases, which
have the potential to skew measurements of $\sigma_{8}$ from RSD [101]. These
effects have so far not been modeled with LPT, though the equivalent Eulerian
framework have been explored in e.g. Ref. [52]. We discuss the current status
of evidence for these effects in Appendix D. Of course, many of these
additional effects will likely be better constrained by combining data: for
example, relative baryon-dark matter perturbations have a potential to bias
BAO measurements but conversely, by including its effects in a theory model,
it may be easier to constrain the size of their effect on galaxy clustering
when post-reconstruction data is included in the analysis. We leave these
developments for future work.

## Acknowledgments

We thank Pat McDonald for useful discussions about, and help with, the window
functions used in this work. We thank Mariana Vargas Magaña for providing data
and mock measurements of the BOSS correlation functions. S.C. is supported by
the National Science Foundation Graduate Research Fellowship (Grant No. DGE
1106400) and by the UC Berkeley Theoretical Astrophysics Center Astronomy and
Astrophysics Graduate Fellowship. M.W. is supported by the U.S. Department of
Energy and by NSF grant number 1713791. Z.V. acknowledges the support by the
Kavli Foundation. We acknowledge the use of Cobaya [102, 103], Class [50],
GetDist [104] and velocileptors [17] and thank their authors for making these
products public. This research used resources of the National Energy Research
Scientific Computing Center (NERSC), a U.S. Department of Energy Office of
Science User Facility operated under Contract No. DE-AC02-05CH11231. This work
made extensive use of the NASA Astrophysics Data System and of the astro-ph
preprint archive at arXiv.org.

##  Appendix A Fast Evaluation via Taylor Series

Figure 9: Differences between unconvolved power spectra computed directly from
CLASS and velocileptors vs. approximated by a 4th-order Taylor series, sampled
from a chain fitting the NGCz3 power spectrum, compared to the data error bars
on that sample. Solid lines show mean deviation while $\pm 1\sigma$ deviations
are shown as dashed lines. The monopole and quadrupole deviations are shown in
blue and orange, respectively.

In order to speed up our model evaluations (obtaining transfer functions from
CAMB alone takes a few seconds per cosmology) we use a Taylor series centered
around a representative cosmology
($\Omega_{m},\,h,\,\sigma_{8}=0.31,0.68,0.73$999This central $\sigma_{8}$
value was chosen prior to discovering the window-function normalization issue
resulting in systematically low $\sigma_{8}$ constraints. The resulting
Taylor-series predictions are nonetheless sufficiently accurate for our final
constraints, as shown below. ) to evaluate the perturbation-theory predictions
of each cosmology in our chains. This technique has previously been applied to
both full-shape forecasts [105] and data analyses of BOSS data [27, 106], and
even at the linear level to approximate the predictions of a hybrid
N-body/Lagrangian bias model of real-space clustering [93]. Specifically, both
the pre-reconstruction power spectrum and post-reconstruction correlation
functions (minus broadband terms) can written as inhomogeneous quadratic
polynomials in the bias parameters, e.g. for a fixed cosmology $\Theta$

$\big{(}P_{0}(\textbf{k}),P_{2}(\textbf{k}),P_{4}(\textbf{k}),\xi_{0}(\textbf{s}),\xi_{2}(\textbf{s})\big{)}(\Theta)=\beta_{i}\beta_{j}\mathcal{M}^{ij}(\Theta)$

where the vector
$\beta=(1,b_{1},b_{2},b_{s},\alpha_{0},\alpha_{2},R_{h}^{3},R_{h}^{3}\sigma^{2},B_{1},F)$.
Each component of the matrix $\mathcal{M}^{ij}$ is a smooth function of the
cosmological parameters so can be approximated as

$\mathcal{M}^{ij}(\Theta)=\sum_{n=0}^{N}\frac{1}{n!}\,(\Theta-\Theta_{0})_{i_{1}}...(\Theta-\Theta_{0})_{i_{n}}\,\partial_{i_{1}...i_{n}}\mathcal{M}^{ij}(\Theta_{0}).$

We evaluate the derivatives numerically using the publicly available
FinDiff101010 https://findiff.readthedocs.io/en/latest/ package. In order to
take advantage of central differences these derivatives are computed using a
grid with $2N+1$ points along each axis, where we find $N=4$ to be sufficient
for any cosmology relevant for constraints from any of the sets of samples
considered in our analysis. In Figure 9 we show differences between the
unconvolved power spectrum multipoles as computed directly from velocileptors
and the Taylor series approximation for 100 representative elements of a
power-spectrum fit to NGCz3, compared to the much larger error bars of the
data itself. Producing the grids used to compute derivatives using one core on
Cori111111https://docs.nersc.gov/systems/cori/ takes about three minutes,
while evaluating the Taylor series itself given the coefficients takes less
than a hundredth of a second.

##  Appendix B Nonlinear Damping of the BAO within RecIso

In this appendix we give the specific form of the BAO damping in the
reconstructed power spectrum within the RecIso scheme. The reconstructed field
is defined to be the difference between the overdensities of displaced
galaxies $d$ and shifted galaxies $s$, i.e. $\delta_{\rm
recon}=\delta_{d}-\delta_{s}$. Theoretical modeling of the nonlinear damping
of the reconstructed field involves calculating cross correlations between the
Lagrangian displacements of the two fields. Thus in general we expect each
piece of

$P^{\rm recon}_{s}(k,\mu)=P^{dd}_{s}-2P^{ds}_{s}+P^{ss}_{s},\quad
P^{ab}_{s}=P^{ab}_{lin,nw}+e^{-\frac{1}{2}k^{2}\Sigma^{2}_{ab}(\mu)}P^{ab}_{lin,w}+...,$

to have a different damping form, where $P^{ab}_{lin}$ refers to the
(undamped) linear-theory predictions for each spectrum and $\Sigma^{2}_{ab}$
its damping parameter. Specifically, $\Sigma^{2}_{ab}$ is the isotropic
component of the displacement two-point function

$A^{ab}_{ij}=\left\langle\Delta^{ab}_{i}\Delta^{ab}_{j}\right\rangle,\;\Delta^{ab}=\Psi^{a}(\textbf{q}_{1})-\Psi^{b}(\textbf{q}_{2}).$
(B.1)

evaluated at the BAO scale, i.e.
$\Sigma^{2}_{ab}=\frac{1}{3}\delta_{ij}A^{ab}_{ij}$ for $q=r_{d}$.

Applying the above logic we then have the linear-theory forms

$\displaystyle P^{dd}_{lin}(k,\mu)$
$\displaystyle=\Big{(}b-\mathcal{S}+f\mu^{2}(1-\mathcal{S})\Big{)}^{2}P_{\rm
lin}(k)$ $\displaystyle P^{ds}_{lin}(k,\mu)$
$\displaystyle=-\mathcal{S}\,\Big{(}b-\mathcal{S}+f\mu^{2}(1-\mathcal{S})\Big{)}P_{\rm
lin}(k)$ $\displaystyle P^{ss}_{lin}(k,\mu)$
$\displaystyle=\mathcal{S}^{2}P_{\rm lin}(k)$

and damping parameters

$\displaystyle\Sigma^{2}_{dd}(\mu)$
$\displaystyle=(1+f(2+f)\mu^{2})\,\Big{(}2\tilde{\Sigma}^{2}_{dd}(0)-2\tilde{\Sigma}^{2}_{dd}(r_{d})\Big{)}$
$\displaystyle\Sigma^{2}_{ds}(\mu)$
$\displaystyle=(1+f(2+f)\mu^{2})\tilde{\Sigma}^{2}_{dd}(0)+\tilde{\Sigma}^{2}_{ss}(0)-2(1+f\mu^{2})\tilde{\Sigma}^{2}_{ds}(r_{d})$
$\displaystyle\Sigma^{2}_{ss}(\mu)$
$\displaystyle=2\tilde{\Sigma}^{2}_{ss}(0)-2\tilde{\Sigma}^{2}_{ss}(r_{d})$

where the tilded $\Sigma$’s are defined as

$\displaystyle\tilde{\Sigma}_{dd}^{2}(r)$
$\displaystyle=\frac{1}{3}\int\frac{dk}{2\pi^{2}}\Big{[}\big{(}1-\mathcal{S}\big{)}^{2}\,P_{\rm
lin}(k)\ j_{0}(kr)\Big{]}$ $\displaystyle\tilde{\Sigma}_{ss}^{2}(r)$
$\displaystyle=\frac{1}{3}\int\frac{dk}{2\pi^{2}}\Big{[}\mathcal{S}^{2}\,P_{\rm
lin}(k)\ j_{0}(kr)\Big{]}$ $\displaystyle\tilde{\Sigma}_{ds}^{2}(r)$
$\displaystyle=\frac{1}{3}\int\frac{dk}{2\pi^{2}}\Big{[}-\mathcal{S}\,\big{(}1-\mathcal{S}\big{)}\,P_{\rm
lin}(k)\ j_{0}(kr)\Big{]}.$ (B.2)

We refer the readers to [23] for a detailed derivation of these damping
parameters.

##  Appendix C Parameters for BAO Fit

Our goal in this appendix is to explicitly spell out the assumptions of the
standard template fits to the BAO (e.g. in [30, 35, 28, 29]). We begin by
writing down Alcock-Paczynski scalings needed to convert between fiducial and
true cosmology before writing down the additional assumptions that result in
the traditional template fit with $r_{d}$-dependent BAO parameters.

###  C.1 Alcock-Paczynski Effect

Galaxy survey data are typically presented in units wherein redshifts and
angles are translated into distances via the Hubble function $E^{\rm fid}(z)$
and angular-diameter distance $D^{\rm fid}_{A}(z)$ of some fiducial
cosmology121212For concreteness we will assume throughout that all distances
are given in $h^{-1}$ units.. Given this choice of coordinates it is necessary
given any cosmological model $\Theta$ to rescale theory predictions from the
true physical separations in the theory into theory units. For the anisotropic
power spectrum this is given by

$P^{\rm obs}_{s}(k^{{\rm obs}}_{\parallel},k^{{\rm
obs}}_{\perp})=\alpha_{\parallel}^{-1}\alpha_{\perp}^{-2}\,P^{\rm true}(k^{\rm
true}_{\parallel},k^{\rm true}_{\perp}),\quad k^{\rm
true}_{\parallel,\perp}=\frac{k^{\rm obs}}{\alpha_{\parallel,\perp}}$ (C.1)

where the Alcock-Paczynski (AP) parameters are defined as

$\alpha_{\parallel}=\frac{E^{\rm
fid}(z)}{E(z)},\quad\alpha_{\perp}=\frac{D_{A}(z)}{D^{\rm fid}_{A}(z)}.$ (C.2)

For covenience we will drop the superscript for the “true” quantities in the
remainder of this appendix. In terms of these quantities we can write the true
wavenumber magnitude and angle as [30]

$\displaystyle k$ $\displaystyle=\alpha_{\perp}^{-1}\sqrt{1+\mu_{\rm
obs}^{2}(F_{\rm AP}^{-2}-1)}\,k^{\rm obs}$ $\displaystyle\mu$
$\displaystyle=\frac{1}{F_{\rm AP}}\Big{(}\sqrt{1+\mu_{\rm obs}^{2}(F_{\rm
AP}^{-2}-1)}\Big{)}^{-1}\,\mu^{\rm obs}$ (C.3)

where $F_{\rm AP}=\alpha_{\parallel}/\alpha_{\perp}$. Note that both $F_{\rm
AP}$ and the $\mu$ are invariant under isotropic scaling
$\alpha_{\parallel,\perp}\rightarrow c\alpha_{\parallel,\perp}$.

###  C.2 BAO in the Power Spectrum

In standard template fits to the BAO the wiggles from a fixed “template” power
spectrum are rescaled to model the observed BAO signal. In this case we must
allow for the possibility that the true linear power spectrum of the universe
has BAO wiggles with a slightly different shape.

The trick is to assume that the linear power spectrum for a given cosmology
$\Theta$ can be split into a smooth “no-wiggle” component $P_{nw}$ and a
“wiggle” component $P_{w}$ whose cosmology dependence can be approximated in
terms of its amplitude $A$ and a template scaled by the BAO radius $r_{d}$:

$P_{\rm lin}(k)=P_{nw}(k|\Theta)+A(\Theta)g(r_{d}(\Theta)k).$ (C.4)

In particular, for a given “template” power spectrum at the fiducial cosmology
we can extract the BAO template

$G^{\rm fid}(k)=A^{\rm fid}g(r^{\rm fid}_{d}k).$ (C.5)

The “wiggle” component should be understood to be unique only up to a smooth
(polynomial) broadband that does not carry scale information.

The anisotropic redshift-space power spectrum, taking into nonlinear BAO
damping due to bulk displacements, is

$P_{s}(k,\mu)=(b+f\mu^{2})^{2}\big{(}P_{nw}(k)+e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}P_{w}(k)\big{)}+...$

where the ellipsis stands for higher order terms, which we will assume can be
absorbed by smooth broadband terms, particularly post-reconstruction.
Combining with the above parametrization of the linear power spectrum we get
that the observed linear power spectrum is

$P^{\rm obs}_{s}(k_{\rm obs},\mu_{\rm
obs})=\alpha_{\parallel}^{-1}\alpha_{\perp}^{-2}(b+f\mu^{2})^{2}\Big{(}P_{nw}(k)+e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}A(\Theta)g(r_{d}k)\Big{)}+...,$
(C.6)

In particular, the damped “wiggle” component takes the form

$\displaystyle P^{\rm obs}_{w}$
$\displaystyle=\alpha_{\parallel}^{-1}\alpha_{\perp}^{-2}(b+f\mu^{2})^{2}e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}Ag(r_{d}k)$
$\displaystyle=\alpha_{\parallel}^{-1}\alpha_{\perp}^{-2}(b+f\mu^{2})^{2}e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}A\,g\Bigg{(}r^{\rm
fid}_{d}\Big{(}\frac{r_{d}}{r^{\rm
fid}_{d}}\Big{)}\alpha_{\perp}^{-1}\sqrt{1+\mu_{\rm obs}^{2}(F_{\rm
AP}^{-2}-1)}k_{\rm obs}\Bigg{)}$ $\displaystyle=\Bigg{(}\frac{r^{\rm
fid}_{d}}{r_{d}}\Bigg{)}^{-3}\,\tilde{\alpha}_{\parallel}^{-1}\tilde{\alpha}_{\perp}^{-2}(b+f\mu^{2})^{2}e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}\Big{(}\frac{A}{A^{\rm
fid}}\Big{)}\,G^{\rm fid}\Bigg{(}\tilde{\alpha}_{\perp}^{-1}\sqrt{1+\mu_{\rm
obs}^{2}(\tilde{F}_{\rm AP}^{-2}-1)}\,k_{\rm obs}\Bigg{)}$
$\displaystyle\equiv\tilde{A}(b+f\mu^{2})^{2}e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}G^{\rm
fid}\Bigg{(}\tilde{\alpha}_{\perp}^{-1}\sqrt{1+\mu_{\rm
obs}^{2}(\tilde{F}_{\rm AP}^{-2}-1)}\,k_{\rm obs}\Bigg{)}$
$\displaystyle\equiv(B+F\mu^{2})^{2}e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}G^{\rm
fid}\Bigg{(}\tilde{\alpha}_{\perp}^{-1}\sqrt{1+\mu_{\rm
obs}^{2}(\tilde{F}_{\rm AP}^{-2}-1)}\,k_{\rm obs}\Bigg{)}$

where we have dropped the cosmology dependence in $A=A(\Theta)$ and
$r_{d}=r_{d}(\Theta)$ and defined the modified AP parameters

$\tilde{\alpha}_{\parallel,\perp}=\Bigg{(}\frac{r^{\rm
fid}_{d}}{r_{d}}\Bigg{)}\ \alpha_{\parallel,\perp},\quad\tilde{F}_{\rm
AP}\equiv\frac{\tilde{\alpha}_{\parallel}}{\tilde{\alpha}_{\perp}}=F_{\rm AP}$
(C.7)

casting the BAO measurement specifically as one of ratios of cosmological
distances with the sound horizon. Note that the ratio $\beta=F/B=f/b$ is equal
to that in the normal Kaiser formula and is invariant to the AP effect and
template normalization. For the sake of brevity we will not repeat the above
derivation for the case of reconstruction but note that the essential features
(i.e. $r_{d}$ scaling with fixed $F_{\rm AP}$) are unchanged in that case.

In practice we would like to expand the power spectrum, broadband included,
about the fiducial cosmology, at which all the ratios $X/X^{\rm fid}$ are
equal to unity. In this case, the “wiggle” component can be exactly rescaled
as above while changes in the broadband power spectrum are expected to be
smooth and degenerate with smooth polynomials, i.e.

$\displaystyle P^{\rm obs}_{s}(k_{\rm obs},\mu_{\rm obs})$
$\displaystyle=(B+F\mu_{\rm obs}^{2})^{2}P^{\rm fid}_{nw}(k_{\rm
obs})+(B+F\mu^{2})^{2}$ $\displaystyle
e^{-\frac{1}{2}k^{2}\Sigma^{2}(\mu)}G^{\rm
fid}\Bigg{(}\tilde{\alpha}_{\perp}^{-1}\sqrt{1+\mu_{\rm
obs}^{2}(\tilde{F}_{\rm AP}^{-2}-1)}\,k_{\rm
obs}\Bigg{)}+\sum_{n,m}\,a_{nm}k^{n}\mu^{2m}.$ (C.8)

where $P^{\rm fid}_{nw}$ is the “no-wiggle” power spectrum at the fiducial
cosmology. In practice we work with multipoles, in which case we can write

$P^{\rm obs}_{\ell}(k_{\rm
obs})=\Big{(}...\Big{)}_{\ell}+\sum_{n}\,a_{\ell,n}k^{n}$ (C.9)

where the ellipses stand for multipoles of the non-polynomial terms.
Equivalently, for configuration space analyses we can simply supplement the
Fourier transform of these terms with a polynomial in the (inverse) radius:

$\xi^{\rm obs}_{\ell}(s_{\rm obs})={\rm
FT}\left\\{(...)\right\\}_{\ell}+\sum_{n}b_{\ell,n}s_{\rm obs}^{-n}$ (C.10)

where we have implicitly used that the Fourier transform of the smooth
broadband is also smooth over the range of interest.

##  Appendix D Anisotropic secondary bias

Our analysis, in common with most other analyses in the field, has assumed
that the probability that a galaxy makes it into the sample is independent of
the large-scale tidal field in which that galaxy sits. Specifically, we assume
that the galaxy overdensity is a function of scalar quantities that can be
constructed from second and higher derivatives of the gravitational potential.
Since several “non-scalar” halo properties, like shapes and angular momenta,
depend upon the large-scale tidal fields in which the halos are situated [107]
there is the possibility that important galaxy properties also inherit this
dependence and if the probability of the galaxy appearing in the catalog with
a successful redshift depends upon those properties we would introduce a non-
scalar component to the bias [101, 52]. This would invalidate our analysis.

The case for or against anisotropic secondary (or “assembly”) bias for BOSS
galaxies is currently uncertain [108, 109, 110]. The strongest claim so far is
that of ref. [109], who argue they have detected such a bias at $5\,\sigma$ by
splitting the galaxies using a combination of stellar mass and line-of-sight
velocity dispersion. Specifically they show that for two subsamples of
galaxies, selected in the $M_{\star}-\sigma_{\star}$ plane, they can obtain
different power spectrum quadrupoles while matching the power spectrum
monopole. Within linear theory and in the absence of anisotropic bias
$P(k,\mu)=(b+f\mu^{2})^{2}P_{\rm lin}(k)$ [3] and so matching the monopoles
should imply that the quadrupoles also match. However in the presence of
anisotropic secondary bias the prefactor becomes
$(b+f\mu^{2}+b_{q}[\mu^{2}-1/3])^{2}$, where $b_{q}$ is the anisotropic bias
[109] and it is possible to have different quadrupoles while matching the
monopoles.

We should treat this claim with caution, since it is based upon a very simple
model. Indeed, we have already seen that samples of galaxies in the NGC and
SGC regions of the BOSS survey can have quite similar monopoles with
relatively different quadrupoles. A careful look at Figs. 3-6 of ref. [109]
shows that most of the evidence for anisotropic secondary bias arises from
$k>0.1\,h\,{\rm Mpc}^{-1}$ where the linear analysis for the quadrupole is
completely inadequate. As one example, our models contain terms going as
$k^{2}\,P_{\rm lin}(k)$ that can have different amplitudes between the
monopole and quadrupole for different samples, showing that the high-$k$ part
of the quadrupole can be quite different than linear theory estimates may
imply. Phrased differently, models that vary significantly in their satellite
content and finger-of-god can cause variations in the quadrupole at high $k$
that are significantly larger than the variations in the monopole that they
induce.

Figure 10: Model power spectra for 100 models chosen at random from a Markov
chain fit to the $z=0.61$ NGC power spectrum with $k_{\rm
max}=0.25\,h\,\mathrm{Mpc}^{-1}$ for the monopole but $k_{\rm
max}=0.05\,h\,\mathrm{Mpc}^{-1}$ for the quadrupole. All of the models are
thus consistent within statistical errors with a constant monopole while being
essentially unconstrained as to the quadrupole. Note the monopoles (blue
dashed lines) form a tight envelope while the quadrupoles (orange dotted
lines) agree only at low $k$. By $k\simeq 0.1\,h\,\mathrm{Mpc}^{-1}$ the
differences are about a factor of $2$.

To get a sense for how much non-linear bias and non-linear dynamics within the
standard model (without anisotropic secondary bias) can modify the quadrupole
at fixed monopole we did the following experiment. We picked the $z=0.61$ NGC
sample, and fit our standard pre-reconstruction power spectrum model to
$k\simeq 0.25\,h\,\mathrm{Mpc}^{-1}$ for the monopole but only $k\simeq
0.05\,h\,\mathrm{Mpc}^{-1}$ for the quadrupole131313Since the quadrupole is
now highly unconstrained we narrowed the prior on $\alpha_{2}$ to be
$\mathcal{N}(0,25)$, corresponding to a finger-of-god velocity dispersion of
$\mathcal{O}(500\mathrm{km}\,\mathrm{s}^{-1})$ which is comparable to the
values of $\alpha_{2}$ we obtain from the fits in the main body of the paper.
at fixed cosmology. We then sampled 100 models from the chain, each model
having equal monopoles within statistical errors, to see how much the
quadrupoles can differ as a function of $k$. The results are shown in Fig. 10.
Note the monopoles (blue dashed lines) form a tight envelope while the
quadrupoles (orange dotted lines) agree only at low $k$. By $k\simeq
0.1\,h\,\mathrm{Mpc}^{-1}$ the differences are about a factor of $2$.

Based upon this calculation we believe the case for anisotropic secondary bias
in the BOSS galaxies remains unproven. While our calculations do not prove the
absence of such an effect, existing measurements are also consistent with
differences expected in currently popular models that neglect these effects.
Given its importance as a source of systematic errors for future redshift
surveys, further investigation is clearly warranted.

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