# Developing a Theoretical Model for the Resummation of Infrared Effects
in the Post-Reconstruction Power Spectrum

Naonori Sugiyama<EMAIL_ADDRESS>National Astronomical Observatory
of Japan, Mitaka, Tokyo 181-8588, Japan

###### Abstract

Since galaxy distribution reconstruction effectively reduces non-Gaussian
terms in the power spectrum covariance matrix, it has attracted interest not
only for Baryon Acoustic Oscillation (BAO) signals but also for various
cosmological signal analyses. To this end, this paper presents a novel
theoretical model that addresses infrared (IR) effects in the post-
reconstruction galaxy power spectrum, including 1-loop corrections. In
particular, we discuss the importance of incorporating non-perturbative
effects arising from IR contributions into the displacement vector $\bm{s}$
used for reconstruction. Consequently, post-reconstruction nonlinear damping
of BAO can be described by a single two-dimensional Gaussian function. This is
a phenomenon not observed when $\bm{s}$ is considered to at a linear order in
the Zel’dovich approximation. Furthermore, we confirm that the cross-power
spectrum of the pre- and post-reconstruction density fluctuations lacks IR
effect cancellations, and shows an exponential decay in both the cross-power
spectrum and the associated shot-noise term.

##  I Introduction

The reconstruction of galaxy distributions [1] has traditionally been a focus
of research to amplify baryon acoustic oscillation (BAO) signals [2, 3].
However, recent findings indicate that these reconstruction techniques can
effectively reduce non-Gaussian terms in the covariance matrix of cosmological
$n$-point statistics. As a result, their application has expanded beyond BAO
signals to encompass a variety of cosmological signals. In particular, there
is a growing interest in improving constraints on cosmological parameters
through post-reconstruction two-point statistics [4, 5]. In addition,
applications in three-point statistics have been reported to strengthen
constraints on primordial non-Gaussianity [6].

Despite the potential of reconstruction techniques, their practical
applications, particularly beyond BAO signal analysis, are still emerging.
This slow advancement is due to a limited understanding of the nonlinear
characteristics of reconstructed density fluctuations. The examination of pre-
reconstruction two-point correlation functions and power spectra primarily
employs advanced theoretical models, which extend beyond Standard Perturbation
Theory (SPT) [e.g., 7], to precisely extract small-scale information. For
instance, sophisticated theoretical models such as the Convolution Lagrangian
Perturbation Theory (CLPT) [8, 9], the Renormalized Perturbation Theory (RPT)
[10], the TNS model [11], and the Effective Field Theory of Large-scale
Structure (EFT of LSS) [12, 13] have been applied to analyze actual galaxy
survey datasets, such as the Baryon Oscillation Spectroscopic Survey Data
Release 12 (BOSS DR12) galaxy data [14, 15, 16, 17]. On the other hand,
theoretical research into post-reconstruction density fluctuations has been
limited to 1-loop calculations in SPT [18, 19, 20] and methods based on the
Zel’dovich approximation [21, 22, 23, 24]. See also [25, 26] for perturbation
theory approaches to iterative reconstruction methods [27, 28, 29, 30, 31, 32,
33, 34].

In this study, we primarily investigate the infrared (IR) effect on the power
spectrum of galaxies. This effect possesses a unique property that allows for
non-perturbative analysis, making it crucial for advancing theories beyond
SPT. For the pre-reconstruction scenario, several IR effects are found, and we
summarize them below. When taking the IR limit and considering only the
nonlinearities arising from IR effects, all IR nonlinear effects are negated
in the calculations of the power spectrum, leaving only the linear power
spectrum [35, 36, 37, 38, 39, 40, 41, 42, 43]. This IR cancellation holds true
when evaluating all density fluctuations at equal times. However, assessing
the correlation of fluctuations at distinct times yields the consistency
relation for the Large Scale Structure (LSS), which uniquely connects higher-
order statistics to their lower-order counterparts [38, 37, 44]. Moreover,
proper consideration of the IR effect can shed light on the nonlinear decay of
the BAO signal [45, 46, 47, 48, 40, 49, 50, 51, 52, 53, 54, 55].

This paper aims to comprehensively elucidate the impact of non-perturbative IR
effects on the BAO signal and the overall power spectrum shape after
reconstruction. In this context, we present a new IR-resummed model of the
post-reconstruction power spectrum, including 1-loop corrections.
Concurrently, we explore its relation to established theories that address
partial IR effects after reconstruction, such as the 1-loop correction in SPT
and the Zel’dovich approximation. We further demonstrate the fundamental
significance of the IR effect in the cross power spectrum between the pre- and
post-reconstruction density fluctuations. The proposed model should be
directly applicable to observational galaxy data as it incorporates Redshift
Space Distortions (RSDs) [e.g., 56] and galaxy bias effects [e.g., 57].

The outline of this paper is as follows. Section II reviews the non-
perturbative treatment of infrared (IR) effects on pre-reconstruction density
fluctuations. Section III explores the IR effects on post-reconstruction
density fluctuations and presents a model for their power spectrum. Section IV
calculates the cross power spectrum of density fluctuations before and after
reconstruction, highlighting the importance of nonlinear IR effects. Section V
concludes with the findings and implications in this paper. Appendix A
introduces a kind of modified gravity theory as an example of breaking the IR
cancellation. Appendices B and C summarize the behavior in the IR limit for
solutions up to third-order fluctuations in SPT before and after
reconstruction, respectively.

##  II Pre-Reconstruction Case

In preparation for the post-reconstruction case, we review the derivation for
the IR-resummed power spectrum model in the pre-reconstruction case. This
section provides an abbreviated version of the proof found in [39, 40, 55].
For a more complete understanding, see the original papers, which detail the
calculations; see e.g., [50, 58, 49, 51, 52, 53, 54] for other approaches to
dealing with IR effects.

###  II.1 Infrared (IR) effects on dark matter density fluctuations

Consider a density field function $\rho(\bm{x})$ defined over three-
dimensional spatial coordinates $\bm{x}$. Let $\delta(\bm{x})$ represent the
deviation from the mean density field $\bar{\rho}$. This relation can be
expressed as $\rho(\bm{x})=\bar{\rho}\left(1+\delta(\bm{x})\right)$. The
Fourier transform of $\delta(\bm{x})$ can be represented as
$\widetilde{\delta}(\bm{k})=\int d^{3}xe^{-i\bm{x}\cdot\bm{k}}\delta(\bm{x})$,
where the tilde denotes a Fourier-transformed quantity.

Let us begin by considering the density fluctuations of dark matter, denoted
$\delta_{\rm m}(\bm{x})$, wherein the subscript “m” represents “matter”. In
perturbation theory, an $n$th-order dark matter density fluctuation in Fourier
space, denoted $\widetilde{\delta}^{\,[n]}_{{\rm m}}(\bm{k})$, can be
represented by the convolution integral of $n$ wave vectors
($\bm{p}_{1},\dots,\bm{p}_{n}$) with an associated $n$th-order kernel function
$F_{n}(\bm{p}_{1},\dots,\bm{p}_{n})$. Hence, the $n$th-order density
fluctuation is expressed as

$\displaystyle\widetilde{\delta}^{\,[n]}_{{\rm m}}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ (1) $\displaystyle\times$ $\displaystyle
F_{n}(\bm{p}_{1},\dots,\bm{p}_{n})\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}_{n})\;,$

where $\bm{p}_{[1,n]}=\bm{p}_{1}+\cdots+\bm{p}_{n}$, and
$\widetilde{\delta}^{\,[1]}_{\rm m}$ is the linear (first-order) dark matter
density fluctuation in Fourier space, and $\delta_{\rm D}$ is the three-
dimensional delta function.

####  II.1.1 Second order fluctuations

For example, the second-order kernel function is given by

$\displaystyle
F_{2}(\bm{p}_{1},\bm{p}_{2})=\frac{17}{21}+\frac{1}{2}\mu_{12}\left(\frac{p_{1}}{p_{2}}+\frac{p_{2}}{p_{1}}\right)+\frac{2}{7}\left(\mu_{12}^{2}-\frac{1}{3}\right)\;,$
(2)

where $p_{n}=|\bm{p}_{n}|$ and $\mu_{12}=\hat{p}_{1}\cdot\hat{p}_{2}$ with
$\hat{p}_{n}=\bm{p}_{n}/p_{n}$. Substituting Eq. (2) into Eq. (1) and
performing the inverse Fourier transform, we obtain

$\displaystyle\delta_{\rm m}^{[2]}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{17}{21}\big{[}\delta_{\rm
m}^{[1]}(\bm{x})\big{]}^{2}-\bm{\Psi}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})$ (3) $\displaystyle+$
$\displaystyle\frac{2}{7}\left[\left(\frac{\partial_{i}\partial_{j}}{\partial^{2}}-\frac{1}{3}\delta_{ij}\right)\delta_{\rm
m}^{[1]}(\bm{x})\right]^{2}\;.$

On the right-hand side of Eq. (3), the three terms from left to right are
respectively named the “nonlinear growth term”, “shift term”, and “tidal term”
[59, 18]. The linear displacement vector, associated with the shift term, is
defined by the linear dark matter density fluctuation as

$\displaystyle\bm{\Psi}^{[1]}(\bm{x})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot\bm{x}}\frac{i\bm{k}}{k^{2}}\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{k})\;.$ (4)

In the IR limit, where one of the wave vectors in
$F_{2}(\bm{p}_{1},\bm{p}_{2})$ is much smaller than the other (specifically,
when $p_{2}\to 0$), the following relation holds:

$\displaystyle F_{2}(\bm{p}_{1},\bm{p}_{2})\xrightarrow[p_{2}\to
0]{}\frac{1}{2}\left(\frac{\bm{p}_{1}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\;.$
(5)

When substituting Eq. (5) into Eq. (1), the following approximation is
considered simultaneously:

$\displaystyle\delta_{\rm
D}(\bm{k}-\bm{p}_{1}-\bm{p}_{2})\xrightarrow[p_{2}\to 0]{}\delta_{\rm
D}(\bm{k}-\bm{p}_{1})\;.$ (6)

Consequently, in the IR limit, we derive:

$\displaystyle\delta_{\rm m}^{[2]}(\bm{x})\xrightarrow[\rm
IR\,limit]{}-\overline{\bm{\Psi}}^{[1]}\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})\;,$ (7)

where

$\displaystyle\overline{\bm{\Psi}}^{[1]}=\bm{\Psi}^{[1]}(\bm{x}=\bm{0})\;.$
(8)

In deriving Eq. (7), we also accounted for the condition $p_{1}\to 0$ and
introduced a pre-factor of $2$. Comparison with Eq. (3) reveals that the IR
limit solution involves isolating only the shift term, and furthermore, the
displacement vector contained in the shift term is evaluated at the origin
$\bm{x}=\bm{0}$.

Decomposing the second-order fluctuation solution into terms manifesting in
the IR limit and the others, the solution can be reformulated as

$\displaystyle\delta_{\rm
m}^{[2]}=\delta^{[2]}_{\rm(S)m}(\bm{x})-\overline{\bm{\Psi}}^{[1]}\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})\;,$ (9)

where the subscript $(\rm S)$ denotes the contribution from short-wavelength
modes, and $\delta_{\rm(S)m}^{[2]}(\bm{x})$ is explicitly given by

$\displaystyle\delta_{\rm(S)m}^{[2]}(\bm{x})$ $\displaystyle=$
$\displaystyle\frac{17}{21}\left[\delta_{\rm
m}^{[1]}(\bm{x})\right]^{2}-\bm{\Psi}_{\rm(\rm
S)}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm m}^{[1]}(\bm{x})$ (10)
$\displaystyle+$
$\displaystyle\frac{2}{7}\left[\left(\frac{\partial_{i}\partial_{j}}{\partial^{2}}-\frac{1}{3}\delta_{ij}\right)\delta_{\rm
m}^{[1]}(\bm{x})\right]^{2}\;.$

In the equation above, we defined the displacement vector contributing from
short-wavelength modes as

$\displaystyle\bm{\Psi}_{\rm(S)}^{[1]}(\bm{x})=\bm{\Psi}^{[1]}(\bm{x})-\overline{\bm{\Psi}}^{[1]}\;,$
(11)

where $\bm{\Psi}_{\rm(S)}^{[1]}(\bm{x}=\bm{0})=\bm{0}$.

####  II.1.2 Non-perturbative treatment

In the IR limit of the $n$th-order kernel function
$F_{n}(\bm{p}_{1},\cdots,\bm{p}_{n})$, one of the $n$ wavenumbers contributing
to the scale of interest $k=|\bm{p}_{[1,n]}|$ possesses a magnitude
substantially smaller than the other $(n-1)$ wavenumbers. For a single dark
matter fluid in General Relativity (GR), the recursion relation allows the
determination of solutions for $F_{n}(\bm{p}_{1},\cdots,\bm{p}_{n})$ up to an
infinite order (e.g., see [7]). Using mathematical induction on this recursion
relation, the relation below can be proven [39]:

$\displaystyle F_{n}(\bm{p}_{1},\dots,\bm{p}_{n-1},\bm{p})$
$\displaystyle\xrightarrow[p\to
0]{}\frac{1}{n}\left(\frac{\bm{p}_{[1,n-1]}\cdot\bm{p}}{p^{2}}\right)$ (12)
$\displaystyle\hskip 14.22636pt\times\hskip
5.69046ptF_{n-1}(\bm{p}_{1},\dots,\bm{p}_{n-1})\;.$

More generally, for $r$ wavenumbers approaching zero,

$\displaystyle
F_{n}(\bm{p}_{1},\dots,\bm{p}_{n-r},\bm{p}_{n-r+1},\dots,\bm{p}_{n})\xrightarrow[p_{n-r+1},\dots,p_{n}\to
0]{}$
$\displaystyle\frac{(n-r)!}{n!}\prod_{\alpha=1}^{r}\left(\frac{\bm{p}_{[1,n-r]}\cdot\bm{p}_{n-r+\alpha}}{p_{n-r+\alpha}^{2}}\right)F_{n-r}(\bm{p}_{1},\dots,\bm{p}_{n-r})\;.$
(13)

In this context, we decompose the linear dark matter density fluctuation into
long-wavelength modes, with wavenumbers approaching zero, and short-wavelength
modes which are independent of the former:

$\displaystyle\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p})=\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}\to
0)+\widetilde{\delta}_{\rm(S)m}^{\,[1]}(\bm{p})\;.$ (14)

Substituting Eqs. (13) and (14) into Eq. (1) and using the approximation

$\displaystyle\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})\xrightarrow[p_{n-r+1},\dots,p_{n}\to 0]{}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n-r]})\;,$ (15)

the dark matter density fluctuation can be formally expressed as [40]

$\displaystyle\delta_{\rm
m}(\bm{x})=\delta_{\rm(S)m}(\bm{x}-\overline{\bm{\Psi}}^{[1]})\;,$ (16)

where the $n$-th order of $\delta_{\rm(S)m}(\bm{x})$ is given by

$\displaystyle\delta^{[n]}_{{\rm(S)m}}(\bm{x})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}\,e^{i(\bm{p}_{1}+\cdots+\bm{p}_{n})\cdot\bm{x}}$
(17) $\displaystyle\times$ $\displaystyle
F_{n}(\bm{p}_{1},\dots,\bm{p}_{n})\widetilde{\delta}_{\rm(S)m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm(S)m}^{\,[1]}(\bm{p}_{n})\;.$

Thus, the IR limit effect in a single dark matter fluid can be interpreted as
a result of a coordinate transformation due to the displacement vector at the
origin.

The IR limit is an operation that extracts nonlinear contributions from large
scales (small wavenumbers). In other words, the wavenumber $k$ of interest is
considerably larger than the wavenumbers contributed in the IR limit.
Therefore, historically, the IR limit solution has also been referred to as
the high-$k$ ($k\to\infty$) limit solution (e.g., see [60]). Conversely, the
ultraviolet (UV) limit corresponds to the low-$k$ ($k\to 0$) limit and is
treated by e.g. EFT of LSS [12, 13].

In summary, this paper makes three approximations related to the IR limit:

  1. 1.

As illustrated in Eqs. (5) and (13), for the kernel function
$F_{n}(\bm{p}_{1},\cdots,\bm{p}_{n})$ that characterizes nonlinear effects, we
employ the approximate kernel function when one or multiple wavenumbers
approach zero. This means that the dominant effect in the IR limit appears as
the coordinate transformation of the short-wavelength density fluctuation
$\delta_{\rm(S)}$ due to $\overline{\bm{\Psi}}$ as in Eq. (16).

  2. 2.

As presented in Eqs. (6) and (15), within the delta function $\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ that characterizes the mode coupling integral with
multiple wave vectors, we assume no mode coupling occurs between the IR effect
and other components when one or more wavenumbers approach zero. This implies
that the displacement vector is evaluated at the origin in Eq. (8), and
furthermore, indicates no correlation between $\delta_{\rm(S)}$ and
$\overline{\bm{\Psi}}$, as will be discussed in Section II.3.

  3. 3.

The nonlinear effects of $\delta_{\rm(S)m}$ are truncated at a finite
perturbation order. Any higher order perturbative effects beyond this are
treated as IR effects through $\overline{\bm{\Psi}}$.

The clearest example in the IR limit occurs when
$\delta_{\rm(S)m}(\bm{x})\sim\delta_{\rm m}^{[1]}(\bm{x})$, approximating the
nonlinear dark matter density fluctuation as

$\displaystyle\delta_{\rm m}(\bm{x})\xrightarrow[\rm IR\,limit]{}\delta_{\rm
m}^{[1]}(\bm{x}-\overline{\bm{\Psi}}^{[1]}).$ (18)

The conclusions of this paper will be mainly based on this simplest case in
Sections III and IV. When higher order correction terms up to the 1-loop order
are to be considered, they will be explained on there as appropriate.

###  II.2 Extensions to RSD and bias effects

In cases affected by RSD and bias effects, such as observed galaxies, the use
of Lagrangian Perturbation Theory (LPT) is advantageous. This is because LPT
naturally accommodates displacement vectors. Within LPT, RSD effects are
integrated into the displacement vector. Bias effects, on the other hand, can
be treated using the equation [61]:

$\displaystyle 1+\delta_{\rm g}(\bm{x},\hat{n})$ (19) $\displaystyle=$
$\displaystyle\int d^{3}q\left[1+\delta_{\rm bias}(\bm{q})\right]\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{\Psi}_{\rm red}(\bm{q},\hat{n})\right),$

where $\delta_{\rm g}$ represents the galaxy density fluctuation, $\delta_{\rm
bias}$ is the density fluctuation including the bias effect in Lagrangian
space, $\bm{\Psi}_{\rm red}$ is the displacement vector including the RSD
effect, and $\hat{n}$ is a unit vector directed along the line of sight (LOS).
The relation between $\bm{\Psi}$ and $\bm{\Psi}_{\rm red}$ is given by

$\displaystyle\bm{\Psi}_{\rm
red}=\bm{\Psi}+\frac{\dot{\bm{\Psi}}\cdot\hat{n}}{aH}\hat{n},$ (20)

where $\dot{\bm{\Psi}}(\bm{q})$ is the time derivative of $\bm{\Psi}(\bm{q})$,
and $a$ and $H$ are the scale factor and the Hubble parameter, respectively.

The $n$th-order of $\bm{\Psi}(\bm{q})$ in Fourier space is represented as

$\displaystyle\widetilde{\bm{\Psi}}^{\,[n]}_{{\rm m}}(\bm{k})$
$\displaystyle=$ $\displaystyle
i\,\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ (21) $\displaystyle\times$
$\displaystyle\bm{L}_{n}(\bm{p}_{1},\dots,\bm{p}_{n})\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}_{n}),$

where the nonlinear kernel vector functions up to the second-order are given
by

$\displaystyle\bm{L}_{1}(\bm{p}_{1})$ $\displaystyle=$
$\displaystyle\frac{\bm{p}_{1}}{p_{1}^{2}},$
$\displaystyle\bm{L}_{2}(\bm{p}_{1},\bm{p}_{2})$ $\displaystyle=$
$\displaystyle\frac{3}{14}\frac{\bm{p}_{[1,2]}}{|\bm{p}_{[1,2]}|^{2}}\left(1-\mu_{12}^{2}\right)^{2}.$
(22)

Note that the second order kernel vector function
$\bm{L}_{2}(\bm{p}_{1},\bm{p}_{2})$ does not include a shift term but consists
of a combination of a growth term and a tidal term, unlike
$F_{2}(\bm{p}_{1},\bm{p}_{2})$.

The displacement vector is decomposed into two parts: the value at the origin
$\bm{q}=\bm{0}$ and the other values. Thus, we can write

$\displaystyle\bm{\Psi}_{\rm red}(\bm{q},\hat{n})=\overline{\bm{\Psi}}_{\rm
red}(\hat{n})+\bm{\Psi}_{\rm(S)red}(\bm{q},\hat{n}).$ (23)

Here, $\bm{\Psi}_{\rm(S)red}=\bm{\Psi}_{\rm
red}(\bm{q})-\overline{\bm{\Psi}}_{\rm red}$ is termed as the short-wavelength
mode of $\bm{\Psi}_{\rm red}$. Consequently, $\delta_{\rm g}$ can be formally
rewritten as

$\displaystyle\delta_{\rm
g}(\bm{x},\hat{n})=\delta_{\rm(S)g}(\bm{x}-\overline{\bm{\Psi}}_{\rm
red}(\hat{n}),\hat{n}),$ (24)

where

$\displaystyle 1+\delta_{\rm(S)g}(\bm{x},\hat{n})$ (25) $\displaystyle=$
$\displaystyle\int d^{3}q\left[1+\delta_{\rm bias}(\bm{q})\right]\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{\Psi}_{\rm(S)red}(\bm{q},\hat{n})\right),$

In the above derivation, we used $1=\int d^{3}q\delta_{\rm
D}(\bm{x}-\bm{q})=\int d^{3}q\delta_{\rm
D}(\bm{x}-\bm{q}-\overline{\bm{\Psi}}_{\rm red}(\hat{n}))$. In Fourier space,
it becomes

$\displaystyle\widetilde{\delta}_{\rm
g}(\bm{k},\hat{n})=e^{-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}(\hat{n})}\,\widetilde{\delta}_{\rm(S)g}(\bm{k},\hat{n}).$ (26)

Note that Eq. (24) is simply a rewrite of Eq. (19) and, unlike Eq. (16), is
not the result of solving the equation to perturbative infinite order in the
IR limit. To consider Eq. (24) as an extension of the solution of Eq. (16)
that takes into account the bias effect, the RSD effect, and even nonlinear
effects on the displacement vector, one must make an assumption. That is, in
the IR limit, all contributions from IR modes appear as the coordinate
transformation through $\overline{\bm{\Psi}}_{\rm red}$. This is expected to
be true for a single dark matter fluid in GR. However, in Appendix A, we
introduce Degenerate Higher-Order Scalar-Tensor (DHOST) theories (for reviews,
see e.g. [62, 63]), a type of modified gravity theory, as an example of how a
simple rewrite of Eq. (19) into Eq. (24) cannot extract only the IR effect
through $\overline{\bm{\Psi}}_{\rm red}$. Also, while this assumption is
considered valid for the standard bias model in GR (e.g., see [57]), there is
a possibility that it may be violated by other bias effects in the context of
modified gravity theories [64, 65] and multicomponent fluids [66].

####  II.2.1 Perturbative expansion

In Fourier space, the $n$th-order term of the galaxy density fluctuation can
be expressed as

$\displaystyle\widetilde{\delta}_{\rm g}^{\,[n]}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ (27) $\displaystyle\times$ $\displaystyle
Z_{n}(\bm{p}_{1},\dots,\bm{p}_{n},\hat{n})\,\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}_{n}).$

Here, $Z_{n}(\bm{p}_{1},\cdots,\bm{p}_{n})$ represents nonlinear kernel
functions that incorporate both the bias effect and the RSD effect.

Approximating both $\delta^{(\rm S)}(\bm{x})$ and $\overline{\bm{\Psi}}_{\rm
red}$ as linear, we then obtain [55]

$\displaystyle\delta_{\rm g}(\bm{x},\hat{n})$ $\displaystyle\xrightarrow[\rm
IR\,limit]{}$ $\displaystyle\delta_{\rm
g}^{[1]}(\bm{x}-\overline{\bm{\Psi}}_{\rm red}^{[1]}(\hat{n}),\hat{n}),$
$\displaystyle\widetilde{\delta}_{\rm g}(\bm{k},\hat{n})$
$\displaystyle\xrightarrow[\rm IR\,limit]{}$ $\displaystyle
e^{-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}^{[1]}(\hat{n})}\widetilde{\delta}_{\rm g}^{\,[1]}(\bm{k},\hat{n}).$ (28)

The linear galaxy density fluctuation is given by

$\displaystyle\widetilde{\delta}_{\rm
g}^{\,[1]}(\bm{k},\hat{n})=Z^{[1]}(\bm{k},\hat{n})\widetilde{\delta}^{\,[1]}_{\rm
m}(\bm{k}),$ (29)

where $Z_{1}(\bm{k},\hat{n})$ is the first order kernel function [56]:

$\displaystyle Z_{1}(\bm{k},\hat{n})=b_{1}+f\mu^{2}.$ (30)

Here, $b_{1}$ is the linear bias parameter, $f$ represents the linear growth
rate function, and $\mu=\hat{k}\cdot\hat{n}$ denotes the cosine of the angle
between $\hat{k}$ and $\hat{n}$.

From Eq. (26), up to the third-order in perturbation theory,
$\widetilde{\delta}^{\,[n]}_{\rm(S)g}(\bm{k},\hat{n})$ are expressed as

$\displaystyle\widetilde{\delta}^{\,[1]}_{\rm(S)g}(\bm{k},\hat{n})$
$\displaystyle=$ $\displaystyle\widetilde{\delta}^{\,[1]}_{\rm
g}(\bm{k},\hat{n})$
$\displaystyle\widetilde{\delta}^{\,[2]}_{\rm(S)g}(\bm{k},\hat{n})$
$\displaystyle=$ $\displaystyle\widetilde{\delta}^{\,[2]}_{\rm
g}(\bm{k},\hat{n})-\left(-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}^{[1]}(\hat{n})\right)\widetilde{\delta}^{\,[1]}_{\rm g}(\bm{k},\hat{n})$
$\displaystyle\widetilde{\delta}^{\,[3]}_{\rm(S)g}(\bm{k},\hat{n})$
$\displaystyle=$ $\displaystyle\widetilde{\delta}^{\,[3]}_{\rm
g}(\bm{k},\hat{n})-\left(-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}^{[2]}(\hat{n})\right)\widetilde{\delta}^{\,[1]}_{\rm g}(\bm{k},\hat{n})$
(31) $\displaystyle-$
$\displaystyle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}^{[1]}(\hat{n})\right)\widetilde{\delta}^{\,[2]}_{\rm g}(\bm{k},\hat{n})$
$\displaystyle+$
$\displaystyle\frac{1}{2}\left(-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
red}^{[1]}(\hat{n})\right)^{2}\widetilde{\delta}^{\,[1]}_{\rm
g}(\bm{k},\hat{n}).$

The $n$th-order displacement vector in redshift space is computed by the
following linear transformation as [61]

$\displaystyle\bm{\Psi}_{\rm
red}^{[n]}(\bm{q},\hat{n})=\bm{R}_{n}(\hat{n})\cdot\bm{\Psi}^{[n]}(\bm{q})\;.$
(32)

The transformation matrix is given by

$\displaystyle[\bm{R}_{n}]_{ij}=\bm{I}_{ij}+n\,f\,\hat{n}_{i}\hat{n}_{j},$
(33)

where $\bm{I}$ is the three-dimensional identity matrix, and $i,j=1,2,3$.

Throughout the remainder of this paper, we always consider both RSD and bias
effects. To simplify notation, the subscripts “g” and “red”, as well as the
LOS-dependence $\hat{n}$, are omitted in the subsequent sections. We retain
the subscript “m” only for quantities related to dark matter: i.e.,
$\bm{\Psi}_{\rm red}(\bm{q},\hat{n})=\bm{\Psi}(\bm{q})$,
$\overline{\bm{\Psi}}_{\rm red}(\hat{n})=\overline{\bm{\Psi}}$, $\delta_{\rm
g}(\bm{x},\hat{n})=\delta(\bm{x})$, and
$Z_{n}(\bm{p}_{1},\cdots,\bm{p}_{n},\hat{n})=Z_{n}(\bm{p}_{1},\cdots,\bm{p}_{n})$.
Furthermore, a simple right arrow will represent the IR limit:
$\xrightarrow[\rm IR\,limit]{}=\to$.

###  II.3 IR-cancellation of two-point statistics in the IR limit

In this subsection, we provide a brief overview of the IR effect cancellation
in the 2PCF and its Fourier-transformed counterpart, the power spectrum. A
fundamental assumption is the complete absence of correlation between
$\delta_{\rm(S)}(\bm{x})$ and $\overline{\bm{\Psi}}$, which corresponds to the
second assumption in the IR limit discussed at the end of Section II.1. Owing
to the translational symmetry of the ensemble average, all IR effects
originating from $\overline{\bm{\Psi}}$ are canceled out, leaving only the
short-wavelength 2PCF:

$\displaystyle\left\langle\delta(\bm{x})\delta(\bm{x}^{\prime})\right\rangle$
$\displaystyle=$
$\displaystyle\big{\langle}\delta_{\rm(S)}(\bm{x}-\overline{\bm{\Psi}})\delta_{\rm(S)}(\bm{x}^{\prime}-\overline{\bm{\Psi}})\big{\rangle}$
(34) $\displaystyle\to$
$\displaystyle\big{\langle}\delta_{\rm(S)}(\bm{x})\delta_{\rm(S)}(\bm{x}^{\prime})\big{\rangle}\;.$

Assuming that all nonlinear effects emerge only from the IR effects through
$\overline{\bm{\Psi}}^{[1]}$, then the nonlinear 2PCF reduces to the linear
2PCF:

$\displaystyle\left\langle\delta(\bm{x})\delta(\bm{x}^{\prime})\right\rangle$
$\displaystyle\to$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x}-\overline{\bm{\Psi}}^{[1]})\delta^{[1]}(\bm{x}^{\prime}-\overline{\bm{\Psi}}^{[1]})\big{\rangle}$
(35) $\displaystyle=$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x})\delta^{[1]}(\bm{x}^{\prime})\big{\rangle}\;.$

In Fourier space, it becomes

$\displaystyle\left\langle\delta(\bm{k})\delta(\bm{k}^{\prime})\right\rangle$
$\displaystyle\to$
$\displaystyle\big{\langle}e^{-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}}e^{-i\bm{k}^{\prime}\cdot\overline{\bm{\Psi}}^{[1]}}\big{\rangle}\big{\langle}\widetilde{\delta}^{\,[1]}(\bm{k})\widetilde{\delta}^{\,[1]}(\bm{k}^{\prime})\big{\rangle}$
(36) $\displaystyle=$ $\displaystyle(2\pi)^{3}\delta_{\rm
D}(\bm{k}+\bm{k}^{\prime})[Z_{1}(\bm{k})]^{2}\,P^{[11]}_{\rm m}(k)\;,$

where $P_{\rm m}^{[11]}(k)$ is the linear matter power spectrum, calculated as

$\displaystyle\langle\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{k})\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{k}^{\prime})\rangle=(2\pi)^{3}\delta_{\rm
D}\left(\bm{k}+\bm{k}^{\prime}\right)P_{\rm m}^{[11]}(k).$ (37)

In the following, we provide a detailed analysis of the exponential function
$\big{\langle}e^{-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}}e^{-i\bm{k}^{\prime}\cdot\overline{\bm{\Psi}}^{[1]}}\big{\rangle}$
in Eq. (36). Using cumulants, denoted by $\langle\cdots\rangle_{\rm c}$ 111The
cumulants of a statistical quantity $X$, denoted $\langle X^{n}\rangle_{\rm
c}$, are obtained from a power series expansion of the cumulant-generating
function $\ln\langle e^{X}\rangle$:  $\displaystyle\ln\left(\langle
e^{X}\rangle\right)=\sum_{n=1}\frac{1}{n!}\langle X^{n}\rangle_{\rm c}.$ (38)
, this can be reconstructed as

$\displaystyle\big{\langle}e^{-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}}e^{-i\bm{k}^{\prime}\cdot\overline{\bm{\Psi}}^{[1]}}\big{\rangle}={\cal
D}(\bm{k}){\cal D}(\bm{k}^{\prime}){\cal E}(\bm{k},\bm{k}^{\prime}),$ (39)

where ${\cal D}(\bm{k})$ and ${\cal E}(\bm{k},\bm{k}^{\prime})$ are defined as

$\displaystyle{\cal D}(\bm{k})$ $\displaystyle=$
$\displaystyle\exp\left(\frac{1}{2}\left\langle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}\right)^{2}\right\rangle_{\rm
c}\right),$ $\displaystyle{\cal E}(\bm{k},\bm{k}^{\prime})$ $\displaystyle=$
$\displaystyle\exp\left(\left\langle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}\right)\left(-i\bm{k}^{\prime}\cdot\overline{\bm{\Psi}}^{[1]}\right)\right\rangle_{\rm
c}\right).$ (40)

The function ${\cal D}(\bm{k})$ is subsequently described as a two-dimensional
exponentially decaying function [45, 46, 47, 48]:

$\displaystyle{\cal
D}(\bm{k})=\exp\left(-\frac{k^{2}(1-\mu^{2})\sigma_{\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\parallel}}{2}\right),$
(41)

where the radial and transverse components of smoothing factors,
$\sigma_{\parallel}^{2}$ and $\sigma_{\perp}^{2}$, are given by

$\displaystyle\sigma^{2}_{\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int\frac{dp}{2\pi^{2}}P_{\rm m}^{[11]}(p),$
$\displaystyle\sigma^{2}_{\parallel}$ $\displaystyle=$
$\displaystyle(1+f)^{2}\,\sigma^{2}_{\perp}.$ (42)

In addition, the function ${\cal E}(\bm{k},\bm{k}^{\prime})$ satisfies

$\displaystyle{\cal E}(\bm{k},\bm{k})$ $\displaystyle=$ $\displaystyle{\cal
D}^{2}(\bm{k}),$ $\displaystyle{\cal E}(\bm{k},-\bm{k}^{\prime})$
$\displaystyle=$ $\displaystyle{\cal E}^{-1}(\bm{k},\bm{k}^{\prime}),$
$\displaystyle{\cal E}(\bm{k},-\bm{k})$ $\displaystyle=$ $\displaystyle{\cal
D}^{-2}(\bm{k}),$ (43)

and generates an exponentially increasing function as the square of the
inverse of Eq. (41) when the statistical translational symmetry
$\bm{k}+\bm{k^{\prime}}=\bm{0}$ is satisfied. Finally, as indicated in Eq.
(36), the nonlinear galaxy power spectrum converges to the linear galaxy power
spectrum in the IR limit:

$\displaystyle P(\bm{k})$ $\displaystyle\to$
$\displaystyle\big{\langle}e^{-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}}e^{i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}}\big{\rangle}[Z_{1}(\bm{k})]^{2}P_{\rm
m}^{[11]}(k)$ (44) $\displaystyle=$ $\displaystyle{\cal D}^{2}(\bm{k}){\cal
D}^{-2}(\bm{k})[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)$ $\displaystyle=$
$\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;.$

###  II.4 Relations to previous works

####  II.4.1 1-loop corrections in SPT

Consider the next leading order, i.e., the 1-loop order contribution, in the
framework of SPT [7]. This is divided into $P^{[22]}(\bm{k})$, which is the
product of the dual second-order density fluctuations, and $P^{[13]}(\bm{k})$,
which is the product of the first- and third-order density fluctuations, where
$P^{\rm 1\mathchar 45\relax loop}(\bm{k})=P^{[22]}(\bm{k})+P^{[13]}(\bm{k})$.

In this context, $P_{22}$ and $P_{13}$ become

$\displaystyle P^{[22]}(\bm{k})$ $\displaystyle=$ $\displaystyle
2\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,2]})$ $\displaystyle\times$
$\displaystyle[Z_{2}(\bm{p}_{1},\bm{p}_{2})]^{2}P_{\rm m}^{[11]}(p_{1})P_{\rm
m}^{[11]}(p_{2}),$ $\displaystyle P^{[13]}(\bm{k})$ $\displaystyle=$
$\displaystyle 6Z_{1}(\bm{k})\,P_{\rm m}^{[11]}(k)$ (45) $\displaystyle\times$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}Z_{3}(\bm{k},\bm{p},-\bm{p})P_{\rm
m}^{[11]}(p).$

In the IR limit, both $Z_{2}(\bm{k},\bm{p})$ and
$Z_{3}(\bm{k},\bm{p},-\bm{p})$ are approximated as (see Appendix B)

$\displaystyle Z_{2}(\bm{k},\bm{p})$ $\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{2}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)Z_{1}(\bm{k})\;,$
$\displaystyle Z_{3}(\bm{k},\bm{p},-\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle-\frac{1}{3!}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)^{2}Z_{1}(\bm{k})\;,$
(46)

where $\bm{R}_{1}$ is the transformation matrix given in Eq. (33).
Consequently, $P^{[22]}$ and $P^{[13]}$ in the IR limit are described as

$\displaystyle P_{\rm IR}^{[22]}(\bm{k})$ $\displaystyle\to$
$\displaystyle\left(k^{2}(1-\mu^{2})\sigma_{\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\parallel}\right)$
$\displaystyle\times$ $\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;,$
$\displaystyle P_{\rm IR}^{[13]}(\bm{k})$ $\displaystyle\to$
$\displaystyle-\left(k^{2}(1-\mu^{2})\sigma_{\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\parallel}\right)$
(47) $\displaystyle\times$ $\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm
m}^{[11]}(k)\;,$

where the subscript “IR” means the values evaluated in the IR limit. When
summing up $P^{[22]}_{\rm IR}$ and $P_{\rm IR}^{[13]}$, they mutually cancel
out, resulting in a net value of zero. These results align with the form
obtained when expanding ${\cal D}(\bm{k}){\cal D}^{-1}(\bm{k})$ up to the
1-loop order in Eq. (44).

####  II.4.2 $\Gamma$-expansion

The nonlinear power spectrum can generally be decomposed into two components.
One component is proportional to the linear matter power spectrum, and the
other is associated with mode-coupling integrals. This relation is presented
in [10, 47]

$\displaystyle P(\bm{k})=G^{2}(\bm{k})P_{\rm m}^{[11]}(k)+P_{\rm MC}(\bm{k}).$
(48)

Here, the function $G(\bm{k})$ appearing in the first term on the right-hand
side is referred to as the propagator, and the second term is known as the
mode-coupling term.

The $\Gamma$-expansion corresponds to the decomposition of the power spectrum
within the framework of mode-coupling integrals (e.g., see [60]). When
compared with Eq. (48), the first-order $\Gamma$-expansion is related to the
propagator, and the sum of the degrees of $\Gamma$-expansion from the second
order up to infinity constitutes the mode-coupling term. In the context of the
$\Gamma$-expansion, the nonlinear power spectrum in the IR limit is decomposed
into [39, 40]:

$\displaystyle P(\bm{k})$ $\displaystyle\to$ $\displaystyle{\cal
D}^{2}(\bm{k})\sum_{n=1}^{\infty}\frac{\left(k^{2}(1-\mu^{2})\sigma_{\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\parallel}\right)^{n-1}}{(n-1)!}$
(49) $\displaystyle\times$ $\displaystyle[Z_{1}(\bm{k})]^{2}\,P_{\rm
m}^{[11]}(k)\;,$

where the summation index $n$ denotes the $n$th order of the
$\Gamma$-expansion. Furthermore, we obtain the corresponding expressions of
$G^{2}(\bm{k})$ and $P_{\rm MC}(\bm{k})$ in the IR limit:

$\displaystyle G^{2}(\bm{k})P_{\rm m}^{[11]}(k)$ $\displaystyle\to$
$\displaystyle{\cal D}^{2}(\bm{k})[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;,$
$\displaystyle P_{\rm MC}(\bm{k})$ $\displaystyle\to$ $\displaystyle{\cal
D}^{2}(\bm{k})\left({\cal E}(\bm{k},-\bm{k})-1\right)[Z_{1}(\bm{k})]^{2}P_{\rm
m}^{[11]}(k)$ (50) $\displaystyle=$ $\displaystyle\big{(}1-{\cal
D}^{2}(\bm{k})\big{)}[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k).$

It is important to note that the function ${\cal E}(\bm{k},\bm{k}^{\prime})$
serves to connect different wave vectors $\bm{k}$ and $\bm{k}^{\prime}$.
Consequently, a series expansion of ${\cal E}(\bm{k},\bm{k}^{\prime})-1$
yields a term related to the mode coupling integrals.

####  II.4.3 Propagator

In this paper, we focus on the impact of the BAO signal due to non-
perturbative IR effects. To that end, we provide a detailed explanation of the
calculation of the propagator, which directly influences the nonlinear decay
of BAO.

The propagator is defined by the correlation between two factors: nonlinear
galaxy density fluctuations and linear dark matter density fluctuations.

$\displaystyle\langle\delta(\bm{x})\delta_{\rm
m}^{[1]}(\bm{x}^{\prime})\rangle=\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot(\bm{x}-\bm{x}^{\prime})}G(\bm{k})P_{\rm
m}^{[11]}(k).$ (51)

Note that this definition assumes the absence of primordial non-Gaussianities.
When the nonlinear density fluctuations are decomposed into long-wavelength
displacement vectors and short-wavelength density fluctuations, as shown in
Eq. (24), the propagator can be computed as

$\displaystyle\langle\delta_{\rm(S)}(\bm{x}-\overline{\bm{\Psi}})\delta_{\rm
m}^{[1]}(\bm{x}^{\prime})\rangle$ (52) $\displaystyle=$
$\displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}e^{i\bm{k}\cdot(\bm{x}-\bm{x}^{\prime})}\left\langle
e^{-i\bm{k}\cdot\overline{\bm{\Psi}}}\right\rangle$ $\displaystyle\times$
$\displaystyle\left[Z_{1}(\bm{k})P_{\rm
m}^{[11]}(k)+\sum_{n=1}^{\infty}P_{\rm(S)}^{[1(2n+1)]}(\bm{k})/(2Z_{1}(\bm{k}))\right],$

where $\delta_{\rm m}^{[1]}(\bm{x})$ corresponds to short-wavelength modes and
thus correlate only with $\delta_{\rm(S)}(\bm{x})$, and
$P_{\rm(S)}^{[1(2n+1)]}$ represents the cross-power spectrum of
$\delta_{\rm(S)}^{[2n+1]}$ and $\delta_{\rm m}^{[1]}$. Note that only odd
orders appear in $P_{\rm(S)}^{[1(2n+1)]}$ since primordial non-Gaussianities
are absent.

By truncating $\delta_{\rm(S)}$ up to the third-order and by further
approximating $\overline{\bm{\Psi}}$ to be of linear order, we obtain

$\displaystyle G(\bm{k})$ (53) $\displaystyle\simeq$ $\displaystyle{\cal
D}(\bm{k})\left[Z_{1}(\bm{k})+P_{\rm(S)}^{[13]}(\bm{k})\Big{/}\left(2Z_{1}(\bm{k})P_{\rm
m}^{[11]}(k)\right)\right].$

Here, $P_{\rm(S)}^{[13]}$ is defined as

$\displaystyle P_{\rm(S)}^{[13]}(\bm{k})=P^{[13]}(\bm{k})-P_{\rm
IR}^{[13]}(\bm{k}),$ (54)

where $P_{\rm IR}^{[13]}$ represents the IR limit value of $P^{[13]}$ as given
in Eq. (47).

In the definition of the propagator (52), it is important to note that
nonlinear density fluctuations and linear density fluctuations possess
different IR effects. In particular, there is no IR cancellation based on
statistical translational symmetry in the calculation of the propagator. The
exponential decay of the propagator is due to the absence of this IR
cancellation. Similarly, other higher-order coefficients in the
$\Gamma$-expansion decay exponentially, as shown in Eq. (49).

This paper repeatedly emphasizes that an exponential decay effect is
consistently observed when calculating cross-correlations between fluctuations
with distinct IR effects. Notably, Section IV will demonstrate the exponential
decay of the cross-power spectrum for both pre- and post-reconstruction
density fluctuations 222As another example, Chisari and Pontzen [67] showed in
the Zel’dovich approximation that the cross-power spectrum of density
fluctuations evaluated at two different times decays exponentially due to IR
cancellation breaking..

###  II.5 IR-resummed model for
the pre-reconstruction power spectrum

In the previous section, we discussed the IR cancellation under the assumption
that $\delta^{[1]}(\bm{x})$ and $\overline{\bm{\Psi}}^{[1]}$ are uncorrelated.
This lack of correlation is due to the three assumptions mentioned at the end
of Section II.1.2, especially the one stating that there is no mode coupling
between the IR effect and the short-wavelength density fluctuation. In
practice, however, there is mode-coupling that prevents complete IR
cancellation. By addressing this imperfection in IR cancellation and isolating
only the effects of BAO, we can construct a theoretical template model that
effectively handles the nonlinear damping effects of BAO.

####  II.5.1 Linear level

As an example, the lowest-order term among the mode-coupling terms is
$P^{[22]}(\bm{k})$. Applying the delta function with the limit $p_{2}\to 0$
(or $p_{1}\to 0$) in Eq. (45), $P^{[22]}(\bm{k})$ becomes

$\displaystyle P^{[22]}(\bm{k})\to 4\,P_{\rm
m}^{[11]}(k)\int\frac{d^{3}p}{(2\pi)^{3}}\big{[}Z^{[2]}(\bm{k},\bm{p})\big{]}^{2}P_{\rm
m}^{[11]}(p).$ (55)

The equation above indicates that when we decouple the large-scale mode from
the small-scale mode, the mode coupling term is proportional to the linear
matter power spectrum, $P_{\rm m}^{[11]}(k)$. However, in reality, the
integral of the mode coupling is expected to smooth out the BAO signal in the
linear power spectrum, resulting in a smoother function. To properly account
for this effect, we decompose the linear matter power spectrum into two
components: the “wiggle” component containing only the BAO signal and the “no-
wiggle” component without BAO [68]. Thus, it is expressed as $P_{\rm
m}^{[11]}(k)=P_{\rm w}(k)+P_{\rm nw}(k)$, where the subscripts “w” and “nw”
denote “wiggle” and “no-wiggle”, respectively. By replacing $P_{\rm
m}^{[11]}(k)$ with $P_{\rm nw}(k)$, and by using Eq. (46), we obtain

$\displaystyle P_{\rm IR}^{[22]}(\bm{k})=-2\ln{\cal
D}(\bm{k})[Z_{1}(\bm{k})]^{2}P_{\rm nw}(k)\;.$ (56)

The approximation used for $P^{[22]}$ can also be applied for the non-
perturbative mode coupling term in the IR limit. Based on Eq. (50), replacing
$P_{\rm m}^{[11]}(k)$ that appears in the mode coupling term with $P_{\rm
nw}(k)$ yields

$\displaystyle P(\bm{k})$ $\displaystyle\to$ $\displaystyle{\cal
D}^{2}(\bm{k})\big{[}Z_{1}(\bm{k})\big{]}^{2}P_{\rm m}^{[11]}(k)$ (57)
$\displaystyle+$ $\displaystyle\big{(}1-{\cal
D}^{2}(\bm{k})\big{)}\big{[}Z_{1}(\bm{k})\big{]}^{2}P_{\rm nw}(k)$
$\displaystyle=$ $\displaystyle\big{[}Z_{1}(\bm{k})\big{]}^{2}\left[{\cal
D}^{2}(\bm{k})P_{\rm w}(k)+P_{\rm nw}(k)\right]\;.$

This form is consistent with the template model proposed by [46], which has
been widely used for BAO analyses. The method used to derive the power
spectrum model in Eq. (57) is based on [40, 55]. On the other hand, for other
approaches concerning the derivation of the IR-resummed power spectrum model,
see also [50, 58, 51, 53].

####  II.5.2 1-loop level

Let us consider the 1-loop order correction terms. In the IR limit, the
nonlinear galaxy power spectrum is described as

$\displaystyle P(\bm{k})\to{\cal D}^{2}(\bm{k}){\cal
D}^{-2}(\bm{k})P_{\rm(S)}(\bm{k})\;,$ (58)

where $P_{\rm(S)}$ is consists of only the short-wavelength density
fluctuations, $\widetilde{\delta}_{\rm(S)}$. At the 1-loop level, we derive
from Eq. (31)

$\displaystyle P_{\rm(S)}^{[22]}(\bm{k})$ $\displaystyle=$ $\displaystyle
P^{[22]}(\bm{k})-P_{\rm IR}^{[22]}(\bm{k})\;,$ $\displaystyle
P_{\rm(S)}^{[13]}(\bm{k})$ $\displaystyle=$ $\displaystyle
P^{[13]}(\bm{k})-P_{\rm IR}^{[13]}(\bm{k})\;,$ (59)

where we used the uncorrelation between $\overline{\bm{\Psi}}$ and
$\delta_{\rm(S)}$. Compared with the propagator including the 1-loop
correction term in Eq. (53), the propagator and the mode-coupling term at the
1-loop level are respectively shown as

$\displaystyle G^{2}(\bm{k})P_{\rm m}^{[11]}(k)$ (60) $\displaystyle=$
$\displaystyle{\cal D}^{2}(\bm{k})\left[\left[Z_{1}(\bm{k})\right]^{2}P_{\rm
m}^{[11]}(k)+P_{\rm(S)}^{[13]}(\bm{k})\right]\;,$

and

$\displaystyle P_{\rm MC}(\bm{k})$ (61) $\displaystyle=$
$\displaystyle\left(1-{\cal
D}^{2}(\bm{k})\right)\left[\left[Z_{1}(\bm{k})\right]^{2}P_{\rm
m}^{[11]}(k)+P_{\rm(S)}^{[13]}(\bm{k})\right]$ $\displaystyle+$ $\displaystyle
P_{\rm(S)}^{[22]}(\bm{k})\;.$

By replacing all linear matter power spectrum $P_{\rm m}^{[11]}$ appearing in
the mode-coupling term with the no-wiggle matter power spectrum $P_{\rm nw}$,
we finally obtain the IR-resummed power spectrum model at the 1-loop level
[40]:

$\displaystyle P(\bm{k})$ $\displaystyle=$ $\displaystyle{\cal
D}^{2}(\bm{k})\left[\left[Z_{1}(\bm{k})\right]^{2}+P_{\rm(S)}^{[13]}(\bm{k})\big{/}P_{\rm
m}^{[11]}(k)\right]P_{\rm w}(k)$ (62) $\displaystyle+$
$\displaystyle\left[Z_{1}(\bm{k})\right]^{2}P_{\rm nw}(k)+P^{[13]}_{\rm
nw}(\bm{k})+P^{[22]}(\bm{k})\;,$

where

$\displaystyle P^{[13]}_{\rm nw}(\bm{k})=P^{[13]}(\bm{k})\left(P_{\rm
nw}(k)/P_{\rm m}^{[11]}(k)\right)\;.$ (63)

####  II.5.3 Mode-coupling contributions

In this paper, we have entirely disregarded the effects of BAO arising from
the mode-coupling terms. However, it is well-known that contributions from
these mode-coupling terms introduce additional corrective terms to the
nonlinear damping of BAO. For instance, the perpendicular dispersion,
$\sigma_{\perp}$, in Eq. (42) should be modified as follows [50, 51, 52]:

$\displaystyle\sigma_{\perp}^{2}=\frac{1}{3}\int\frac{dp}{2\pi^{2}}P_{\rm
m}^{[11]}(p)\left[1-j_{0}(pr_{\rm s})\right]\;,$ (64)

where $j_{0}$ is the zeroth-order spherical Bessel function, and $r_{\rm s}$
is the sound horizon scale. The correction term in the above equation only
slightly changes the value of the smoothing parameter $\sigma^{2}$ in the
Gaussian decaying function ${\cal D}$ (41), and does not change the form of
the function itself. Therefore, for the sake of simplicity of the theoretical
model, we did not calculate these mode-coupling terms in detail in this paper.
The contributions from the mode-coupling terms are naturally accounted for in
methods like CLPT [8, 9], which involve appropriate calculations of mode-
coupling integrals. Applying further detailed calculations on these mode
coupling terms to the post-reconstruction case will be a topic for future
work.

##  III Post-Reconstruction Case

We apply the method for constructing the IR-resummed power spectrum model from
the pre-reconstruction to the post-reconstruction case.

###  III.1 Post-reconstruction density fluctuations

To reconstruct the galaxy distribution, the displacement vector for
reconstruction is calculated from the observed galaxy density fluctuations as
follows [1]:

$\displaystyle\bm{s}(\bm{x})=\int\frac{d^{3}p}{(2\pi)^{3}}e^{i\bm{p}\cdot\bm{x}}\left(\frac{i\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{1,\rm fid}}\right)\widetilde{\delta}_{\rm obs}(\bm{p}),$
(65)

where $b_{1,\rm fid}$ is an input fiducial linear bias parameter for
reconstruction, $W_{\rm G}(pR_{\rm s})=\exp\left(-p^{2}R_{\rm s}^{2}/2\right)$
is a Gaussian filter function, and $R_{\rm s}$ is an input smoothing scale.
This $\bm{s}(\bm{x})$ is derived from the observed galaxy fluctuation
$\delta_{\rm obs}(\bm{x})$. By substituting $\delta_{\rm obs}$ with the
nonlinear density fluctuation $\delta$ in Eq. (19), which includes the bias
and RSD effects, we can make a theoretical prediction of $\bm{s}(\bm{x})$.

The reconstruction scheme uses $\bm{s}(\bm{x})$ to move each of the galaxy
data particles and the corresponding random particles. Such an operation is
expressed as

$\displaystyle 1+\delta_{\rm d}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}\left(1+\delta(\bm{x})\right)\delta_{\rm
D}\left(\bm{x}-\bm{x}^{\prime}-\bm{s}(\bm{x}^{\prime})\right)$ $\displaystyle
1+\delta_{\rm s}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}\delta_{\rm
D}\left(\bm{x}-\bm{x}^{\prime}-\bm{s}(\bm{x}^{\prime})\right).$ (66)

Here, $\delta_{\rm d}(\bm{x})$ is the density fluctuation of the reconstructed
galaxy data particles, and $\delta_{\rm s}(\bm{x})$ is the density fluctuation
of the reconstructed random particles. The observed post-reconstruction galaxy
density fluctuation is then given by [55, 6]

$\displaystyle\delta_{\rm rec}(\bm{x})$ $\displaystyle=$
$\displaystyle\delta_{\rm d}(\bm{x})-\delta_{\rm s}(\bm{x})$ (67)
$\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}\delta(\bm{x}^{\prime})\delta_{\rm
D}\left(\bm{x}-\bm{x}^{\prime}-\bm{s}(\bm{x}^{\prime})\right).$

###  III.2 IR effects on the post-reconstruction density fluctuation

The displacement vector for reconstruction, $\bm{s}(\bm{x})$, is derived from
the nonlinear density fluctuation, which includes the shift term that should
be considered in the IR limit (e.g., see Eq. 2). This characteristic of
$\bm{s}(\bm{x})$ is unique, contrasting with the displacement vector
$\bm{\Psi}(\bm{q})$, where the shift term is excluded (e.g., Eq. 22). In this
subsection, we focus on examining the non-perturbative behavior of the shift
term within $\bm{s}(\bm{x})$ and demonstrate its impact on the post-
reconstruction density fluctuation.

Following the approach introduced in Section II, we decompose the galaxy
density fluctuation $\delta(\bm{x})$ into two components: the IR effect
represented by $\overline{\bm{\Psi}}$, and the short-wavelength density
fluctuation $\delta_{\rm(S)}(\bm{x})$. By substituting Eq. (24) and Eq. (26)
into Eq. (65) and Eq. (67), we obtain

$\displaystyle\delta_{\rm rec}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime}\delta_{\rm(S)}(\bm{x}^{\prime}-\overline{\bm{\Psi}})\delta_{\rm
D}\left(\bm{x}-\bm{x}^{\prime}-\bm{s}_{\rm(S)}(\bm{x}^{\prime}-\overline{\bm{\Psi}})\right)$
(68) $\displaystyle=$ $\displaystyle\int
d^{3}x^{\prime\prime}\delta_{\rm(S)}(\bm{x}^{\prime\prime})\delta_{\rm
D}\left(\bm{x}-\overline{\bm{\Psi}}-\bm{x}^{\prime\prime}-\bm{s}_{\rm(S)}(\bm{x}^{\prime\prime})\right).$

In the second line, we used the coordinate transformation:
$\bm{x}^{\prime\prime}=\bm{x}^{\prime}-\overline{\bm{\Psi}}$. Here,
$\bm{s}_{\rm(S)}(\bm{x})$ is defined as

$\displaystyle\bm{s}_{\rm(S)}(\bm{x})=\int\frac{d^{3}p}{(2\pi)^{3}}e^{i\bm{p}\cdot\bm{x}}\left(\frac{i\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{1,\rm fid}}\right)\widetilde{\delta}_{\rm(S)}(\bm{p}).$
(69)

In the IR limit, the short-wavelength displacement vector for reconstruction,
$\bm{s}_{\rm(S)}$, no longer receives contributions from the shift term,
leading to its analogy with the displacement vector $\bm{\Psi}$. As
demonstrated in Eqs. (23) and (24), the IR effect within the nonlinear density
fluctuation is represented by $\bm{\Psi}$ at the origin. Similarly to
$\bm{\Psi}$, by decomposing $\bm{s}_{\rm S}$ into its origin value and other
components, we can elucidate the IR effect within the post-reconstruction
density fluctuation:

$\displaystyle\bm{s}_{\rm(S)}(\bm{x})=\overline{\bm{s}}+\bm{s}_{\rm(SS)}(\bm{x}),$
(70)

where $\overline{\bm{s}}=\bm{s}_{\rm S}(\bm{x}=\bm{0})$. Finally, $\delta_{\rm
rec}(\bm{x})$ can be formally rewritten as

$\displaystyle\delta_{\rm
rec}(\bm{x})=\delta_{\rm(S)rec}(\bm{x}-\overline{\bm{\Psi}}_{\rm rec}),$ (71)

where

$\displaystyle\overline{\bm{\Psi}}_{\rm
rec}=\overline{\bm{\Psi}}+\overline{\bm{s}},$ (72)

and

$\displaystyle\delta_{\rm(S)rec}(\bm{x})=\int
d^{3}x^{\prime}\delta_{\rm(S)}(\bm{x}^{\prime})\delta_{\rm
D}\left(\bm{x}-\bm{x}^{\prime}-\bm{s}_{\rm(SS)}(\bm{x}^{\prime})\right).$ (73)

This result shows that the IR effect manifests in the post-reconstruction
density fluctuation as the coordinate transformation via
$\overline{\bm{\Psi}}_{\rm rec}$, like the pre-reconstruction case in Eq.
(24). Consequently, the arguments applicable to the pre-reconstruction IR
effect can similarly be applied to the post-reconstruction context.

###  III.3 Lagrangian space

We can also perform the calculation of density fluctuations after
reconstruction in Lagrangian space. To facilitate a comparison with the
Zel’dovich approximation, set to be discussed in Section III.5, we provide
proof for Eq. (71) in Lagrangian space.

In Lagrangian space, $\delta_{\rm d}(\bm{x})$ and $\delta_{\rm s}(\bm{x})$ are
each expressed as

$\displaystyle 1+\delta_{\rm d}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\left(1+\delta_{\rm bias}(\bm{q})\right)$ $\displaystyle\times$
$\displaystyle\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{\Psi}(\bm{q})-\bm{s}(\bm{q}+\bm{\Psi}(\bm{q}))\right),$
$\displaystyle 1+\delta_{\rm s}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\delta_{\rm D}\left(\bm{x}-\bm{q}-\bm{s}(\bm{q})\right).$ (74)

Similar to Eq. (68), they can be rewritten as

$\displaystyle 1+\delta_{\rm d}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\left(1+\delta_{\rm bias}(\bm{q})\right)$ $\displaystyle\times$
$\displaystyle\delta_{\rm
D}\left(\bm{x}-\bm{q}-\overline{\bm{\Psi}}-\bm{\Psi}_{\rm(S)}(\bm{q})-\bm{s}_{\rm(S)}(\bm{q}+\bm{\Psi}_{\rm(S)}(\bm{q}))\right),$
$\displaystyle 1+\delta_{\rm s}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{s}_{\rm(S)}(\bm{q}-\overline{\bm{\Psi}})\right)$
(75) $\displaystyle=$ $\displaystyle\int d^{3}q^{\prime}\delta_{\rm
D}\left(\bm{x}-\bm{q}^{\prime}-\overline{\bm{\Psi}}-\bm{s}_{\rm(S)}(\bm{q}^{\prime})\right),$

where $\bm{\Psi}_{\rm(S)}(\bm{q})=\bm{\Psi}(\bm{q})-\overline{\bm{\Psi}}$ in
Eq. (23). For $\delta_{\rm s}(\bm{x})$, we used the coordinate transformation:
$\bm{q}^{\prime}=\bm{q}-\overline{\bm{\Psi}}$.

Note that $\bm{s}(\bm{q}+\bm{\Psi}(\bm{q}))$ within $\delta_{\rm d}(\bm{x})$
does not include the contribution from the shift term in the IR limit. This is
because the coordinate transformation, represented by
$\bm{q}+\bm{\Psi}(\bm{q})$, cancels out the IR effect of
$\overline{\bm{\Psi}}$. As a result, it can be expressed only in terms of
short-wavelength modes as
$\bm{s}_{\rm(S)}(\bm{q}+\bm{\Psi}_{\rm(S)}(\bm{q}))$.

Furthermore, we decompose $\bm{s}_{\rm(S)}$ into the value at the origin
$\bm{q}=\bm{0}$ and other parts:

$\displaystyle\bm{s}_{\rm(S)}(\bm{q}+\bm{\Psi}_{\rm(S)}(\bm{q}))$
$\displaystyle=$
$\displaystyle\overline{\bm{s}}+\bm{s}_{\rm(SS)}(\bm{q}+\bm{\Psi}_{\rm(S)}(\bm{q}))$
$\displaystyle\bm{s}_{\rm(S)}(\bm{q})$ $\displaystyle=$
$\displaystyle\overline{\bm{s}}+\bm{s}_{\rm(SS)}(\bm{q}),$ (76)

where we used $\bm{\Psi}_{\rm S}(\bm{q}=\bm{0})=\bm{0}$.

Finally, we obtain

$\displaystyle\delta_{\rm rec}(\bm{x})$ $\displaystyle=$
$\displaystyle\delta_{\rm(S)d}(\bm{x}-\overline{\bm{\Psi}}_{\rm
rec})-\delta_{\rm(S)s}(\bm{x}-\overline{\bm{\Psi}}_{\rm rec})$ (77)
$\displaystyle=$
$\displaystyle\delta_{\rm(S)rec}(\bm{x}-\overline{\bm{\Psi}}_{\rm rec}),$

where

$\displaystyle 1+\delta_{\rm(S)d}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\left(1+\delta_{\rm bias}(\bm{q})\right)$ $\displaystyle\times$
$\displaystyle\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{\Psi}_{\rm(S)}(\bm{q})-\bm{s}_{\rm(SS)}(\bm{q}+\bm{\Psi}_{\rm(S)}(\bm{q}))\right),$
$\displaystyle 1+\delta_{\rm(S)s}(\bm{x})$ $\displaystyle=$ $\displaystyle\int
d^{3}q\delta_{\rm D}\left(\bm{x}-\bm{q}-\bm{s}_{\rm(SS)}(\bm{q})\right).$ (78)

###  III.4 IR cancellation after reconstruction

We can show that the IR cancellation occurs even after reconstruction:

$\displaystyle\left\langle\delta_{\rm rec}(\bm{x})\delta_{\rm
rec}(\bm{x}^{\prime})\right\rangle$ $\displaystyle=$
$\displaystyle\big{\langle}\delta_{\rm(S)rec}(\bm{x}-\overline{\bm{\Psi}}_{\rm
rec})\delta_{\rm(S)rec}(\bm{x}^{\prime}-\overline{\bm{\Psi}}_{\rm
rec})\big{\rangle}$ (79) $\displaystyle\to$
$\displaystyle\big{\langle}\delta_{\rm(S)rec}(\bm{x})\delta_{\rm(S)rec}(\bm{x}^{\prime})\big{\rangle}\;.$

Approximating both $\delta_{\rm(S)rec}(\bm{x})$ and $\overline{\bm{\Psi}}_{\rm
rec}$ as linear, we obtain

$\displaystyle\left\langle\delta_{\rm rec}(\bm{x})\delta_{\rm
rec}(\bm{x}^{\prime})\right\rangle$ $\displaystyle\to$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x}-\overline{\bm{\Psi}}^{[1]}_{\rm
rec})\delta^{[1]}(\bm{x}^{\prime}-\overline{\bm{\Psi}}^{[1]}_{\rm
rec})\big{\rangle}$ (80) $\displaystyle=$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x})\delta^{[1]}(\bm{x}^{\prime})\big{\rangle}\;.$

Under this approximation, the propagator term and the mode-coupling term are
respectively represented as

$\displaystyle G^{2}(\bm{k})P_{\rm m}^{[11]}(k)\to{\cal D}_{\rm
rec}^{2}(\bm{k})[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;,$ (81)

and

$\displaystyle P_{\rm MC}(\bm{k})\to(1-{\cal D}_{\rm
rec}^{2}(\bm{k}))[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;.$ (82)

These specifically lead to the IR cancellation in Fourier space:

$\displaystyle P_{\rm rec}(\bm{k})$ $\displaystyle\to$ $\displaystyle{\cal
D}_{\rm rec}^{2}(\bm{k})[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)$ (83)
$\displaystyle+$ $\displaystyle(1-{\cal D}_{\rm
rec}^{2}(\bm{k}))[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)$ $\displaystyle=$
$\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k)\;.$

Here, ${\cal D}_{\rm rec}(\bm{k})$ is the two-dimensional exponentially
decaying function after reconstruction, given by

$\displaystyle{\cal D}_{\rm rec}(\bm{k})$ $\displaystyle=$
$\displaystyle\exp\left(\frac{1}{2}\left\langle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}_{\rm
rec}^{[1]}\right)^{2}\right\rangle_{\rm c}\right),$ (84) $\displaystyle=$
$\displaystyle\exp\left(-\frac{k^{2}(1-\mu^{2})\sigma^{2}_{\rm
rec,\perp}+k^{2}\mu^{2}\sigma^{2}_{\rm rec,\parallel}}{2}\right).$

The radial and transverse components of the smoothing factors are further
decomposed into

$\displaystyle\sigma_{\rm rec,\perp}^{2}$ $\displaystyle=$
$\displaystyle\sigma_{\perp}^{2}+\sigma_{\rm ps,\perp}^{2}+\sigma_{\rm
ss,\perp}^{2}\;,$ $\displaystyle\sigma_{\rm rec,\parallel}^{2}$
$\displaystyle=$ $\displaystyle\sigma_{\parallel}^{2}+\sigma_{\rm
ps,\parallel}^{2}+\sigma_{\rm ss,\parallel}^{2}\;,$ (85)

where $\sigma_{\perp}^{2}$ and $\sigma_{\parallel}^{2}$ are given in Eq. (42).
The subscript “ps” means the correlation between $\overline{\bm{\Psi}}$ and
$\overline{\bm{s}}$, and “ss” denotes the auto-correlation of
$\overline{\bm{s}}$. These smoothing factors are calculated as

$\displaystyle\sigma^{2}_{\rm ps,\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)\left[2\left(b_{1}+\frac{f}{5}\right)\right]P_{\rm m,lin}(p)\;,$
$\displaystyle\sigma^{2}_{\rm ps,\parallel}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)\Bigg{[}2\left(1+f\right)\left(b_{1}+\frac{3}{5}f\right)\Bigg{]}P_{\rm
m,lin}(p)\;,$ $\displaystyle\sigma^{2}_{\rm ss,\perp}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\int_{k_{\rm min}}^{k_{\rm
max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm s})}{b_{1,\rm
fid}}\right)^{2}\left[\left(b_{1}^{2}+\frac{2}{5}b_{1}f+\frac{3}{35}f^{2}\right)P_{\rm
m,lin}(p)+\frac{1}{\bar{n}}\right]\;,$ $\displaystyle\sigma^{2}_{\rm
ss,\parallel}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\int_{k_{\rm
min}}^{k_{\rm max}}\frac{dp}{2\pi^{2}}\left(-\frac{W_{\rm G}(pR_{\rm
s})}{b_{1,\rm
fid}}\right)^{2}\Bigg{[}\left(b_{1}^{2}+\frac{42}{35}b_{1}f+\frac{3}{7}f^{2}\right)P_{\rm
m,lin}(p)+\frac{1}{\bar{n}}\Bigg{]}\;.$ (86)

Here, we summarize the key considerations when performing the above
calculations:

  1. 1.

The integral range should depend on the scale at which $\bm{s}(\bm{x})$ is
measured.

  2. 2.

Since $\bm{s}(\bm{x})$ is derived from the power spectrum of the measured
galaxy density fluctuations, we should consider the shot-noise term
$1/\bar{n}$ in $\sigma_{\rm ss,\perp}^{2}$ and $\sigma_{\rm
ss,\parallel}^{2}$, where $\bar{n}$ is the mean number density [69].

###  III.5 Comparisons with the Zel’dovich approximation

Previous studies [21, 22, 23, 24] have often used a simple linear
approximation for $\bm{s}(\bm{x})$, known as the Zel’dovich approximation for
reconstruction. This approximation, however, does not account for nonlinear IR
effects in $\bm{s}(\bm{x})$. In this subsection, we present a detailed
comparison between the Zel’dovich approximation and our findings in Section
III.4.

In the Zel’dovich approximation, Eq. (74) becomes

$\displaystyle 1+\delta_{\rm d}(\bm{x})$ $\displaystyle\sim$
$\displaystyle\int d^{3}q\left(1+\delta_{\rm bias}^{[1]}(\bm{q})\right)$
$\displaystyle\times$ $\displaystyle\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{\Psi}^{[1]}(\bm{q})-\bm{s}^{[1]}(\bm{q})\right)\;,$
$\displaystyle 1+\delta_{\rm s}(\bm{x})$ $\displaystyle\sim$
$\displaystyle\int d^{3}q\delta_{\rm
D}\left(\bm{x}-\bm{q}-\bm{s}^{[1]}(\bm{q})\right)\;.$ (87)

In the IR limit, they are represented as

$\displaystyle\delta_{\rm d}(\bm{x})$ $\displaystyle\to$
$\displaystyle\delta_{\rm
d}^{[1]}\left(\bm{x}-\overline{\bm{\Psi}}^{[1]}-\overline{\bm{s}}^{[1]}\right)\;,$
$\displaystyle\delta_{\rm s}(\bm{x})$ $\displaystyle\to$
$\displaystyle\delta_{\rm
s}^{[1]}\left(\bm{x}-\overline{\bm{s}}^{[1]}\right)\;.$ (88)

We then derive

$\displaystyle\delta_{\rm rec}(\bm{x})\to\delta_{\rm
d}^{[1]}\left(\bm{x}-\overline{\bm{\Psi}}^{[1]}-\overline{\bm{s}}^{[1]}\right)-\delta_{\rm
s}^{[1]}\left(\bm{x}-\overline{\bm{s}}^{[1]}\right)\;.$ (89)

The key difference of this equation from Eq. (77) is how $\bm{s}(\bm{q})$ is
treated within $\delta_{\rm s}(\bm{x})$. In Eq. (77), non-perturbative IR
effects are incorporated into $\bm{s}(\bm{q})$, revealing both
$\overline{\bm{\Psi}}$ and $\overline{\bm{s}}$ effects in $\delta_{\rm
s}(\bm{x})$. In contrast, the Zel’dovich approximation assumes a linear
$\bm{s}(\bm{q})$, resulting in only the $\overline{\bm{s}}$ effect.

In Fourier space, Eq. (89) becomes

$\displaystyle\widetilde{\delta}_{\rm rec}(\bm{k})$ (90) $\displaystyle\to$
$\displaystyle\Big{[}e^{-i\bm{k}\cdot\left(\overline{\bm{\Psi}}^{[1]}+\overline{\bm{s}}^{[1]}\right)}Z_{{\rm
d},1}(\bm{k})-e^{-i\bm{k}\cdot\overline{\bm{s}}^{[1]}}Z_{{\rm
s},1}(\bm{k})\Big{]}\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{k})\;,$

where

$\displaystyle Z_{{\rm d},1}(\bm{k})$ $\displaystyle=$
$\displaystyle\left[1+\left(-\frac{W_{\rm G}(kR_{\rm s})}{b_{1,\rm
fid}}\right)\right]Z_{1}(\bm{k})\;,$ $\displaystyle Z_{{\rm s},1}(\bm{k})$
$\displaystyle=$ $\displaystyle\left(-\frac{W_{\rm G}(kR_{\rm s})}{b_{1,\rm
fid}}\right)\,Z_{1}(\bm{k})\;.$ (91)

The propagator is then given by

$\displaystyle G(\bm{k})={\cal D}_{\rm rec}(\bm{k})Z_{{\rm d},1}(\bm{k})-{\cal
D}_{\rm ss}(\bm{k})Z_{{\rm s},1}(\bm{k})\;,$ (92)

with

$\displaystyle{\cal D}_{\rm
ss}(\bm{k})=\exp\left(\frac{1}{2}\left\langle\left(-i\bm{k}\cdot\overline{\bm{s}}^{[1]}\right)^{2}\right\rangle_{\rm
c}\right)\;.$ (93)

Equation (92) shows that the propagator in the Zel’dovich approximation
comprises two Gaussian decay functions: ${\cal D}_{\rm rec}(\bm{k})$ and
${\cal D}_{\rm ss}(\bm{k})$. This contrasts with Eq. (81), where only ${\cal
D}_{\rm rec}(\bm{k})$ appears. The specific expression for ${\cal D}_{\rm
ss}(\bm{k})$ is

$\displaystyle{\cal D}_{\rm
ss}(\bm{k})=\exp\left(-\frac{k^{2}(1-\mu^{2})\sigma^{2}_{\rm
ss,\perp}+k^{2}\mu^{2}\sigma^{2}_{\rm ss,\parallel}}{2}\right),$ (94)

where the radial and transverse components of the smoothing factors are given
in Eq. (86).

We can also demonstrate that Eq. (89) breaks the IR cancellation:

$\displaystyle\langle\delta_{\rm rec}(\bm{x})\delta_{\rm
rec}(\bm{x}^{\prime})\rangle$ $\displaystyle\to$
$\displaystyle\langle\delta_{\rm d}^{[1]}(\bm{x})\delta_{\rm
d}^{[1]}(\bm{x}^{\prime})\rangle$ (95) $\displaystyle-$
$\displaystyle\langle\delta_{\rm
d}^{[1]}(\bm{x}-\overline{\bm{\Psi}}^{[1]})\delta^{[1]}_{\rm
s}(\bm{x}^{\prime})\rangle$ $\displaystyle-$ $\displaystyle\langle\delta_{\rm
s}^{[1]}(\bm{x})\delta^{[1]}_{\rm
d}(\bm{x}^{\prime}-\overline{\bm{\Psi}}^{[1]})\rangle$ $\displaystyle+$
$\displaystyle\langle\delta_{\rm s}^{[1]}(\bm{x})\delta^{[1]}_{\rm
s}(\bm{x}^{\prime})\rangle\;.$

In the cross-correlation functions between $\delta_{\rm d}(\bm{x})$ and
$\delta_{\rm s}(\bm{x}^{\prime})$, the IR cancellation does not occur, and the
contribution of $\overline{\bm{\Psi}}^{[1]}$ remains. In Fourier space, the
cross-power spectrum between $\widetilde{\delta}_{\rm d}(\bm{k})$ and
$\widetilde{\delta}_{\rm s}(\bm{k}^{\prime})$ is represented as

$\displaystyle P_{\rm ds}(\bm{k})\to{\cal D}(\bm{k})Z_{{\rm
d},1}(\bm{k})Z_{{\rm s},1}(\bm{k})P_{\rm m}^{[11]}(k)\;.$ (96)

Hence, breaking the IR cancellation leads to an exponential decay of the power
spectrum. This finding deviates from our result where the IR cancellation is
satisfied even after reconstruction.

###  III.6 Comparison with the 1-loop solution in SPT

In this subsection, we investigate the IR limit properties of the 1-loop
solution in SPT for the post-reconstruction power spectrum.

In Fourier space, the $n$th-order term of the redshift-space density
fluctuations after reconstruction is given by

$\displaystyle\widetilde{\delta}_{\rm rec}^{\,[n]}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ (97) $\displaystyle\times$ $\displaystyle Z_{{\rm
rec},n}(\bm{p}_{1},\dots,\bm{p}_{n})\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}_{n}).$

The function $Z_{{\rm rec},n}(\bm{p}_{1},\cdots,\bm{p}_{n})$ denotes nonlinear
kernel functions accounting for the bias effect, the RSD effect, and the
reconstruction effect. Consequently, the post-reconstruction terms
corresponding to $P^{[22]}$ and $P^{[13]}$ are expressed as

$\displaystyle P_{\rm rec}^{[22]}(\bm{k})$ $\displaystyle=$ $\displaystyle
2\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\int\frac{d^{3}p_{2}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,2]})$ $\displaystyle\times$ $\displaystyle[Z_{{\rm
rec},2}(\bm{p}_{1},\bm{p}_{2})]^{2}P_{\rm m}^{[11]}(p_{1})P_{\rm
m}^{[11]}(p_{2}),$ $\displaystyle P_{\rm rec}^{[13]}(\bm{k})$ $\displaystyle=$
$\displaystyle 6Z_{1}(\bm{k})\,P_{\rm m}^{[11]}(k)$ (98) $\displaystyle\times$
$\displaystyle\int\frac{d^{3}p}{(2\pi)^{3}}Z_{{\rm
rec},3}(\bm{k},\bm{p},-\bm{p})P_{\rm m}^{[11]}(p).$

In the IR limit, both $Z_{{\rm rec},2}(\bm{k},\bm{p})$ and $Z_{{\rm
rec},3}(\bm{k},\bm{p},-\bm{p})$ are approximated as (see Appendix C)

$\displaystyle Z_{{\rm rec},2}(\bm{k},\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle\frac{1}{2}Z_{1}(\bm{k})\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W(pR_{\rm
s})}{b_{\rm 1,fid}}\right)Z_{1}(\bm{p})\Bigg{\\}}$ $\displaystyle Z_{{\rm
rec},3}(\bm{k},\bm{p},-\bm{p})$ $\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle-\frac{1}{3!}Z_{1}(\bm{k})\Bigg{\\{}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)$
(99) $\displaystyle+$
$\displaystyle\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W(pR_{\rm
s})}{b_{\rm 1,fid}}\right)Z_{1}(\bm{p})\Bigg{\\}}^{2}\;.$

Using Eq. (99), $P_{\rm rec}^{[22]}$ and $P_{\rm rec}^{[13]}$ in the IR limit
are described as

$\displaystyle P_{\rm rec,IR}^{[22]}(\bm{k})$ $\displaystyle=$
$\displaystyle\left(k^{2}(1-\mu^{2})\sigma_{\rm
rec,\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\rm rec,\parallel}\right)$
$\displaystyle\times$ $\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k),$
$\displaystyle P_{\rm rec,IR}^{[13]}(\bm{k})$ $\displaystyle=$
$\displaystyle-\left(k^{2}(1-\mu^{2})\sigma_{\rm
rec,\perp}^{2}+k^{2}\mu^{2}\sigma^{2}_{\rm rec,\parallel}\right)$ (100)
$\displaystyle\times$ $\displaystyle[Z_{1}(\bm{k})]^{2}P_{\rm m}^{[11]}(k).$

These terms cancel each other out, leading to the post-reconstruction 1-loop
power spectrum approaching zero in the IR limit: $P_{\rm rec}^{\rm 1\mathchar
45\relax loop}(\bm{k})\to 0$. This result is consistent with those obtained by
expanding both the propagator term and the mode-coupling term up to the 1-loop
order, as presented in Eqs. (81) and (82). Given that SPT ensures accurate
calculations for each respective order, this finding further reinforces the
validity of our main result: i.e., the IR cancellation occurs even after
reconstruction.

###  III.7 IR-resummed model for
the post-reconstruction power spectrum

In the same way as Eq. (57), we replace the linear matter power spectrum,
which appears in the mode-coupling term in the IR limit, with the
corresponding no-wiggle power spectrum. We then derive

$\displaystyle P_{\rm rec}(\bm{k})=\big{[}Z_{1}(\bm{k})\big{]}^{2}\left[{\cal
D}_{\rm rec}^{2}(\bm{k})P_{\rm w}(k)+P_{\rm nw}(k)\right]\;.$ (101)

This form is identical to the one before reconstruction, with the exception of
the values of the smoothing parameters that characterize the exponential decay
of the BAO signal. Template models using a single Gaussian function have been
widely used in post-reconstruction BAO analysis (e.g., see [70, 71, 72, 73]).
The findings in this paper confirm the validity of these previous post-
reconstruction BAO analyses.

At the 1-loop level, we obtain

$\displaystyle P_{\rm rec}(\bm{k})$ $\displaystyle=$ $\displaystyle{\cal
D}_{\rm
rec}^{2}(\bm{k})\left[\left[Z_{1}(\bm{k})\right]^{2}+P_{\rm(S)rec}^{[13]}(\bm{k})\big{/}P_{\rm
m}^{[11]}(k)\right]P_{\rm w}(k)$ (102) $\displaystyle+$
$\displaystyle\left[Z_{1}(\bm{k})\right]^{2}P_{\rm nw}(k)+P_{\rm
rec,nw}^{[13]}(\bm{k})+P_{\rm rec}^{[22]}(\bm{k})\;,$

where

$\displaystyle P^{[13]}_{\rm rec,nw}(\bm{k})=P_{\rm
rec}^{[13]}(\bm{k})\left(P_{\rm nw}(k)/P_{\rm m}^{[11]}(k)\right)\;,$ (103)

and

$\displaystyle P_{\rm(S)rec}(\bm{k})^{[13]}=P_{\rm rec}^{[13]}(\bm{k})-P_{\rm
rec,IR}^{[13]}(\bm{k})\;.$ (104)

##  IV Cross-Power Spectrum between Pre- and Post-Reconstruction Density
Fluctuations

We now turn to the cross power spectrum between the pre- and post
reconstruction density fluctuations. In this section, we demonstrate that the
pre- and post-reconstruction cross-power spectrum does not occur the IR
cancellation, resulting in an overall exponential decay.

In the IR limit, the cross 2PCF of $\delta(\bm{x})$ and $\delta_{\rm
rec}(\bm{x})$ can be expressed as

$\displaystyle\langle\delta(\bm{x})\delta_{\rm rec}(\bm{x}^{\prime})\rangle$
$\displaystyle\to$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x}-\overline{\bm{\Psi}}^{[1]})\delta_{\rm
rec}^{[1]}(\bm{x}^{\prime}-\overline{\bm{\Psi}}_{\rm
rec}^{[1]})\big{\rangle}\;.$ (105)

Due to the statistical translation symmetry, this expression transforms to

$\displaystyle\langle\delta(\bm{x})\delta_{\rm rec}(\bm{x}^{\prime})\rangle$
$\displaystyle\to$
$\displaystyle\big{\langle}\delta^{[1]}(\bm{x}+\overline{\bm{s}}^{[1]})\delta_{\rm
rec}^{[1]}(\bm{x}^{\prime})\big{\rangle}\;.$ (106)

As a result, the contributions from the IR effect related to
$\overline{\bm{s}}^{[1]}$ are retained in the cross 2PCF.

In Fourier space, the cross power spectrum of $\widetilde{\delta}(\bm{k})$ and
$\widetilde{\delta}_{\rm rec}(\bm{k})$ in the IR limit is given by

$\displaystyle P_{\rm cross}(\bm{k})$ (107) $\displaystyle\to$
$\displaystyle{\cal D}(\bm{k}){\cal D}_{\rm rec}(\bm{k}){\cal E}_{\rm
pd}(\bm{k},-\bm{k})\big{[}Z_{1}(\bm{k})\big{]}^{2}P_{\rm m}^{[11]}(k),$

where the exponential function ${\cal E}_{\rm pd}(\bm{k},\bm{k}^{\prime})$ is
defined as

$\displaystyle{\cal E}_{\rm pd}(\bm{k},\bm{k}^{\prime})$ $\displaystyle=$
$\displaystyle\exp\left(\left\langle\left(-i\bm{k}\cdot\overline{\bm{\Psi}}^{[1]}\right)\left(-i\bm{k}^{\prime}\cdot\overline{\bm{\Psi}}_{\rm
rec}^{[1]}\right)\right\rangle_{\rm c}\right),$ (108)

This function satisfies the following relation:

$\displaystyle{\cal E}_{\rm pd}(\bm{k},-\bm{k})={\cal D}_{\rm ss}(\bm{k}){\cal
D}^{-1}_{\rm rec}(\bm{k}){\cal D}^{-1}(\bm{k}).$ (109)

Substituting Eq. (109) into Eq. (107) leads to

$\displaystyle P_{\rm cross}(\bm{k})$ $\displaystyle\to$ $\displaystyle{\cal
D}_{\rm ss}(\bm{k})\big{[}Z_{1}(\bm{k})\big{]}^{2}P_{\rm m}^{[11]}(k).$ (110)

Thus, the contribution from the remaining IR effect in the cross 2PCF, shown
in Eq. (106), appears as the exponential decay function in the cross power
spectrum.

To account for the nonlinear damping effect of BAO, as was done in Eqs. (57)
and (101), we make the following substitutions in Eq. (107):

$\displaystyle{\cal E}_{\rm pd}(\bm{k},-\bm{k})P_{\rm m}^{[11]}(k)$
$\displaystyle\xrightarrow[\rm replace]{}P_{\rm w}(k)+{\cal E}_{\rm
pd}(\bm{k},-\bm{k})P_{\rm nw}(k).$ (111)

Subsequently, we obtain

$\displaystyle P_{\rm cross}(\bm{k})$ (112) $\displaystyle\to$
$\displaystyle\big{[}Z_{1}(\bm{k})\big{]}^{2}\big{[}{\cal D}(\bm{k}){\cal
D}_{\rm rec}(\bm{k})P_{\rm w}+{\cal D}_{\rm ss}(\bm{k})P_{\rm
nw}(\bm{k})\big{]}\;.$

###  IV.1 Shot-noise in the cross-power spectrum

The shot-noise term in the power spectrum arises when the same particle is
counted multiple times during power spectrum calculations. Usually, the shot-
noise term is absent in the cross power spectrum of different density
fluctuations. However, in the case of the cross power spectrum between the
pre- and post-reconstruction density fluctuations, the same galaxies can be
identified, making the shot-noise term significant.

In the discrete picture, the number density in Fourier space is given by

$\displaystyle n(\bm{k})=\sum_{i}^{N_{\rm g}}e^{-i\bm{k}\cdot\bm{x}_{i}},$
(113)

where $\bm{x}_{i}$ denotes the position of the $i$-th galaxy, and $N_{\rm g}$
represents the total number of galaxies. The auto power spectrum in this
discrete picture is expressed as

$\displaystyle\frac{V}{N^{2}_{\rm g}}|n(\bm{k})|^{2}$ $\displaystyle=$
$\displaystyle\frac{V}{N^{2}_{\rm g}}\sum_{i\neq
j}e^{-i\bm{k}\cdot(\bm{x}_{i}-\bm{x}_{j})}+\frac{V}{N^{2}_{\rm
g}}\sum_{i=j}^{N_{\rm g}}$ (114) $\displaystyle=$ $\displaystyle
P(\bm{k})+\frac{1}{\bar{n}},$

where $V$ denotes the survey volume. The first and second terms on the right-
hand side correspond to the power spectrum and the shot-noise term,
respectively:

$\displaystyle P(\bm{k})$ $\displaystyle=$ $\displaystyle\frac{V}{N^{2}_{\rm
g}}\sum_{i\neq j}e^{-i\bm{k}\cdot(\bm{x}_{i}-\bm{x}_{j})},$
$\displaystyle\frac{1}{\bar{n}}$ $\displaystyle=$
$\displaystyle\frac{V}{N^{2}_{\rm g}}\sum_{i=j}^{N_{\rm g}},$ (115)

where $\bar{n}=N_{\rm g}/V$ is the mean number density.

Let $n_{\rm rec}(\bm{k})$ be the post-reconstruction density field in Fourier
space. It can be expressed in the discrete picture as

$\displaystyle n_{\rm rec}(\bm{k})=\sum_{i}^{N_{\rm
g}}e^{-i\bm{k}\cdot\bm{x}_{{\rm rec},i}},$ (116)

where

$\displaystyle\bm{x}_{{\rm rec},i}=\bm{x}_{i}+\bm{s}(\bm{x}_{i}).$ (117)

Therefore, the shot-noise term in the cross-power spectrum is given by

$\displaystyle\frac{V}{N^{2}_{\rm g}}\sum_{i=j}^{N_{\rm
g}}e^{-i\bm{k}\cdot\left(\bm{x}_{i}-\bm{x}_{{\rm rec},i}\right)}$
$\displaystyle=$ $\displaystyle\frac{V}{N^{2}_{\rm g}}\sum_{i=j}^{N_{\rm
g}}e^{i\bm{k}\cdot\bm{s}(\bm{x}_{i})}$ (118)

Calculating the ensemble average of the above equation and taking the
continuous limit, we obtain

$\displaystyle\left\langle\frac{V}{N^{2}_{\rm g}}\sum_{i=j}^{N_{\rm
g}}e^{i\bm{k}\cdot\bm{s}(\bm{x}_{i})}\right\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{N_{\rm g}}\int d^{3}x\left\langle
e^{i\bm{k}\cdot\bm{s}(\bm{x})}\right\rangle$ (119) $\displaystyle=$
$\displaystyle\frac{1}{\bar{n}}\left\langle
e^{i\bm{k}\cdot\bm{s}(\bm{x}=\bm{0})}\right\rangle$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\bar{n}}\left\langle
e^{i\bm{k}\cdot\overline{\bm{s}}^{[1]}}\right\rangle,$

In the final line, we made an approximation of the linear $\bm{s}(\bm{x})$.
Finally, we derive

$\displaystyle\left\langle\frac{V}{N_{\rm g}^{2}}n(\bm{k})n_{\rm
rec}^{*}(\bm{k})\right\rangle=P_{\rm cross}(\bm{k})+{\cal D}_{\rm
ss}(\bm{k})\left(\frac{1}{\bar{n}}\right).$ (120)

This result indicates that the shot-noise term is proportional to ${\cal
D}_{\rm ss}(\bm{k})$ and exponentially decays, as is $P_{\rm cross}(\bm{k})$.

##  V Conclusions

In this paper, we present a new theoretical model addressing the resummation
of infrared (IR) effects in the power spectrum of galaxy density fluctuations
after reconstruction. Our model accurately describes the nonlinear damping of
the post-reconstruction BAO signal, while including the 1-loop correction term
in SPT. The first main results are summarized in Eqs. (101) and (102).

We point out in Section III.2 that it is crucial to consider the nonlinear IR
effects contained within the displacement vector for reconstruction $\bm{s}$
(65). As a result, the IR effects on the post-reconstruction density
fluctuation can be described as a coordinate transformation of the density
fluctuations (71), like the pre-reconstruction case (24). This result leads to
the cancellation of the IR effects on the power spectrum in the IR limit even
after reconstruction. Furthermore, the nonlinear behavior of BAO is
characterized by the two-dimensional Gaussian decaying function of BAO,
supporting the theoretical foundation of the commonly used single two-
dimensional Gaussian decay function in post-reconstruction power spectrum and
2PCF analyses [e.g., 70, 71, 72, 73].

We compare our findings regarding the post-reconstruction infrared (IR)
effects with those obtained using the Zel’dovich approximation and SPT in
Section III.5 and SectionIII.6. Notably, in the Zel’dovich approximation,
which linearly approximates $\bm{s}$, IR cancellation does not occur, and two
Gaussian damping functions are required to describe the nonlinear BAO effects.
Conversely, in the post-reconstruction SPT, IR cancellation is observed in the
IR limit, corroborating our results.

We further clarify the behavior of the cross-power spectrum between pre- and
post-reconstruction density fluctuations, noting its overall exponential decay
due to the absence of IR cancellation. This observation also applies to the
corresponding shot-noise term. While these phenomena have been observed in
previous simulation studies [5], our work thus provides a mathematical
foundation for these observations. Additionally, as our second main result, we
present a model, as shown in Eq. (112), to elucidate the nonlinear effects of
BAO on the cross-power spectrum.

In conclusion, our work presents a novel direction in developing theoretical
models for the post-reconstruction power spectrum beyond standard perturbation
theory. The IR-resummed models presented in this paper, which consider the
galaxy bias and RSD effects, are directly applicable to the analysis of real
galaxy data.

During the preparation of this paper, we learned that the DESI collaboration
[74] was planning to submit a study with conclusions similar to ours,
suggesting that the BAO signal in the post-reconstruction power spectrum
should be described by a single Gaussian damping function. To ensure a
coordinated approach, we arranged to submit our paper close to the DESI
collaborators’ paper submission. Their study investigates the theoretical and
numerical systematics of the post-reconstruction BAO signal in detail and,
furthermore, discusses the contributions of mode-coupling terms to the BAO
signal, an aspect not covered in our paper. In contrast, our work focuses on
the IR effects before and after reconstruction, providing a more detailed
theoretical analysis. Therefore, these studies complement each other well,
each enriching the understanding of the topic from a different perspective. We
refer readers interested in the numerical aspects to the paper by the DESI
collaboration.

###### Acknowledgements.

NS acknowledges financial support from JSPS KAKENHI Grant Number 19K14703. NS
thanks Shun Saito, Hee-Jong Seo, Florian Beutler, and Shi-Fan Stephen Chen for
useful comments and discussion.

##  Appendix A Degenerate Higher-Order Scalar-Tensor (DHOST) Theories

In DHOST theories, the second-order kernel density fluctuation of dark matter
is given by [75]

$\displaystyle\delta_{\rm m}^{[2]}(\bm{x})$ $\displaystyle=$
$\displaystyle\left(\kappa-\frac{4}{21}\lambda\right)\big{[}\delta_{\rm
m}^{[1]}(\bm{x})\big{]}^{2}-\kappa\,\bm{\Psi}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})$ (121) $\displaystyle+$
$\displaystyle\frac{2}{7}\,\lambda\,\left[\left(\frac{\partial_{i}\partial_{j}}{\partial^{2}}-\frac{1}{3}\delta_{ij}\right)\delta_{\rm
m}^{[1]}(\bm{x})\right]^{2}.$

Here, in $\Lambda$CDM assuming $f^{2}=\Omega_{\rm m}$, $\kappa=\lambda=1$.
From Eqs. (9) and (10), the short-wavelength density fluctuation then becomes

$\displaystyle\delta_{\rm(S)m}^{[2]}(\bm{x})$ $\displaystyle=$
$\displaystyle\left(\kappa-\frac{4}{21}\lambda\right)\big{[}\delta_{\rm
m}^{[1]}(\bm{x})\big{]}^{2}-\bm{\Psi}_{\rm(S)}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})$ (122) $\displaystyle-$
$\displaystyle\Delta\kappa\,\bm{\Psi}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})$ $\displaystyle+$
$\displaystyle\frac{2}{7}\,\lambda\,\left[\left(\frac{\partial_{i}\partial_{j}}{\partial^{2}}-\frac{1}{3}\delta_{ij}\right)\delta_{\rm
m}^{[1]}(\bm{x})\right]^{2}.$

where $\kappa=1+\Delta\kappa$. There remains a shift term,
$\Delta\kappa\,\bm{\Psi}^{[1]}(\bm{x})\cdot\nabla\delta_{\rm
m}^{[1]}(\bm{x})$, in $\delta^{[2]}_{\rm(S)m}(\bm{x})$ that does not go to
zero in the IR limit. Therefore, it is not possible to simply separate IR
effects from the rest based on Eq. (24). This unique feature of DHOST theories
leads to a violation of the IR cancellation [76]. This deviation in the
coefficient of the shift term from $1$ in DHOST theories is also related to
the consistency relation for the large-scale structure [77, 78, 65].

##  Appendix B Standard Perturbation Theory

In this appendix, we show the specific form of the nonlinear kernel functions
$Z_{n}$ given in Eq. (27) up to the third-order in SPT and investigate their
behaviors in the IR limit.

In Fourier space, Eq. (19) becomes

$\displaystyle\widetilde{\delta}_{\rm g}(\bm{k})=\int
d^{3}qe^{-i\bm{k}\cdot\bm{q}}\Big{\\{}\left[1+\delta_{\rm
bias}(\bm{q})\right]e^{-i\bm{k}\cdot\bm{\Psi}_{\rm
red}(\bm{q})}-1\Big{\\}}\;.$ (123)

We assume that the $n$th-order of the biased density fluctuation can be
represented as

$\displaystyle\widetilde{\delta}_{\rm bias}^{\,[n]}(\bm{k})$ $\displaystyle=$
$\displaystyle\int\frac{d^{3}p_{1}}{(2\pi)^{3}}\cdots\frac{d^{3}p_{n}}{(2\pi)^{3}}(2\pi)^{3}\delta_{\rm
D}(\bm{k}-\bm{p}_{[1,n]})$ (124) $\displaystyle\times$ $\displaystyle
B_{n}(\bm{p}_{1},\dots,\bm{p}_{n})\widetilde{\delta}_{\rm
m}^{\,[1]}(\bm{p}_{1})\cdots\widetilde{\delta}_{\rm m}^{\,[1]}(\bm{p}_{n})\;,$

where $B_{1}(\bm{p}_{1})=(b_{1}-1)$ with $b_{1}$ being the Eulerian linear
bias. We follow the standard bias theory [e.g., 57] and assume that $B_{n\geq
2}(\bm{p}_{1},\cdots,\bm{p}_{n})$ do not include the shift terms. Then, the
nonlinear kernel functions up to the third order are given by [79, 61]

$\displaystyle Z_{1}(\bm{k})=B_{1}(\bm{k})+\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{k})\;,$ (125) $\displaystyle Z_{2}(\bm{p}_{1},\bm{p}_{2})$
$\displaystyle=$ $\displaystyle
B_{2}(\bm{p}_{1},\bm{p}_{2})+\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},2}(\bm{p}_{1},\bm{p}_{2})\big{]}$ (126) $\displaystyle+$
$\displaystyle\frac{1}{2}\big{\\{}B_{1}(\bm{p}_{1})\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{2})\big{]}+\mbox{1 perm.}\big{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{2}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{1})\big{]}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{2})\big{]}\;,$

and

$\displaystyle Z_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$ (127) $\displaystyle=$
$\displaystyle
B_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})+\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})\big{]}$ $\displaystyle+$
$\displaystyle\frac{1}{3}\big{\\{}B_{2}(\bm{p}_{1},\bm{p}_{2})\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{3})\big{]}+\mbox{2 perms.}\big{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{3}\big{\\{}B_{1}(\bm{p}_{1})\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},2}(\bm{p}_{2},\bm{p}_{3})\big{]}+\mbox{2 perms.}\big{\\}}$
$\displaystyle+$
$\displaystyle\frac{1}{6}\big{\\{}B_{1}(\bm{p}_{1})\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{2})\big{]}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{3})\big{]}+\mbox{2 perms.}\big{\\}}$ $\displaystyle+$
$\displaystyle\frac{1}{6}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{1})\big{]}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{2})\big{]}\big{[}\bm{k}\cdot\bm{L}_{{\rm
red},1}(\bm{p}_{3})\big{]}\;,$

where $\bm{k}=\bm{p}_{1}+\bm{p}_{2}$ for $Z_{2}$,
$\bm{k}=\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3}$ for $Z_{3}$, and $L_{{\rm
red},n}(\bm{p}_{1},\cdots,\bm{p}_{n})$ are calculated through the
transformation matrix given in Eq. (33):

$\displaystyle\bm{L}_{{\rm
red},n}(\bm{p}_{1},\cdots,\bm{p}_{n})=\bm{R}_{n}\cdot\bm{L}_{n}(\bm{p}_{1},\cdots,\bm{p}_{n}).$
(128)

Since $\bm{L}_{n\geq 2}(\bm{p}_{1},\cdots,\bm{p}_{n})$ do not include the
shift term, the functions $Z_{2}(\bm{k},\bm{p})$ and
$Z_{3}(\bm{k},\bm{p},-\bm{p})$, which are needed to compute the 1-loop power
spectrum in the IR limit, are approximated as

$\displaystyle Z_{2}(\bm{k},\bm{p})$ $\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle\frac{1}{2}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)Z_{1}(\bm{k})\;,$
$\displaystyle Z_{3}(\bm{k},\bm{p},-\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle-\frac{1}{3!}\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)^{2}Z_{1}(\bm{k})\;.$
(129)

##  Appendix C Standard Perturbation Theory
after Reconstruction

We investigate the IR limit properties of the post-reconstruction density
fluctuations up to the third order.

In Fourier space, Eq. (67) becomes

$\displaystyle\widetilde{\delta}_{\rm rec}(\bm{k})=\int
d^{3}xe^{-i\bm{k}\cdot\bm{x}}e^{-i\bm{k}\cdot\bm{s}(\bm{x})}\delta(\bm{x}).$
(130)

The nonlinear kernel function up to the third order are then given by [20]

$\displaystyle Z_{{\rm rec},1}(\bm{p}_{1})$ $\displaystyle=$ $\displaystyle
Z_{1}(\bm{p}_{1})\;,$ $\displaystyle Z_{{\rm rec},2}(\bm{p}_{1},\bm{p}_{2})$
$\displaystyle=$ $\displaystyle
Z_{2}(\bm{p}_{1},\bm{p}_{2})+\frac{1}{2}\left\\{\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{\rm
1,fid}}\right)+\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{\rm
1,fid}}\right)\right\\}Z_{1}(\bm{p}_{1})Z_{1}(\bm{p}_{2})\;,$ $\displaystyle
Z_{{\rm rec},3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$ $\displaystyle=$
$\displaystyle Z_{3}(\bm{p}_{1},\bm{p}_{2},\bm{p}_{3})$ (131) $\displaystyle+$
$\displaystyle\frac{1}{3}\left\\{\left(\frac{\bm{k}\cdot\bm{p}_{[1,2]}}{p_{[1,2]}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{[1,2]}R_{\rm s})}{b_{\rm
1,fid}}\right)Z_{2}(\bm{p}_{1},\bm{p}_{2})Z_{1}(\bm{p}_{3})+\mbox{2
perms.}\right\\}$ $\displaystyle+$
$\displaystyle\frac{1}{3}\left\\{\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{\rm
1,fid}}\right)Z_{1}(\bm{p}_{1})Z_{2}(\bm{p}_{2},\bm{p}_{3})+\mbox{2
perms.}\right\\}$ $\displaystyle+$
$\displaystyle\frac{1}{6}\left\\{\left(\frac{\bm{k}\cdot\bm{p}_{1}}{p_{1}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{1}R_{\rm s})}{b_{\rm
1,fid}}\right)\left(\frac{\bm{k}\cdot\bm{p}_{2}}{p_{2}^{2}}\right)\left(-\frac{W_{\rm
G}(p_{2}R_{\rm s})}{b_{\rm
1,fid}}\right)Z_{1}(\bm{p}_{1})Z_{1}(\bm{p}_{2})Z_{1}(\bm{p}_{3})+\mbox{2
perms.}\right\\}\;,$

where $\bm{k}=\bm{p}_{1}+\bm{p}_{2}$ for $Z_{{\rm rec},2}$, and
$\bm{k}=\bm{p}_{1}+\bm{p}_{2}+\bm{p}_{3}$ for $Z_{{\rm rec},3}$. The above
equations include the bias effects as a slight extension of those given by
Hikage  _et al._ [20]. Approximations in the IR limit of $Z_{{\rm rec},2}$ and
$Z_{{\rm rec},3}$ in the form needed to compute the 1-loop power spectrum are
given by

$\displaystyle Z_{{\rm rec},2}(\bm{k},\bm{p})$ $\displaystyle\xrightarrow[p\to
0]{}$
$\displaystyle\frac{1}{2}\left\\{\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm 1,fid}}\right)Z_{1}(\bm{p})\right\\}Z_{1}(\bm{k})\;,$
$\displaystyle Z_{{\rm rec},3}(\bm{k},\bm{p},-\bm{p})$
$\displaystyle\xrightarrow[p\to 0]{}$
$\displaystyle-\frac{1}{3!}\left\\{\left(\frac{\bm{k}\cdot\bm{R}_{1}\cdot\bm{p}}{p^{2}}\right)+\left(\frac{\bm{k}\cdot\bm{p}}{p^{2}}\right)\left(-\frac{W_{\rm
G}(pR_{\rm s})}{b_{\rm
1,fid}}\right)Z_{1}(\bm{p})\right\\}^{2}Z_{1}(\bm{k})\;.$ (132)

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