RBI-ThPhys-2023-14

# The observed power spectrum & frequency-angular power spectrum

Alvise Raccanelli<EMAIL_ADDRESS>Dipartimento di Fisica Galileo
Galilei, Università di Padova, I-35131 Padova, Italy  INFN Sezione di Padova,
I-35131 Padova, Italy  INAF-Osservatorio Astronomico di Padova, Italy
Theoretical Physics Department, CERN, 1 Esplanade des Particules, 1211 Geneva
23, Switzerland  Zvonimir Vlah<EMAIL_ADDRESS>Ruđer Bošković Institute,
Bijenička cesta 54, 10000 Zagreb, Croatia  Kavli Institute for Cosmology,
University of Cambridge, Cambridge CB3 0HA, UK  Department of Applied
Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3
0WA, UK

###### Abstract

The two-point summary statistics is one of the most commonly used tools in the
study of cosmological structure. Starting from the theoretical power spectrum
defined in the 3D volume and obtained via the process of ensemble averaging,
we establish the construction of the observed 3D power spectrum, folding the
unequal-time information around the average position into the wave modes along
the line of sight. We show how these unequal-time cross-correlation effects
give rise to scale-dependent corrections in the observable 3D power spectrum.
We also introduce a new dimensionless observable, the frequency-angular power
spectrum, which is a function of dimensionless and directly observable
quantities corresponding to Fourier counterparts of angles and redshifts.
While inheriting many useful characteristics of the canonical observed power
spectrum, this newly introduced statistic does not depend on physical
distances and is hence free of so-called Alcock-Paczyński effects. Such
observable thus presents a clear advantage and simplification over the
traditional power spectrum.

Moreover, relying on linear theory calculations, we estimate that unequal-time
corrections, while generally small, can amount to a few percent on large
scales and high redshifts. Interestingly, such corrections depend on the bias
of the tracers, the growth rate, but also their time derivatives, opening up
the possibility of new tests of cosmological models. These radial mode effects
also introduce anisotropies in the observed power spectrum, in addition to the
ones arising from redshift-space distortions, generating non-vanishing odd
multiples and imaginary contributions. Lastly, we investigate the effects of
unequal-time corrections in resumming long displacements (IR-resummation) of
the observed power spectrum.

##  I Introduction

The cosmological large-scale structure offers a competitive and promising
avenue for extracting physical information on our Universe from the
distribution of matter. Next-generation galaxy surveys, like Euclid [1], DESI
[2], Rubin [3], Roman [4], SPHEREx [5], SKAO [6], MegaMapper [7], ATLAS [8]
and others, aim to address various cosmological questions, ranging from
uncovering the nature of dark energy and tests of general relativity on large
scales [9, 10], to constraining the properties of the initial conditions of
the universe by measuring signals of primordial non-Gaussianity [11, 12]. In
order to successfully extract accurate cosmological information, robust
measurements and reliable statistical tools are of paramount importance. For
this purpose, the two-point statistics in Fourier space, be it in 3D (the
power spectrum) or in 2D (the angular power spectrum), has been the observable
of choice for a wide range of these surveys; alternative statistics are e.g.,
the configuration space two-point function and the spherical-Fourier Bessel
(for a (incomplete) list of measurements using different procedures see, e.g.,
[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34]). The primary motivation for the choice of Fourier statistics is
the linearity of wave modes on the largest cosmological scales, thus ensuring
the independence of theoretical errors, subsequently reflected in the diagonal
form of the corresponding covariance matrix.

$\delta^{\rm K}_{ij}$ |  Kronecker symbol
---|---
$\delta^{\rm D}(\bm{x})$ |  Dirac delta function
$W(\chi)$ |  Window function; related to the specific observable and survey
$\delta(\bm{x})$ |  3D density field of matter or biased tracer
$\hat{\delta}(\bm{\theta})$ |  2D projected filed in the real space coordinates on the sky
$\mathcal{P}(\bm{k};\,z,z^{\prime})$ |  Unequal-time theoretical power spectrum of the 3D density field (unobservable)
$P(\bm{k};\,z)$ |  Equal-time observed power spectrum (constructed from observable fields)
$\mathbb{C}_{\ell}(z,z^{\prime})$ |  Unequal-time angular power spectrum (in the narrow window function limit)
$C_{\ell}$ |  Projected angular power spectrum (with finite size window functions)
$\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})$ |  Frequency-angular power spectrum (dimensionless, equivalent to the observed power spectrum)
Table 1: Notation used for the most important quantities in this paper.

However, due to the complex nature of relating the actual observable to
theoretical predictions in most of the analyses (sometimes also called
lightcone effects), in practice, one has to resort to a series of
simplifications and auxiliary modelling. One such approximation that we focus
on in this paper is related to the fact that for a tracer (e.g., galaxies)
observed in a redshift bin, a 3D power spectrum necessarily needs to
incorporate the unequal-time effects related to the redshift difference of the
correlated points. Moreover, even in the case when we observe angular
positions as well as redshifts of individual tracers, the observables that can
be constructed never correspond to the actual 3D unequal-time power spectrum.
The reason for this is that the wave modes along the line-of-sight and
unequal-time effects are inevitably mixed up and projected on top of each
other (see [35] for a illustration). Such projection effects are ignored in
all contemporary practical applications of the 3D power spectrum, relying on
existing approximations and beliefs that the correction will be small and
negligible. At any rate, this is a notion that ought to be scrutinized,
especially in light of the ever-increasing depth and area coverage of
forthcoming surveys.

In this paper, we focus on the ab-initio construction of the two-point
observables, taking into account these projection effects. On our path to
addressing this matter, we find that the issue at hand naturally fragments
into the following set of questions:

  * •

How to construct the flat-sky approximation to the angular power spectrum
taking into account unequal time effects?

  * •

Fixed observer breaks the translation invariance in the plane parallel
approximation. How can we quantify these effects?

  * •

How to construct the observed 3D power spectrum from the given projected
angular correlation statistics?

  * •

What are the correction to the observed 3D power spectrum induced by these
projections and unequal time effects?

  * •

Is there an alternative statistic to the 3D power spectrum and angular power
spectrum capturing the same information content?

This paper is organised as follows. In Section II we first re-derive the
projected angular power spectrum in terms of the unequal-time theoretical
power spectrum in the flat-sky approximation. From there, we derive the
observed 3D equal-time power spectrum and introduce the new statistics called
frequency-angular power spectrum that is free of so-called Alcock-Paczynśki
effects. Once these observable statistics are defined, we consider the
corrections generated by the unequal-time effects. This is done in Section
III. In the same section, we also consider the unequal-time effects arising
due to the long displacement field via the IR resummation mechanism. We close
the discussion with our concluding remarks in Section IV. Details of our
analysis are presented in the series of Appendixes A, B , C and D.

We use Planck cosmology [36], where $\Omega_{c}h^{2}=0.11933$,
$\Omega_{b}h^{2}=0.02242$, $\Omega_{K}h^{2}=0$, $h=0.6766$, $n_{s}=0.9665$,
and $\sigma_{8}=0.81027941$. In Table 1 we provide a short summary of the key
physical quantities featured in the paper. The linear 3D power spectrum can be
obtained using the CAMB [37] or CLASS [38] codes.

##  II From the theoretical to the observed power spectrum

•••$\delta(\bm{x},z^{\prime\prime})$$\delta(\bm{x},z^{\prime})$$\delta(\bm{x},z)$$\hat{\delta}(\bm{\theta},\chi)$$\hat{\delta}(\bm{\theta},\chi^{\prime})$$\mathcal{O}$$\mathcal{O}$$\chi$$\chi^{\prime}$$\bar{\chi}$$\delta\chi$$\mathcal{V}$${\bm{z}}$Theoretical
Power Spectrum$\mathcal{P}(\bm{k},z,z^{\prime})$Angular
Spectrum$C_{\ell}\big{(}z[\chi],z[\chi^{\prime}]\big{)}$Observed Power
Spectrum$P\big{(}k_{\hat{n}},\bm{\ell}/\bar{\chi},z[\bar{\chi}]\big{)}$ Figure
1: Scheme showing the three stages we follow in constructing the observed
power spectrum. We start by correlating the 3D density field
$\delta(\bm{x},z)$, which provides us with the theoretical, unobservable,
unequal-time 3D power spectrum $\mathcal{P}(\bm{k},z,z^{\prime})$. From this,
we can construct the observable angular power spectrum
$C_{\ell}(z,z^{\prime})$. Using the flat-sky approximation we can translate
the $C_{\ell}(z,z^{\prime})$ into the observable equal-time power spectrum
$P(\bm{k},\bar{z})$ at the mean redshift $\bar{z}$.

In this section, we delineate several different statistical, two-point,
observables. We start from the usual theoretical power spectrum (see Table 1
and [35] for details), $\mathcal{P}(\bm{k})$ defined as the ensemble average
power spectrum. This power spectrum is not observable as it is properly
defined only in a fully 4D space and could be accessible only by a meta-
observer outside of the system.

However, from there, we can define a procedure such that by introducing the
observer and, therefore, the lightcone for our observations, by taking into
account the projections on the sky, we can compute the observable angular
spectrum $C_{\ell}$ (for the derivation of lightcone effects on galaxy
clustering, see the pioneering work of [39] and e.g., [40, 41, 42, 43, 44, 45,
46] for angular, Fourier, configuration space and spherical-Fourier Bessel).
We note that when defining observable quantities in this work, we do not
include observational effects such as masking and even more instrument-related
issues, which go beyond the scope of our work. As we show, this step folds the
$\mathcal{P}(\bm{k})$ information coming from the modes along the line of
sight into the unequal-time structure of $C_{\ell}$, thus, effectively
performing the compression from 4D into 3D. We construct the observed power
spectrum $P_{\rm obs}(\bm{k})$ (and we will omit the $obs$ indication from now
on), transforming the unequal-time information of $C_{\ell}$ and
reconstructing the modes along the line of sight. Lastly, we introduce the
observable dimensionless frequency-angular power spectrum
$\widetilde{\mathbb{C}}(\omega,\ell)$, which carries information analogous to
the observed power spectrum $P(\bm{k})$, but free of the assumed fiducial
cosmology. We will derive the expression for such observable and argue that it
presents several advantages over the standard power spectra used in
literature. A summary of the different two-point statistics, both theoretical
and observable ones, is given in Table 1.

###  II.1 Angular power spectrum

The simplest and most natural projected observable we can construct is the
angular power spectrum $C_{\ell}$. For a given 3D density field
$\delta(\bm{r})$, we can introduce the projected statistics using a general
window function $W$ as:

$\displaystyle\hat{\delta}(\bm{\theta})$ $\displaystyle=\int
d\chi\leavevmode\nobreak\
W(\chi)\delta\big{(}\chi\bm{\hat{n}},\chi\bm{\theta},z[\chi]\big{)}$ (1)
$\displaystyle=\int d\chi\leavevmode\nobreak\
W(\chi)\int\frac{d^{3}k}{(2\pi)^{3}}\leavevmode\nobreak\
e^{-i\chi\bm{k}.(\bm{\hat{n}}+\bm{\theta})}\delta\big{(}\bm{k},z[\chi]\big{)}\,,$

where $\chi$ is the comoving distance,
$\chi(z)=c\int_{0}^{z}dz\leavevmode\nobreak\ H^{-1}(z)$, and we use the flat-
sky geometric set-up, as depicted in Figure 1. Given that the projected field
is limited to a single plane, we can introduce 2D Fourier coordinates:

$\hat{\delta}(\bm{\ell})=\int d^{2}\theta\leavevmode\nobreak\
e^{i\bm{\ell}\cdot\bm{\theta}}\hat{\delta}(\bm{\theta})\,,$ (2)

for which we have the corresponding momentum representation:

$\hat{\delta}(\bm{\ell})=\int\frac{d\chi}{\chi^{2}}\leavevmode\nobreak\
W(\chi)\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\ e^{-i\chi
k_{\hat{n}}}\delta\big{(}k_{\hat{n}}\bm{\hat{n}},\bm{\tilde{\ell}},z[\chi]\big{)}\,,$
(3)

where we used $\bm{\tilde{\ell}}=\bm{\ell}/\chi$. If we consider the two-point
correlator, we get:

$\displaystyle\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}^{*}(\bm{\ell}^{\prime})\right\rangle$
$\displaystyle=(2\pi)^{2}\int\frac{d\chi}{\chi^{2}}\frac{d\chi^{\prime}}{\chi^{\prime
2}}\leavevmode\nobreak\ W(\chi)W^{\prime}(\chi^{\prime})$ (4)
$\displaystyle\times\delta^{\rm
2D}\left(\bm{\tilde{\ell}}-\bm{\tilde{\ell}^{\prime}}\right)\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
e^{i\delta\chi
k_{\hat{n}}}\mathcal{P}\big{(}k_{\hat{n}}\bm{\hat{n}},\bm{k}_{\perp},z[\chi],z[\chi^{\prime}]\big{)}\,,$

where we introduced the relative radial distance variable
$\delta\chi=\chi^{\prime}-\chi$, and where $\bm{k}_{\perp}$ is constrained by
the relation $\bm{k}_{\perp}=\bm{\tilde{\ell}}=\bm{\tilde{\ell}^{\prime}}$ (as
we are in flat-sky). Given that we have introduced the relative distance
variable $\delta\chi$, we should choose the corresponding mean distance
$\bar{\chi}$. We have the freedom to choose the mean distance, and some
natural options are:

$\bar{\chi}_{\rm
a}\equiv\frac{1}{2}\left(\chi+\chi^{\prime}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \bar{\chi}_{\rm
g}\equiv\sqrt{\chi\chi^{\prime}},\leavevmode\nobreak\ \leavevmode\nobreak\
\bar{\chi}_{\rm h}\equiv\frac{2\chi\chi^{\prime}}{\chi+\chi^{\prime}},$ (5)

which correspond to the arithmetic, geometric and harmonic mean, respectively.
Without specifying the choice of the mean distance $\bar{\chi}$, we can
transform the coordinates to obtain:

$\displaystyle\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}^{*}(\bm{\ell}^{\prime})\right\rangle=$
$\displaystyle(2\pi)^{2}\int d\chi
d\chi^{\prime}\;\frac{\bar{\chi}^{2}}{\chi\chi^{\prime}}W\left(\chi\right)W^{\prime}\left(\chi^{\prime}\right)$
(6) $\displaystyle\times\mathcal{A}(\delta)\delta^{\rm
2D}\big{(}\bm{\ell}-\bm{\ell}^{\prime}+\varphi(\delta)\bm{\Delta}\big{)}\mathbb{C}\left(\ell,\bar{\chi},\delta\chi\right)\,,$

where $\delta=\delta\chi/(2\bar{\chi})$, and $\varphi(\delta)$ is an off-
diagonal phase of the Dirac delta function, and we introduced
$\bm{\Delta}=\bm{\ell}^{\prime}+\bm{\ell}$ and the unequal-time angular power
spectrum:

$\mathbb{C}\left(\ell,\bar{\chi},\delta\chi\right)=\frac{1}{\chi\chi^{\prime}}\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
e^{i\delta\chi
k_{\hat{n}}}\mathcal{P}\big{(}k_{\hat{n}}\bm{\hat{n}},\bm{k}_{\perp},\bar{\chi},\delta\chi\big{)}\,.$
(7)

The factor $\mathcal{A}(\delta)$ originates from the Dirac delta function and
takes a different form depending on the definition of $\bar{\chi}$. The
detailed derivation and a few explicit choices are shown in Appendix A.

Since the observer position is fixed, we see that the exact translation
invariance in the observer plane does not hold. This is reflected in the fact
that the two-dimensional Dirac delta function, besides depending on the wave
vectors $\bm{\ell}$ and $\bm{\ell}^{\prime}$, also depends on the distances
$\chi$ and $\chi^{\prime}$. Nonetheless, since we are interested in geometries
where the mean distance is much larger than the relative distance
$\delta\chi$, it is natural (and it considerably simplifies calculations) to
expand around the leading solution that preserves the translation invariance.
We can thus express the two-point flat-sky correlators as a sum:

$\displaystyle\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}^{*}(\bm{\ell}^{\prime})\right\rangle=(2\pi)^{2}\delta^{\rm
2D}\big{(}\bm{\ell}-\bm{\ell}^{\prime}\big{)}\sum_{n=0}^{\infty}\frac{\big{(}\overset{\leftarrow}{\partial}_{\bm{\ell}^{\prime}}\cdot\bm{\Delta}\big{)}^{n}}{2^{n}n!}C^{(n)}(\ell)\,,$
(8)

where the partial derivative in the Taylor expansion acts on the left,
producing the derivatives of the delta function, and we introduced the
contributing angular spectra:

$\displaystyle C^{(n)}(\ell)=\int d\chi
d\chi^{\prime}\,WW^{\prime}\frac{\bar{\chi}^{2}}{\chi\chi^{\prime}}\mathcal{A}(\delta)\varphi(\delta)^{n}\mathbb{C}(\ell,\bar{\chi},\delta\chi)\,.$
(9)

Note that the higher derivatives of the Dirac delta function introduce off-
diagonal contributions to the angular correlations of
$\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}(\bm{\ell}^{\prime})\right\rangle$,
i.e., $C^{(n)}(\ell)$ for $n>0$ can be interpreted as a measure of non-
diagonal contributions to the usual angular power spectrum. What is the source
of these non-diagonal contributions? As we know, in the full-sky treatment,
isotropy guarantees the proportionality of the angular power spectrum to the
Kronecker delta $\delta^{\rm K}_{\ell\ell^{\prime}}$, while in the flat-sky
approximation, we have obtained this condition from the translational
invariance in the plane. However, for two physical modes $\bm{k}_{\perp}$
lying on two different redshift planes to agree, we have to readjust the
corresponding angles, as stated by the Dirac delta $\delta^{\rm
2D}\big{(}\bm{\ell}-\bm{\ell}^{\prime}\big{)}$. This generates the off-
diagonal contributions as a consequence of the fixed observer. As expected,
the $C^{(n)}(\ell)$ contributions for $n$ values higher than the leading
$n=0$, either vanish due to the geometric considerations (e.g., if
$W=W^{\prime})$, or are suppressed by the
$\varphi(\delta)^{n}\sim\left(\delta\chi/\chi\right)^{n}$ factor. In the next
section, we focus only on the $n=0$ and simply drop the $n$ order label, i.e.,
we define the flat-sky angular power spectrum:

$C(\ell)\equiv C^{(0)}(\ell)\,.$ (10)

The contribution of these higher-order $C^{(n)}(\ell)$ have been numerically
investigated in more detail in paper [35], where we show that they tend to be
suppressed by at least an order of magnitude on all scales. We note that the
proper physical interpretation of these contributions is not as corrections to
be added to the flat-sky $C(\ell)$ that would bring it close to the full-sky
result. Rather, these should be considered as the error estimates of the flat-
sky results, asymptotically approaching the full-sky.

Before continuing our investigation, let us discuss the options for the choice
of $\bm{k}_{\perp}$. Firstly, we can constrain our considerations to the
scalar case $k_{\perp}$, as is given by the isotropy in the plane
perpendicular to the line of sight. This also holds when redshift space
distortions are included in the 3D power spectrum. Moreover, from the Dirac
delta function constraint $k_{\perp}=\tilde{\ell}=\tilde{\ell}^{\prime}$ we
again have the freedom to construct our choice for $k_{\perp}$. This choice
determines at which order in $\delta$ the corrections in
$\mathbb{C}\left(\ell,\bar{\chi},\delta\chi\right)$ appear. Choosing
$k_{\perp}=\ell/\chi$ or $k_{\perp}=\ell/\chi^{\prime}$ is thus suboptimal as
it leads to the $\delta$ corrections we saw above. What is the alternative? We
can again choose one among the arithmetic, geometric and harmonic
combinations:

$k_{\perp}=\frac{\ell}{2}\left(\frac{1}{\chi}+\frac{1}{\chi^{\prime}}\right)\,,\leavevmode\nobreak\
\leavevmode\nobreak\
k_{\perp}=\frac{\ell}{\sqrt{\chi\chi^{\prime}}}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ k_{\perp}=\frac{2\ell}{\chi+\chi^{\prime}}\,,$ (11)

which all provide corrections that are of order $\delta^{2}$. These have to be
again evaluated in the chosen $\bar{\chi}$, $\delta\chi$ coordinates. For
concreteness, if we choose the arithmetic mean $\bar{\chi}_{\rm a}$, this
gives us $\mathcal{A}(\delta)=\chi^{2}_{\rm a}(1-\delta^{2})^{2}$, and
$\varphi(\delta)=\delta)$. Choosing the geometric definition gives
$k_{\perp}=\ell/(\bar{\chi}_{\rm a}\sqrt{1-\delta^{2}})$. However, if we
choose the harmonic $k_{\perp}=\ell/\bar{\chi}_{\rm a}$, introducing a
shorthand notation:

$\displaystyle\overline{\mathbb{C}}\left(\ell,\bar{\chi}_{\rm
a},\delta\chi\right)$
$\displaystyle=\frac{\bar{\chi}^{2}}{\chi\chi^{\prime}}\mathcal{A}(\delta)\mathbb{C}\left(\ell,\bar{\chi}_{\rm
a},\delta\chi\right)$ (12) $\displaystyle=\frac{1}{\bar{\chi}_{\rm
a}^{2}}\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\ e^{i\delta\chi
k_{\hat{n}}}\mathcal{P}\left(k_{\hat{n}},\ell/\bar{\chi}_{\rm
a},\bar{\chi}_{\rm a},\delta\chi\right)\,,$

thus eliminating the dependence on the $\delta$ parameter, besides the
explicit dependence in the unequal-time power spectrum. In the rest of the
paper, unless otherwise specified, we will adopt this geometry choice and drop
the index on the $\bar{\chi}$ and overline on $\overline{\mathbb{C}}$.

The full-sky version of the unequal-time angular power spectrum is well known
and given by (neglecting projections effects and window functions):

$\mathbb{C}^{\rm full}_{\ell}(\chi_{1},\chi_{2})\equiv
4\pi\int\frac{k^{2}dk}{2\pi^{2}}\;\mathcal{P}(k;\,\chi_{1},\chi_{2})\,j_{\ell}(k\chi_{1})j_{\ell}(k\chi_{2})\,.$
(13)

We shall refer back to this full-sky angular power spectrum results when
generalizing the 3D power spectrum from the flat-sky approximation to the
full-sky case.

###  II.2 Unequal-time power spectrum

The theoretical power spectrum $\mathcal{P}$ is an object that we construct
out of the ensemble average of density fields given in the 3D space
hypersurface at given time-slices and positions. We can define the cross-
correlation of two such density fields, not necessarily at the same
hypersurface; this gives us the unequal-time, theoretical, power spectrum:

$\left\langle\delta(\bm{k},z)\delta(\bm{k}^{\prime},z^{\prime})\right\rangle=(2\pi)^{3}\delta^{\rm
D}(\bm{k}+\bm{k}^{\prime})\mathcal{P}(\bm{k},z,z^{\prime})\,,$ (14)

where $z$ is the redshift and $\bm{k}$ is the Fourier wave mode corresponding
to the 3D position vector $\bm{x}$. This power spectrum is obtained by cross-
correlating the 3D density field at two different times of the evolution of
the ensemble, as shown on the left-hand side of Figure 1. It is accessible,
e.g., by meta-observers and in N-body simulations and theoretical
calculations, but not as a real observable, given that every realistic
observer has access only to a lightcone-projected subset of information, thus
being limited to specific observables, constructed from measured quantities.

Constructed observed power spectrum:
$P\big{(}q_{\hat{n}},\ell/\bar{\chi},z[\bar{\chi}]\big{)}$•$\mathcal{O}$$\bar{\chi}$$\chi^{\prime}$$\chi$$\delta\chi$$\mathcal{V}$$\hat{n}$$q_{\hat{n}}\sim
2\pi/\delta\chi$$\bar{\ell}/\bar{\chi}$ Figure 2: Construction of the observed
3D power spectrum using the perpendicular and line of sight Fourier modes,
$q_{\perp}$ and $q_{\hat{n}}$. $q_{\perp}$ corresponds to $\ell$ divided by
the mean comoving distance, i.e., $q_{\perp}=\ell/\bar{\chi}$, while the along
the line of sight mode is the Fourier counterpart of the $\delta\chi$
variable, i.e., $q_{\hat{n}}\sim 2\pi/\delta\chi$.

####  II.2.1 Tomography with narrow windows

Let us look now at expressions for specific choices of narrow window
functions. If we introduce infinitely thin redshift slices as
$W(\chi)\to\delta^{\rm D}\left(\chi-\chi_{*}\right)$, integrating over the
windows gives us
$C\big{(}\ell,\chi_{*},\chi^{\prime}_{*}\big{)}=\mathbb{C}\left(\ell,\chi_{*},\chi^{\prime}_{*}\right)$,
i.e., in the limit of narrow window functions we recover the true unequal-time
angular power spectrum as an observable. This gives us the relation of the
unequal time angular spectrum, measured in the infinitely thin redshift
slices, with theoretical power spectrum $\mathcal{P}(\bm{k},z,z^{\prime})$.
Note that we did not specify if our theoretical power spectrum is in real or
redshift space since none of the above depends on it. Indeed, we will be able
to keep this generality for a while; let us just mention again that in either
case, $\mathcal{P}$ does not depend on the azimuthal angle, which allowed us
to drop the angular dependence of the $\bm{k}_{\perp}$, i.e.,
$\mathcal{P}(k_{\hat{n}},\bm{k}_{\perp},z,z^{\prime})=\mathcal{P}(k_{\hat{n}},k_{\perp},z,z^{\prime})$.
The most common approach in literature at this stage is to resort to the
Limber approximation [47, 48]. Let us remind ourselves how it can be
recovered. We can assume that $\mathcal{P}$, when integrating over
$k_{\hat{n}}$, predominantly depends on $k_{\perp}$, neglecting other scale
dependencies, i.e.,
$\mathcal{P}\big{(}k_{\hat{n}},k_{\perp},z,z^{\prime}\big{)}\simeq\mathcal{P}\big{(}k_{\perp},z,z^{\prime}\big{)}$
and therefore we have:

$\mathbb{C}_{\rm
Lim}\left(\ell,\bar{\chi},\delta\chi\right)=\frac{1}{\bar{\chi}^{2}}\delta^{\rm
D}(\delta\chi)\mathcal{P}\big{(}\ell/\bar{\chi},\bar{\chi},\bar{\chi}\big{)}\,,$
(15)

and thus, only the equal-time correlations are not forced to vanish, i.e.,
$\mathbb{C}(\ell)$ is nonzero only when $z\approx z^{\prime}$. After restoring
the window $W(\chi)$ dependence, we obtain the familiar result:

$C\big{(}\ell\big{)}=\int
d\chi\frac{W(\chi)W^{\prime}(\chi)}{\chi^{2}}\mathcal{P}\big{(}\ell/\chi,\chi,\chi\big{)}\,.$
(16)

On the other hand, if we can assume that the power spectrum $\mathcal{P}$ has
negligible $\delta\chi$ dependence over the relevant integration volume, i.e.,
$z\approx z^{\prime}$, we can invert Equation (12) to obtain:

$\mathcal{P}\big{(}k_{\hat{n}},\ell/\bar{\chi},\bar{\chi}\big{)}=\bar{\chi}^{2}\int
d(\delta\chi)\;e^{-i\delta\chi
k_{\hat{n}}}\mathbb{C}\big{(}\ell,\bar{\chi},\delta\chi\big{)}\,.$ (17)

In Section II.3, we quantify the errors arising due to this approximation and
show the scale-dependent corrections neglected here.

Lastly, let us repeat the statement about the theoretical 3D power spectrum
$\mathcal{P}(\bm{k},z,z^{\prime})$: while it is the quantity that encapsulates
the dynamical and stochastic information about the system, it is not
observable from the point of view of an observer sitting in some point
$\mathcal{O}$ “in the box”, as shown in Figure 1 (see also the discussion in
[35]). What observers can measure are unequal-time angular correlations that
contain the information imprinted in the ensemble power spectrum, projected
into an observable. In the next subsection, we show how the observer can
construct the corresponding 3D power spectrum from measurements of the
unequal-time angular power spectrum.

###  II.3 Observed 3D equal-time power spectrum

Our task in this subsection is to construct the 3D equal time observed power
spectrum, which we call $P(q_{\hat{n}},q_{\perp},\bar{z})$, that an observer
at some position $\mathcal{O}$ could observe. In addition to constructing the
power spectrum, we need to define the corresponding Fourier modes
$q_{\hat{n}}$ and $q_{\perp}$ (which we label with variable $\bm{q}$ in order
to distinguish it from the ensemble power spectrum variable $\bm{k}$), as well
as the mean redshift $z$ (see the set up in Figure 2). The expectation is that
the constructed spectrum is, of course, related to the 3D unequal-time
ensemble power spectrum $\mathcal{P}(\bm{k},z,z^{\prime})$; however, let us
proceed step by step in the construction. We use the information from the
unequal-time angular power spectra
$\mathbb{C}\left(\ell,\bar{\chi},\delta\chi\right)$ in order to construct the
Fourier modes along the line of sight $\hat{n}$, while at the same time keep
the information on the mean distance between the observer $\mathcal{O}$ and
the survey volume $\mathcal{V}$ (or equivalently, the redshift bin analyzed).
Such a fixed volume, depicted in Figure 2, is characterised by the maximal and
minimal comoving distances $\chi_{\rm min}$ and $\chi_{\rm max}$. The mean
distance $\bar{\chi}$ can thus take the value between $\chi_{\rm min}$ and
$\chi_{\rm max}$, while $(\chi_{\rm
min}-\bar{\chi})\leq\delta\chi/2\leq(\chi_{\rm max}-\bar{\chi})$. Sticking to
the center of the box with the mean distance $\bar{\chi}$, to ensure a wide
enough range for $\delta\chi$, we can define a constructed wave mode along the
line of sight as the Fourier counterpart of $\delta\chi$. We can then use the
observable unequal-time angular power spectrum to obtain what we can define as
the observed, equal-time, 3D power spectrum as:

$P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})\equiv\bar{\chi}^{2}\int
d(\delta\chi)\;e^{-i\delta\chi
q_{\hat{n}}}\mathbb{C}(\ell,\bar{\chi},\delta\chi)\,,$ (18)

where $\mathbb{C}$ is given in Equation (12). Relying on the set-up shown in
Figure 2, we define the perpendicular and line of sight Fourier modes,
$q_{\perp}$ and $q_{\hat{n}}$. With $q_{\perp}$ we identify modes
corresponding to $\ell$ divided by the mean comoving distance, i.e.,
$q_{\perp}=\ell/\bar{\chi}$, while for the line of sight mode we can take the
Fourier counterpart of the $\delta\chi$ variable, i.e., $q_{\hat{n}}\sim
2\pi/\delta\chi$.

When does the, so constructed, observable, 3D power spectrum
$P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})$ match the theoretical 3D ensemble
power spectrum $\mathcal{P}(\bm{k},\chi,\chi)$? As argued in [35], this
happens when the observed system (the survey) is so small in width and depth
compared to the full sky and the distance from the observer that we can
approximate the observer as a meta-observer. This here can be compared to
assuming negligible dependence of $\mathcal{P}$ on $\delta\chi$; combining
Equations (18) and (12) we obtain:

$\displaystyle\bar{\chi}^{2}$
$\displaystyle\int_{-\delta\chi_{<}}^{\delta\chi_{>}}d(\delta\chi)\;e^{-i\delta\chi
q_{\hat{n}}}\mathbb{C}(\ell,\bar{\chi},\delta\chi)$ (19) $\displaystyle\hskip
8.5359pt=\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
\mathcal{P}\left(k_{\hat{n}},\ell/\bar{\chi},\bar{z}\right)\int_{-\delta\chi_{<}}^{\delta\chi_{>}}d(\delta\chi)\leavevmode\nobreak\
e^{-i(q_{\hat{n}}-k_{\hat{n}})\delta\chi}$ $\displaystyle\hskip
8.5359pt=\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
\mathcal{B}\left(q_{\hat{n}}-k_{\hat{n}},\delta\chi_{<},\delta\chi_{>}\right)\mathcal{P}\left(k_{\hat{n}},\ell/\bar{\chi},\bar{z}\right)\,,$

where $\chi_{<}$ and $\chi_{>}$ are the closest and farthest integration
distances within the survey, and:

$\mathcal{B}\left(k,\delta\chi_{<},\delta\chi_{>}\right)=\frac{i}{k}\left(e^{-i\delta\chi_{<}}-e^{i\delta\chi_{>}}\right)\,.$
(20)

Given the dependence of $\mathbb{C}$ on $\delta\chi$ (that we discuss in
Section III), we argue that for large enough survey volumes, the steep decline
of $\mathbb{C}$ from $\delta\chi=0$ values guarantees that we can extend and
symmetrize the integration region of $\chi_{<}$ and $\chi_{>}$. This is indeed
the case for a $\Lambda$CDM universe (see Section III), and thus, as long as
$\bar{\chi}$ is not close to the edge of the survey, we can adapt the limit:

$\mathcal{B}\left(k,\Delta\chi\right)=2\frac{\sin(k\Delta\chi)}{k}\approx
2\pi\delta^{\rm D}(k),\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm
as}\leavevmode\nobreak\ \leavevmode\nobreak\ \Delta\chi\to\infty\,,$ (21)

where $\chi_{<}\approx\chi_{>}\approx\Delta\chi$. Using this approximation,
the expression above yields:

$P\left(q_{\hat{n}},\bar{\ell}/\bar{\chi},\bar{z}\right)\approx\mathcal{P}\left(q_{\hat{n}},\bar{\ell}/\bar{\chi},\bar{z}\right)\,,$
(22)

as expected. The definition given in Equation (18) thus recovers the original
ensemble power spectrum. More generally, the proper 3D observed power
spectrum, as defined in Equation (18), is sensitive to unequal-time
contributions from the 3D theoretical power spectrum
$\mathcal{P}(\bm{k},\chi,\chi^{\prime})$. Before quantifying these unequal-
time contributions, we introduce an alternative 3D observable, one that does
not rely on the construction of dimensional wave modes $q_{\hat{n}}$ and
$q_{\perp}$, but still retains the desirable properties of the observed power
spectrum $P(q_{\hat{n}},q_{\perp},\bar{z})$.

Let us recap. Cosmological and dynamical information is encapsulated in the 3D
unequal-time theoretical power spectrum $\mathcal{P}(k,z,z^{\prime})$, and
therefore, it is not directly accessible to observations. From this, we can
compute observable quantities like unequal time 2D angular power spectra
$\mathbb{C}(\ell,z,z^{\prime})$. However, the information is spread out (along
the line of sight) over different redshift shells. The question is thus
whether we can use this angular power spectrum to construct the corresponding
equal-time 3D power spectrum that matches the original one as best as
possible. The constructed perpendicular modes $q_{\perp}$ are related to the
inverse angular multipoles $\ell$ and the mean distance $\bar{\chi}$. The
modes along the line of sight $q_{\hat{n}}$ are constructed by Fourier
transforming along the unequal-time dependence of $\mathbb{C}(\ell)$. This
construction provides us with the result that (to a very good approximation)
corresponds to the familiar equal-time 3D power spectrum. Unequal-time
contributions give rise to sub-leading corrections that we quantify in the
remainder of the paper.

Figure 3: Errors on the comoving distance differences, $\delta\chi$. On the
left panel, we show the relative errors of the linear approximation
$\delta\chi\approx\delta z/H$ and next-to-linear approximation
$\delta\chi\approx\delta z/H(1-c_{3}(\delta z)^{2})$ for several mean
redshifts $\bar{z}$, assuming $\Omega_{m0}=0.3$. The right panel shows the
size and the redshift dependence of the $c_{3}(z)$ coefficient for the
fiducial value of $\Omega_{m0}$, with the grey band indicating 10% variations.
In all cases, we assume the flat $\Lambda$CDM cosmology.

###  II.4 Frequency-angular power spectrum: the “new observable"

So far, we have managed to connect the theoretical with the observed power
spectrum. As shown, the connection was achieved via the unequal time angular
power spectrum $\mathbb{C}(\ell,z,z^{\prime})$. Here we revisit the motivation
for the construction of the observed power spectrum. We have seen that the
information content of the unequal time angular spectrum is equivalent to the
observed power spectrum $P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})$, so why
bother with the additional step of constructing the observed power spectrum?
The reason lies in the compactification of information, i.e., the covariance
matrix of the observed power spectrum is well described with a diagonal
Gaussian approximation. The disadvantage is in the need to construct the
observable wave modes $q_{\hat{n}}$ and $\bm{q}_{\perp}$, using the fiducial
cosmology to determine the comoving distance. This gives rise to the well-
known Alcock-Paczynśki effect [49].

The observed power spectrum is obtained by performing a simple Fourier
transform in the $\delta\chi$ variable. However, one can imagine doing the
same procedure, as defined in Equation (18), without involving the comoving
distance, i.e., we can define a frequency-angular power spectrum as:

$\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})\equiv\int d\delta
z\leavevmode\nobreak\ e^{-i\omega\delta z}\mathbb{C}(\ell,\chi(\bar{z}),\delta
z)\,,$ (23)

where the new Fourier frequency variable $\omega$ plays the role previously
done by $q_{\hat{n}}$. The statistical properties of the covariance matrix
should inherit all the properties of the 3D power spectrum (approximate
diagonal structure and Gaussianity). We highlight that this new observable,
given its functional dependence on only observable quantities, does not
exhibit any Alcock-Paczyśki effects, i.e., we do not need a fiducial cosmology
to compute physical distances, typically required in computing the 3D observed
power spectrum. We can generalize this definition even further by introducing
the variable frequency $\omega(\ell,\bar{z})$, which can also depend on
cosmological parameters. We thus obtain a generalized frequency-angular power
spectrum, defined as:

$\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})\equiv\int d\delta
z\leavevmode\nobreak\ e^{-i\omega(\ell,\bar{z})\delta
z}\mathbb{C}(\ell,\chi(\bar{z}),\delta z)\,.$ (24)

We shall see how to best utilize this generalized form further below.

Later on, we investigate the properties of this observable, assuming some
concrete form of the 3D theoretical power spectrum. However, before that, we
can again look at the simplifying case when we assumed a negligible unequal-
time dependence of the 3D theoretical power spectrum, i.e., we assume
$\mathcal{P}(\bm{k},\bar{z},\delta z)=\mathcal{P}(\bm{k},\bar{z})$. We have:

$\displaystyle\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})$ $\displaystyle=\int
d\delta z\leavevmode\nobreak\ e^{-i\omega\delta
z}\mathbb{C}(\ell,\bar{\chi}(\bar{z}),\delta z)$ (25)
$\displaystyle=\frac{1}{\bar{\chi}^{2}}\int\frac{dk_{\hat{n}}}{2\pi}\;\Omega(\omega,k_{\hat{n}})\mathcal{P}\left(k_{\hat{n}},\ell/\bar{\chi},\bar{z}\right)\,,$
(26)

where:

$\Omega(\omega,k_{\hat{n}})=\int d\delta z\;\exp\left(i\delta\chi
k_{\hat{n}}-i\omega\delta z\right).$ (27)

Assuming linear dependence of $\delta\chi$ on $\delta z$, i.e.,
$\delta\chi\approx\frac{d\chi}{dz}\delta z=\delta z/H$, where $H$ is the
Hubble parameter, evaluated at the mean redshift $\bar{z}$, we have:

$\Omega(\omega,k_{\hat{n}})=(2\pi)\delta^{\rm
D}\left(k_{\hat{n}}/H-\omega\right)\,\,,$ (28)

which gives the frequency-angular power spectrum:

$\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})=\frac{H}{\bar{\chi}^{2}}\mathcal{P}\left(H\omega,\ell/\bar{\chi},\bar{z}\right)\,,$
(29)

that is a dimensionless quantity, depending only on observable variables.
Adding the next order correction to $\delta\chi$, we have
$\delta\chi\approx\frac{d\chi}{dz}\delta z=\frac{\delta z}{H}[1-c_{3}(\delta
z)^{2}]$, where the expression for the $c_{3}(\bar{z})$ in $\Lambda$CDM is
obtained by expanding $\delta\chi(\bar{z},\delta z)$ around the equal time
case, we have:

$c_{3}(z)=\frac{1}{8}\frac{\Omega_{m}(z)\left[1-\tfrac{9}{4}\Omega_{m}(z)\right]}{(1+\bar{z})^{2}}\;,$
(30)

and the full derivation can be found in Appendix B. This gives us:

$\displaystyle\widetilde{\mathbb{C}}(\omega,\ell,\bar{z})$
$\displaystyle=e^{-c_{3}\omega\frac{d^{3}}{d\omega^{3}}}\frac{H}{\bar{\chi}^{2}}\mathcal{P}\left(H\omega,\ell/\bar{\chi},\bar{z}\right)\,$
(31)
$\displaystyle\approx\frac{H}{\bar{\chi}^{2}}\left(1-c_{3}\omega\frac{d^{3}}{d\omega^{3}}\right)\mathcal{P}\left(H\omega,\ell/\bar{\chi},\bar{z}\right)\,.$

In Figure 3, we show the dependence of the $\delta\chi$ variable on $\delta
z$. The purpose of this is to establish the approximations leading to the
result in Equation (29). On the left panel, we show the relative errors of the
linear approximation $\delta\chi\approx\delta z/H$ for several redshifts in
$\Lambda$CDM cosmology, where we see that for most low-redshift spectroscopic
surveys, the error caused by neglecting higher order $\mathcal{O}(\delta
z^{3})$ corrections is suppressed to a fraction of a percent. Moreover, once
the $O(\delta z^{3})$ corrections are added, corrections are suppressed below
the $0.1\%$.

On the right panel, we show the redshift dependence of the $c_{3}$ coefficient
describing the $O(\delta z^{3})$ to the $\delta\chi\leftrightarrow\delta z$
relation. The size of the coefficient in a flat $\Lambda$CDM universe depends
only on one cosmological parameter, $\Omega_{m}$, its value being bounded
$|c_{3}|\lesssim 0.02$ for values of $\Omega_{m}$ currently allowed, and it
asymptotes to zero as $\sim 1/z^{2}$. These considerations prompt us to
believe that for all practical purposes, in $\Lambda$CDM cosmology, accounting
for the leading corrections in $\delta z$ as done in Equation (31) should
suffice for low-redshift surveys, while future surveys that aim to measure
galaxy clustering at $z>3$ might need to account for the higher order
correction term.

Let us also entertain the fact that the 3D power spectrum $\mathcal{P}$ is
typically a function of $k=\sqrt{k^{2}_{\hat{n}}+k_{\perp}^{2}}$, and once
redshift-space distortions are added, also even powers of the orientation
angle $\mu=k_{\hat{n}}/k$. In terms of our observable quantities, $\omega$ and
$\ell$, this means that we have:

$k^{2}=\frac{1}{{\bar{\chi}}^{2}}\left[\ell^{2}+(H\bar{\chi})^{2}\omega^{2}\right]\,.$
(32)

Motivated by the multipole expansion usually performed in the observed power
spectrum, let us look at how we could reproduce this in the case of the
frequency-angular power spectrum $\widetilde{\mathbb{C}}$. First let us define
the total momentum $L\equiv\sqrt{\ell^{2}+(H\bar{\chi})^{2}\omega^{2}}$. We
can immediately notice that $L$ is no longer a cosmology-independent variable
as $\omega$ and $\ell$ are, as it depends on the $H\bar{\chi}$ product. This
is where our generalization introduced in Equation (24) becomes useful. We can
use the freedom introduced in the generalized frequency $\omega(\ell,\bar{z})$
to cancel the cosmology dependence introduced in $k^{2}$, i.e., we can
introduce:

$\omega(\ell,\bar{z})\to\frac{1}{(H\chi)(\bar{z})}\omega\,,$ (33)

which gives us the wave modes:

$k^{2}=\frac{1}{{\bar{\chi}}^{2}}\left(\omega^{2}+\ell^{2}\right)=\frac{L^{2}}{{\bar{\chi}}^{2}}\,,$
(34)

where the total momentum is simply $L\equiv\sqrt{\omega^{2}+\ell^{2}}$. This
allows us to rewrite the frequency-angular power spectrum given in Equation
(29) as:

$\widetilde{\mathbb{C}}(L,\bar{z})=\frac{H}{\bar{\chi}^{2}}\mathcal{P}\left(L/\bar{\chi},\bar{z}\right)\,.$
(35)

We note that a different choice of $\omega$ than the one given in Equation
(33), or even fixing the fiducial cosmology of the $(H\chi)(\bar{z})$ weight,
would lead to anisotropic dependencies in $\ell$ and $\omega$, which is
equivalent to the Alcock-Paczyśki effect [49]. These anisotropies can then be
used to constrain cosmological models; this is also possible since such
anisotropies have a different shape dependence than RSD, assuming enough
dynamic range is captured.

We can also look at the frequency-angular power spectrum in redshift-space,
assuming for now just the usual linear Kaiser formula [48]. Neglecting, for
now, unequal-time effects, we have:

$\mathcal{P}_{\rm
lin}(k,\mu,\bar{z})=D^{2}\left[1+f(\bar{z})\mu^{2}\right]^{2}\mathcal{P}_{0}(k)\,,$
(36)

where we assume that the linear growth factor and rate are evaluated at the
mean redshift $\bar{z}$, i.e., $D=D(\bar{z})$ and $f=f(\bar{z})$.
$\mathcal{P}_{0}$ gives the shape dependence of the linear power spectrum.
There is a simple relation between the wave mode angular variable and a newly
introduced angular variable $\nu=\omega/L$, that is:

$\mu=\frac{k_{\hat{n}}}{k}=(H\bar{\chi})\frac{\omega(\ell,\bar{z})}{L}\to\frac{\omega}{L}=\nu\,.$
(37)

Using the expression given in Equation (29) we obtain the Kaiser frequency-
angular power spectrum in redshift-space:

$\widetilde{\mathbb{C}}_{\rm
lin}(L,\nu,\bar{z})=\frac{HD^{2}}{\bar{\chi}^{2}}\left(1+f\nu^{2}\right)^{2}\mathcal{P}_{0}(L/\bar{\chi})\,.$
(38)

In this observable, the multipoles, obtained by expanding in Legendre
polynomials in $\nu$, retain the same form as in the usual 3D observed power
spectrum.

Figure 4:  Linear theory results for the equal-time theory power spectrum
(left panels) and the frequency-angular power spectrum
$\widetilde{\mathbb{C}}$ (right panels), for three different redshifts,
$z=0.5,\leavevmode\nobreak\ 1.0$ and $2.0$. The bottom panels show the ratio
with the smooth (no BAO wiggles) version of the respective power spectra. For
the equal-time theory power spectrum, the BAO signal is independent of
redshift, while for the frequency-angular power spectrum
$\widetilde{\mathbb{C}}$, the signal shifts proportionally to the comoving
distance $\chi$.

In Figure 4, we show the comparison of the equal-time linear theory power
spectrum and the frequency-angular power spectrum $\widetilde{\mathbb{C}}(L)$.
We show the results for three redshifts $z=0.5,\leavevmode\nobreak\ 1.0$, and
$2.0$, also showing the ratio with the smooth (no-wiggles) version of the
spectrum in the bottom panels. As also indicated in the plots, the usual
observed power spectrum $P(k)$ depends on chosen units, while the new power
spectrum $\widetilde{\mathbb{C}}(L)$ is a unit-independent quantity. We also
notice that, while in the linear theory power spectrum the BAO signal does not
shift with different redshifts, for our new observable the BAO signal shifts
with the comoving distance $\chi(z)$. This happens because of the relation
between the average comoving distance $\chi$ and the multipole scale. From the
bottom panels, which show the ratio to the smooth spectrum, we can see that
the features shift from $z_{1}$ to $z_{2}$, as
$L_{2}/\chi_{2}=L_{1}/\chi_{1}$. This is, of course, similar to the standard
Alcock-Paczynśki effect [49], where the anisotropies in the BAO are used to
calibrate geometric distances. However, the novelty of the approach taken here
is in the relinquishing of the need to use fiducial cosmology and reliance on
isotropy. In our formalism, the comoving distance information can be directly
established from the position of the BAO wiggles.

We now estimate the corrections to the power spectrum
$\widetilde{\mathbb{C}}(L)$ arising from the $(\delta z)^{3}$ term in the
$\delta\chi\leftrightarrow\delta z$ relation. From Equation (31), we see that
the leading correction to the power spectrum takes the form:

$\displaystyle\delta\widetilde{\mathbb{C}}_{\rm lin}(\omega,\ell,\bar{z})$
$\displaystyle\approx-
c_{3}\frac{H}{\bar{\chi}^{2}}\omega\frac{d^{3}}{d\omega^{3}}\mathcal{P}_{\rm
lin}\left(H\omega,\ell/\bar{\chi},\bar{z}\right)\,,$ (39)
$\displaystyle=-c_{3}D^{2}H^{3}\frac{\nu^{2}}{L^{2}}\delta\mathcal{P}_{3}\left(L/\bar{\chi},\nu\right)\,,$

where we used the linear matter power spectrum $\mathcal{P}_{\rm
lin}(k,\bar{z})=D(\bar{z})^{2}\mathcal{P}_{0}(k)$ (neglecting here redshift-
space distortions) and we introduced:

$\displaystyle\delta\mathcal{P}_{3}\left(k,\mu\right)=\mu^{2}$ $\displaystyle
k^{3}\mathcal{P}^{\prime\prime\prime}_{0}(k)$ (40)
$\displaystyle+3\left(1-\mu^{2}\right)\left[k^{2}\mathcal{P}^{\prime\prime}_{0}(k)-k\mathcal{P}^{\prime}_{0}(k)\right]\,.$

The corrections in Equation (39) are proportional to $\nu^{2}$, therefore
introducing an anisotropy even when we start with the isotropic power spectrum
$\mathcal{P}$. This originates in the fact that the introduction of an
observer breaks some symmetries in the system, and it shows how the ensemble
average power spectrum is not accessible as an observed power spectrum.

Moreover, the corrections depend on the derivatives of the theoretical power
spectrum up to the third derivative. In order for the frequency-angular power
spectrum $\widetilde{\mathbb{C}}(L)$ to be useful observable, comparatively to
the observed power spectrum $P(k)$ as defined in Equation (18), these
$\delta\widetilde{\mathbb{C}}$ corrections should be negligible in all
practical cases. We are thus interested in estimating the size of the
$\delta\widetilde{\mathbb{C}}$ corrections. The maximal contribution is
expected on large scales, given the $1/L^{2}$ dependence. Approximating the
power spectrum with the power law on large scales, we have $\nu_{\rm
max}\approx\sqrt{3/(2(4-n_{s}))}$, which turns out to be a good approximation
on all scales. Using this $\nu_{\rm max}$ value, we can provide an estimate
for the corrections to the

$\displaystyle\left|\frac{\delta\widetilde{\mathbb{C}}_{\rm
lin}(L,\nu,\bar{z})}{\widetilde{\mathbb{C}}_{\rm lin}(L,\bar{z})}\right|$
$\displaystyle\leq|c_{3}|\frac{(\bar{\chi}H)^{2}}{L^{2}}\frac{\delta\mathcal{P}_{3}\left(L/\bar{\chi},\nu_{\rm
max}\right)}{\mathcal{P}_{0}\left(L/\bar{\chi}\right)}\,.$ (41)

We can estimate these effects to be of order percent for $L\lesssim 10$ at
high redshifts $z\sim 5$, while their size drops quickly for lower redshifts
and higher $L$. Their impact thus might be relevant only when considering
future wide and deep high redshift surveys.

The conclusion is thus that the frequency-angular power spectrum
$\widetilde{\mathbb{C}}$, as introduced in Equation (24), is a well-behaved
observable with small to negligible sub-leading corrections. In that respect,
it is equivalent to the observed 3D power spectrum $P(k)$, with the additional
advantages already highlighted above.

In this derivation, we adopted the flat-sky approximation and used the
corresponding angular power spectrum $\mathbb{C}_{\ell}$. However, our newly
defined observable, the frequency-angular power spectrum
$\widetilde{\mathbb{C}}$ as defined in Equation (24), does not require a flat-
sky approximation. On the contrary, we are free to extend the relationship and
introduce the full-sky version of the generalized frequency-angular power
spectrum, defined as:

$\widetilde{\mathbb{C}}^{\rm full}_{\ell}(\omega,\bar{z})\equiv\int d\delta
z\leavevmode\nobreak\ e^{-i\omega(\ell,\bar{z})\delta z}\mathbb{C}^{\rm
full}_{\ell}\left(z_{1},z_{2}\right)\,,$ (42)

where $\mathbb{C}_{\ell}^{\rm full}$ is the full-sky unequal-time angular
power spectrum, given in Equation (13). Moreover, the observed, equal-time, 3D
power spectrum, as introduced in Equation (18), can analogously be extended to
its full-sky version. We can simply replace the corresponding angular power
spectrum $\mathbb{C}_{\ell}$:

$P^{\rm
full}\left(q_{\hat{n}},\tfrac{\ell}{\bar{\chi}},\bar{\chi}\right)\equiv\bar{\chi}^{2}\int
d(\delta\chi)\;e^{-i\delta\chi q_{\hat{n}}}\mathbb{C}^{\rm
full}_{\ell}(\bar{\chi},\delta\chi)\,.$ (43)

By adopting this definition for the $P^{\rm full}$, we are abandoning any
notion of the construction of the observable 3D power spectrum in a
rectangular box and have a procedure to go from the theoretical power spectrum
to the observed power spectrum. Besides, this definition naturally
incorporates the so-called wide-angle effects, i.e., effects arising from
deviations from the flat-sky approximation (see, e.g., [50, 51, 52, 45, 53,
54]). Quantifying these deviations basically boils down to estimating the
difference of using flat-sky vs the full-sky version of unequal-time angular
power spectrum $\mathbb{C}_{\ell}$ in Equation (43). We shall address the
quantification of this difference in future work.

##  III Effects of unequal-time cross-correlations

In this section, we look at the results of the observed power spectrum, taking
into account unequal time effects. To compute them, we start from the unequal
time 3D ensemble power spectrum given by linear theory (and the Kaiser formula
when including the redshift space distortions). These are chosen as
representative of two instructive cases while still being computationally
simple. They can also be straightforwardly generalized to include non-linear
corrections using canonical perturbation theory approaches (be it in the EFT
suit or others, see e.g., [55] for a review). In addition to these two linear
theory results, we also consider the linear power spectrum in Lagrangian
perturbation theory as a prototypical example of the resummation of the long
displacement field contributions [56, 57]. Resumming these long displacements,
even in the case of the equal time power spectrum, is important since it
affects the damping and shape of the BAO oscillations [58, 57, 59, 60].
Moreover, in the case of the unequal time correlators, these displacements are
the primary cause of the rapid decorrelation of radial modes and the
suppression of unequal time power relative to the equal time correlators (see
e.g., [61], and also [62] for a related discussion).

Figure 5:  Linear theory results in a real-space power spectrum for two
different redshifts ($z=0.5$, dashed lines, $z=2$, solid lines). Left Panel:
ratio of the correction $\delta P_{\rm lin}$ over the equal time case for the
first two even multipoles for real space dark matter spectra. Right Panel:
ratio of the first order imaginary part correction, $\delta P_{\rm lin}$ for
$\ell=1$, in a real-space multi-tracer case with $\Delta\eta=0.2$. The ratio
is over the amplitude of the (real) monopole, for a comparison of amplitudes.

Combining Equation (18), that defines that observed, equal-time, 3D power
spectrum, with the flat-sky approximation for the angular power spectrum given
in Equation (12), we arrive at the following expression:

$P\left(q_{\hat{n}},\frac{\ell}{\bar{\chi}},\bar{\chi}\right)=\int\frac{dk_{\hat{n}}}{2\pi}d(\delta\chi)e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}\mathcal{P}\left(k_{\hat{n}},\frac{\ell}{\bar{\chi}},\bar{\chi},\delta\chi\right).$
(44)

This expression gives us a direct relationship between the observed equal-time
power spectrum $P$ and the theoretical 3D unequal-time power spectrum
$\mathcal{P}$. We note that the unequal-time effects, encapsulated in the
$\delta\chi$ power spectrum dependence, are folded in together with the
dependence on the modes along the line of sight $k_{\hat{n}}$. This folding is
finally combined in the cumulative (effective) line of sight mode
$q_{\hat{n}}$. In the rest of this section, we investigate the consequences of
this folding, investigating the Equation (44).

###  III.1 Linear Power Spectrum

####  III.1.1 Dark matter results

We start our investigation of unequal time effects by first considering just
dark matter linear theory results, where the 3D unequal time power spectrum is
given by:

$\mathcal{P}_{\rm lin}(k,z_{1},z_{2})=D(z_{1})D(z_{2})\mathcal{P}_{0}(k)\,,$
(45)

and we can separate the time dependence into two linear growth factors
$D(z_{i})$ and the time-independent $k$-dependent term. In order to proceed,
we want to expand around the equal time solution; we follow the analogous
procedure as in [35] and expand the product of the two growth factors up to
quadratic order in $\delta\chi$ to obtain:

$D(z_{1})D(z_{2})=D^{2}(\bar{z})+\frac{1}{8}\Delta^{(0)}_{2}(\bar{z})\left[H(\bar{z})\delta\chi\right]^{2}\,,$
(46)

where we introduced the mean redshift-dependent factor:

$\Delta^{(0)}_{2}(\bar{z})=-2D^{2}(\bar{z})\left[1+f(\bar{z})-\frac{3}{2}\frac{\Omega_{m}(\bar{z})}{f(\bar{z})}\right]\frac{f(\bar{z})}{(1+\bar{z})^{2}}\,,$
(47)

obtained using the arithmetic definitions for $\bar{\chi}$ and $\delta\chi$;
for the extensive calculation we refer to Appendix C. It is worth noting that
the first order correction vanishes (but as we will see later on, this does
not always happen). Using the expression given in Equation (44), we first
evaluate the integral over $\delta\chi$ to obtain:

$\displaystyle\int d(\delta\chi)\;$ $\displaystyle
e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}\left[1-\frac{1}{4}\gamma_{\times}(H\delta\chi)^{2}\right]$
(48)
$\displaystyle=\left[1+\frac{\gamma_{\times}}{4}\left(H\partial_{q_{\hat{n}}}\right)^{2}\right](2\pi)\delta^{\rm
D}(q_{\hat{n}}-k_{\hat{n}})\,,$

where we use the factor $\gamma_{\times}=-\frac{1}{2}\Delta^{(0)}_{2}/D^{2}$;
this gives us the observed power spectrum expression:

$P_{\rm
lin}(k_{\hat{n}},k_{\perp},\bar{z})=\left[1+\frac{\gamma_{\times}}{4}H^{2}\left(\hat{n}\cdot\bm{\nabla}\right)^{2}\right]\mathcal{P}_{\rm
lin}\left(k,\bar{z}\right)\,.$ (49)

The deviation from the leading result obtained in Equation (22), i.e., the
canonical linear theory results, is then given by:

$\displaystyle\delta P_{\rm lin}(k,\mu,\bar{z})$ $\displaystyle=P_{\rm
lin}(k,\mu,\bar{z})-\mathcal{P}_{\rm lin}\left(k,\bar{z}\right)$ (50)
$\displaystyle=\frac{\gamma_{\times}}{4}\left(\frac{H}{k}\right)^{2}D(\bar{z})^{2}\delta\mathcal{P}_{2}\left(k,\mu\right)\,,$

with:

$\delta\mathcal{P}_{2}\left(k,\mu\right)=\mu^{2}k^{2}\mathcal{P}^{\prime\prime}_{0}(k)+\left(1-\mu^{2}\right)k\mathcal{P}^{\prime}_{0}(k)\,,$
(51)

where $\mu$ is the usual cosine of the angle between the wave mode $\bm{k}$
and the line of sight, $\mu=k_{\hat{n}}/k$, and the derivatives
$\mathcal{P}^{(n)}_{0}$ are to be taken w.r.t. the wave mode $k$. The unequal-
time effects can thus give rise to anisotropies in the observed 3D power
spectrum, generating higher multipole contributions. Besides contributing to
the monopole, Equation (51) also contributes to the quadruple, and we can
write:

$\displaystyle\delta P^{(0)}_{\rm lin}$
$\displaystyle=\frac{\gamma_{\times}}{12}\left(\frac{H}{k}\right)^{2}D(\bar{z})^{2}\big{(}k^{2}\mathcal{P}^{\prime\prime}_{0}+2k\mathcal{P}^{\prime}_{0}\big{)}\,,$
(52) $\displaystyle\delta P^{(2)}_{\rm lin}$
$\displaystyle=\frac{\gamma_{\times}}{6}\left(\frac{H}{k}\right)^{2}D(\bar{z})^{2}\big{(}k^{2}\mathcal{P}^{\prime\prime}_{0}-k\mathcal{P}^{\prime}_{0}\big{)}\,.$

In the left panel of Figure 5, we show the ratio of these corrections for the
monopole and quadrupole relative to the linear theory at redshifts $z=0.5$ and
$2.0$. It is interesting to notice that even without considering redshift-
space distortions, these corrections introduced quadrupole corrections, which
is expected as a consequence of breaking the statistical isotropy. In a
follow-up work, we intend to compare such corrections to the ones introduced
by the Doppler term and relativistic corrections (see, e.g., [63, 64, 65,
66]).

####  III.1.2 Multi-tracer analyses

Until now, in this section, we considered correlations in real space and for
the dark matter case. In the case of biased tracers, the results have to
obviously take into account the fact that sources are biased tracers of the
underlying matter distribution; however, the structure of the corrections is
the same. The result is instead different, as introduced in [35] when we
consider the correlation of two different tracers in the so-called multi-
tracer analysis (see e.g. [67]). If we consider the cross-correlation of two
different tracers in linear theory, the 3D unequal time power spectrum is
given by:

$\mathcal{P}_{\rm lin}^{\rm AB}(k,z_{1},z_{2})=D(z_{1})D(z_{2})b_{\rm
A}(z_{1})b_{\rm B}(z_{2})\mathcal{P}_{0}(k)\,,$ (53)

where this is the cross-power spectrum of two sources types $\\{A,B\\}$, at
redshifts $\\{z_{1},z_{2}\\}$ respectively. Expanding in unequal-time (see
Appendix C) gives us a non-vanishing linear $\delta\chi$ contribution, unlike
the case for single tracer:

$\displaystyle b_{\rm A}(z_{1})b_{\rm
B}(z_{2})=\left(1+\frac{\Delta_{b}}{2}H\delta\chi\right)b_{\rm
A}(\bar{z})b_{\rm B}(\bar{z})+\dots\,,$ (54)

where we have defined:

$\Delta_{b}=\frac{d}{dz}\ln\left(\frac{b_{\rm A}}{b_{\rm B}}\right)\,.$ (55)

As an example computation of the deviation from the leading order results in
Equation (44), we can calculate:

$\displaystyle\int d(\delta\chi)\;$ $\displaystyle
e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}\left(1+\frac{1}{2}\Delta_{b}H\delta\chi\right)$
(56)
$\displaystyle=\left(1+\frac{i}{2}\Delta_{b}H\partial_{q_{\hat{n}}}\right)(2\pi)\delta^{\rm
D}(q_{\hat{n}}-k_{\hat{n}})\,,$

which gives us the observed 3D power spectrum:

$P_{\rm
lin}(k_{\hat{n}},k_{\perp},\bar{z})=\left[1+\frac{i}{2}\Delta_{b}H\left(\hat{n}\cdot\bm{\nabla}\right)\right]\mathcal{P}_{\rm
lin}\left(k,\bar{z}\right)\,.$ (57)

This thus gives rise to an imaginary component of the observable 3D power
spectrum. Even though arising from different origins, similar effects are
present when gravitational redshift effects are included in the Kaiser formula
[68]. The presence of odd multipoles was also found and discussed in
configuration space in [65]. However, the advantage of this formalism lies in
the fact that here multipoles can be calculated using only one Legendre
polynomial (as some of the geometrical dependencies are folded during the
conversions in Figure 1), with the introduced error being very small, while in
configuration space there are two angles over which we need to integrate.

We can write this first-order deviation from the equal-time case, in the
multi-tracer power spectrum as:

$\displaystyle\delta P_{\rm lin}(k,\mu,\bar{z})$ $\displaystyle=P_{\rm
lin}(k,\mu,\bar{z})-\mathcal{P}_{\rm lin}\left(k,\bar{z}\right)$ (58)
$\displaystyle=\frac{i}{2}\Delta_{b}\,b_{\rm A}b_{\rm
B}\frac{H}{k}\delta\mathcal{P}_{1}\left(k,\mu\right)\,,$

where:

$\delta\mathcal{P}_{1}\left(k,\mu\right)=\mu
k\mathcal{P}^{\prime}_{0}\left(k\right)\,.$ (59)

The angular dependence in this term arises from the single derivative along
the line of sight and thus gives rise only to a dipole contribution $\delta
P^{(1)}_{\rm lin}$. This is an important result, as in the standard
calculations, odd multipoles are zero; this could therefore result in a new
observable or even a new tool to measure galaxy bias and its evolution [69].

To continue our investigation, let us focus on a concrete case; we assume that
our two tracers have evolution similar to the dark matter but still deviating
slightly from each other, i.e., we assume a simple model $b=b_{0}D^{\eta}$. We
can then write:

$\Delta_{b}=\frac{b^{\prime}_{\rm A}}{b_{\rm A}}-\frac{b^{\prime}_{\rm
B}}{b_{\rm B}}\approx\frac{(\eta_{\rm B}-\eta_{\rm A})f}{(1+\bar{z})}.$ (60)

This result is also a special case of the expressions given in Appendix C. For
concreteness, here we will show results for some examples, and surveys-
motivated cases will be presented in a follow-up paper [69].

We start with a rather conservative case where we assume $\eta_{\rm
B}-\eta_{\rm A}=0.2$, which would correspond to a $\sim 10\%$ deviation of the
time-evolution of each of the tracers from the dark matter case. On the right
panel of Figure 5, we show the amplitude of the first-order imaginary
correction to the equal time power spectrum. We plot the ratio of $\delta
P_{\rm lin}$ of Equation (58), with $\Delta\eta=0.2$. The ratio is over the
amplitude of the (real) monopole and should not be intended as an estimate for
detection but as a comparison of amplitudes. We will see below that in the
case of redshift-space distortions, this generalizes to higher angular
contributions, thus giving rise to higher-order odd multipoles.

###  III.2 Redshift Space Distortions

Figure 6:  Linear theory results, as in Figure 5 in redshift-space for biased
tracers, for different multipoles and redshifts, for a particular choice of
bias parameters and their time dependence. Two different redshifts are shown:
$z=0.5$ (dashed lines) and $z=2$ (solid lines). Left Panel: Even multipoles
(real part). Right Panel: Odd multipoles (imaginary part). In the case of odd
multiples, lines are divided by the linear theory, as in Figure 5.

In this Section, we extend the linear cross-correlation model used in Equation
(53), by adding redshift-space distortions. In this work, as a first proof of
principle and in the spirit of leaving the theoretical part as simplified as
possible in order to have full control of the procedure, we consider just the
Kaiser factor [48, 70], where we can replace:

$b_{\rm X}(z_{i})\to D(z_{i})\left[b_{\rm X}(z_{i})+f(z_{i})\mu^{2}\right]\;.$
(61)

Considering the corrections up to second order in $\delta\chi$, we can expand
the product of two Kaiser factors in the unequal time power spectrum as:

$\displaystyle\left(b_{A}+fD\mu^{2}\right)_{z_{1}}\left(b_{B}+fD\mu^{2}\right)_{z_{2}}=\left(b_{A}+fD\mu^{2}\right)\left(b_{B}+fD\mu^{2}\right)$
$\displaystyle+\frac{1}{2}H\left(\Delta^{(0)}_{1}+\Delta^{(1)}_{1}fD\mu^{2}\right)\delta\chi$
(62)
$\displaystyle+\frac{1}{8}H^{2}\left(\Delta^{(0)}_{2}+\Delta^{(1)}_{2}fD\mu^{2}+\Delta^{(2)}_{2}f^{2}D^{2}\mu^{4}\right)\left(\delta\chi\right)^{2}+\ldots\,,$

where we followed the same procedure of Equation (46) for the additional
terms. This expression is derived in more detail in Appendix C. After
performing the integrals over $\delta\chi$ and $k_{\hat{n}}$ in Equation (44),
the deviation from the usual Kaiser observed redshift-space power spectrum can
be expressed as follows:

$\delta P_{\rm
lin}=\frac{i}{2}\frac{H}{k}\Delta_{1}\delta\mathcal{P}_{1}-\frac{1}{8}\left(\frac{H}{k}\right)^{2}\Delta_{2}\delta\mathcal{P}_{2}\,,$
(63)

where we have the two factors:

$\displaystyle\Delta_{1}$
$\displaystyle=\Delta^{(0)}_{1}+\Delta^{(1)}_{1}fD\mu^{2}\,,$ (64)
$\displaystyle\Delta_{2}$
$\displaystyle=\Delta^{(0)}_{2}+\Delta^{(1)}_{2}fD\mu^{2}+\Delta^{(2)}_{2}f^{2}D^{2}\mu^{4}\,,$

and the $\delta\mathcal{P}_{1}$ and $\delta\mathcal{P}_{2}$ are the $k$
dependent contributions given in Equations (59), (51) respectively. The time
dependent functions $\Delta^{(i)}_{1}$ and $\Delta^{(i)}_{2}$ are derived in
Appendix (C).

Collecting together the different contributions, we can therefore express the
deviations from the standard equal time power spectrum multipoles as:

$\delta P^{(\ell)}_{\rm lin}=i\frac{H}{k}{\rm
Im}\left[\tau^{(1)}_{\ell}\right](k\mathcal{P}^{\prime}_{0})-\left(\frac{H}{k}\right)^{2}{\rm
Re}\left[\tau^{(2)}_{\ell}k^{2}\mathcal{P}^{\prime\prime}_{0}+\tau^{(1)}_{\ell}k\mathcal{P}^{\prime}_{0}\right],$
(65)

where $\tau_{\ell}$ are the multipole coefficients, and their expressions are
presented in Table 2. The real part gives rise to even, and the imaginary part
to odd multipoles.

In Figure 6, we show results for the case presented above; once again, we
leave a detailed estimation of the magnitude of these corrections and their
detectability (and/or need to be included in the galaxy clustering modeling)
for a dedicated separated study. These results show that unequal-time
corrections are generally small; however, there are scenarios in which they
could become relevant. At large scales and high-z, they can contribute up to
several percent to the total observed power spectrum, making it relevant for
planned future surveys that aim to target exactly large cosmological volumes
at very high redshifts.

Moreover, such corrections depend on the different ways galaxies trace the
underlying matter distribution, introducing anisotropies even without
considering redshift-space distortions, and they generate odd multiples.
Interestingly, unequal-time corrections depend on the derivative of the power
spectrum, which enables the study of its slope and runnings in a novel way.
Finally, we note that the presented results rely on a specific choice of bias
values and their evolution, and we used the Kaiser formula for redshift-space
distortions. We intend to extend this analysis to include relativistic effects
in a hypothetical multi-tracer future analysis in a follow-up paper.

Real part of $\tau^{(m)}_{\ell}$ |
---|---
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{0}=\frac{1}{12}\Delta_{2}^{(0)}+\frac{1}{60}fD\Delta_{2}^{(1)}+\frac{1}{140}(fD)^{2}\Delta_{2}^{(2)}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(2)}_{0}=\frac{1}{24}\Delta_{2}^{(0)}+\frac{1}{40}fD\Delta_{2}^{(1)}+\frac{1}{56}(fD)^{2}\Delta_{2}^{(2)}$
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{2}=-\frac{1}{12}\Delta_{2}^{(0)}+\frac{1}{84}fD\Delta_{2}^{(1)}+\frac{1}{84}(fD)^{2}\Delta_{2}^{(2)}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(2)}_{2}=\frac{1}{12}\Delta_{2}^{(0)}+\frac{1}{14}fD\Delta_{2}^{(1)}+\frac{5}{84}(fD)^{2}\Delta_{2}^{(2)}$
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{4}=-\frac{1}{35}fD\Delta_{2}^{(1)}-\frac{4}{385}(fD)^{2}\Delta_{2}^{(2)}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(2)}_{4}=\frac{1}{35}fD\Delta_{2}^{(1)}+\frac{3}{77}(fD)^{2}\Delta_{2}^{(2)}$
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{6}=-\frac{2}{231}(fD)^{2}\Delta_{2}^{(2)}$ | $\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(2)}_{6}=\frac{2}{231}(fD)^{2}\Delta_{2}^{(2)}$
Imaginary part of $\tau^{(m)}_{\ell}$ |
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{1}=\frac{1}{2}\Delta_{1}^{(0)}+\frac{3}{10}fD\Delta_{1}^{(1)}$ |
$\leavevmode\nobreak\ \leavevmode\nobreak\ \tau^{(1)}_{3}=\frac{1}{5}fD\Delta_{1}^{(1)}$ |
Table 2: Real and imaginary parts of the multipole coefficients
$\tau^{(m)}_{\ell}$ used in the linear redshift-space distortions given in
Equation (65). Explicit forms of the $\Delta^{(i)}_{1}$ and $\Delta^{(i)}_{2}$
functions are given in Appendix (C).

###  III.3 IR resummation of Power Spectrum

The unequal-time theoretical power spectrum in Lagrangian perturbation theory
(LPT) can be schematically written as [56, 57]:

$\mathcal{P}(k,z,z^{\prime})=\int d^{3}q\
e^{i\bm{k}\cdot\bm{q}-\frac{1}{2}k_{i}k_{j}A_{ij}(\bm{q},z,z^{\prime})}\xi^{(0)}(q,\bar{z})+\ldots$
(66)

where the two-point displacement cumulant is given by

$A_{ij}(\bm{q},z,z^{\prime})=\left\langle\Delta_{i}\Delta_{j}\right\rangle_{c},$
(67)

and $\Delta_{i}=\psi_{i}(\bm{q}_{2},z^{\prime})-\psi_{i}(\bm{q}_{1},z)$. We
have also written the leading order two-point correlator $\xi^{(0)}(q)$, which
can take the form of the linear correlation function in the case of IR
resummed linear theory or simply unity in the case of the Zeldovich power
spectrum. The dots in Equation (66) give us the higher-order perturbative
corrections as well as the higher unequal-time corrections. For a more
detailed discussion and derivation of the unequal-time LPT results, we refer
the reader to Appendix D. Here we just note that the unequal-time corrections
given as the dotted expansion in $\delta\chi$ lead to $(H/k)$ type of
corrections in the observed power spectrum as given in Subsections III.1 and
III.2, and we thus rather focus on the unequal-time effects of the two-point
displacement cumulant $A_{ij}$. In Appendix D we show that we can expand the
second displacement cumulant around the mean redshift as

$A_{ij}(\bm{q},z,z^{\prime})\approx A_{ij}(\bm{q},\bar{z})+\delta
A_{ij}(\bm{q},\bar{z})(H\delta\chi)^{2}\,.$ (68)

Consequently, the observed power spectrum, given by the expression in Equation
(44), is

$\displaystyle P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})$
$\displaystyle=\int\frac{dk_{\hat{n}}}{2\pi}d^{3}q\;e^{i\bm{k}\cdot\bm{q}}e^{-\frac{1}{2}k_{i}k_{j}A_{ij}(\bm{q},z)}$
(69) $\displaystyle\times\int
d(\delta\chi)e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}e^{-\frac{1}{2}k_{i}k_{j}\delta
A_{ij}(\bm{q},z)(H\delta\chi)^{2}},$

where multiplicative terms like $\xi^{(0)}$ can be easily added into
consideration, as they do not affect the $\delta\chi$ integral; similarly, the
higher $\delta\chi$ terms arising from the unequal-time part of the
$\xi^{(0)}$-like operators can be added, as shown earlier in this section.

Using the quadratic expansion of the displacement cumulant $A_{ij}$ given in
Equation (68), the $\delta\chi$ integral can be done analytically. We obtain
the Gaussian integral form

$\displaystyle\int d(\delta\chi)\;$ $\displaystyle
e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}e^{-\frac{1}{2}k_{i}k_{j}\delta
A_{ij}(\bm{q},z)(H\delta\chi)^{2}}$ (70)
$\displaystyle=\frac{\sqrt{2\pi}}{H\sqrt{|k_{i}k_{j}\delta
A_{ij}|}}\exp\left(-\frac{1}{H^{2}}\frac{(q_{\hat{n}}-k_{\hat{n}})^{2}}{2k_{i}k_{j}\delta
A_{ij}}\right)\;.$

If we assume that $\delta A_{ij}(\bm{q},z)$ is approximately scale-
independent, as argued in Appendix D, we can write $\delta
A_{ij}(\bm{q},z)\simeq 2\delta\sigma(\bar{z})^{2}\delta^{K}_{ij}$, where
$\delta\sigma^{2}$ can be interpreted as the average long displacement
dispersion due to the unequal-time effects. Estimated lower and upper bounds
of $\delta\sigma^{2}$ values are in Equation (107). Using this approximation,
we have

$\displaystyle P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})$
$\displaystyle\simeq\int\frac{dk_{\hat{n}}}{2\pi}d(\delta\chi)\;e^{-i\delta\chi(q_{\hat{n}}-k_{\hat{n}})}$
(71) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \times
e^{-k^{2}\delta\sigma^{2}(H\delta\chi)^{2}}\int
d^{3}q\;e^{i\bm{k}\cdot\bm{q}}e^{-\frac{1}{2}k_{i}k_{j}A_{ij}(\bm{q},z)}$
$\displaystyle=\frac{1}{H\delta\sigma}\int\frac{dk_{\hat{n}}}{2\sqrt{\pi}k}\;e^{-\frac{(q_{\hat{n}}-k_{\hat{n}})^{2}}{4k^{2}(H\delta\sigma)^{2}}}\mathcal{P}\left(k_{\hat{n}},\ell/\bar{\chi},\bar{\chi}\right)\,.$

This result tells us that the unequal time effects of long displacement modes
on the observed power spectrum is to smear the theoretical 3D power spectrum
on scales corresponding to $\sim kH\delta\sigma$. For parameters of the
$\Lambda$CDM cosmology that we are using here, $2H\delta\sigma$ peaks at
$z\sim 0.55$, achieving values of $2H\delta\sigma\sim 0.001$. This provides us
with the smoothing kernel of width smaller than any feature in the 3D power
spectrum of the $\Lambda$CDM universe, i.e. we can treat $\mathcal{P}$
effectively as a constant over the integration region where the exponential
function has support. Moreover, in the integrand, we can approximate
$k=\sqrt{k_{\hat{n}}^{2}+k^{2}_{\perp}}\approx\sqrt{q_{\hat{n}}^{2}+k^{2}_{\perp}}$,
which makes the integral of a simple Gaussian form. Consequently, these
simplifications give us

$P(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi})\approx\mathcal{P}\left(q_{\hat{n}},\ell/\bar{\chi},\bar{\chi}\right)\,,$
(72)

i.e. we can neglect the unequal time effects due to the IR resummation. This
picture changes if we want to discuss the deviation from the $\Lambda$CDM
model, where the power spectrum would exhibit some additional features on
scales $k_{*}\lesssim kH\delta\sigma$. This has an immediate consequence for
cosmological models predicting the linear power spectrum with “features” -
either imprinted during inflation or induced by non-standard expansion
histories (see, e.g. [71, 72, 73] for recent reviews). Current results suggest
that future surveys will be able to detect or tightly constrain features in
the primordial spectrum below the one percent level across a wide range of
scales [74, 75]. This is a far larger effect than the limit imposed due to the
long displacement smearing we are considering here.

##  IV Conclusion

In this paper, we develop a framework for observables of galaxy clustering; in
particular, we investigate the role of unequal-time effects in the observed
power spectrum $P$. Namely, when constructing the observed power spectrum, we
use different redshift slice information to construct the modes along the line
of sight. However, the 3D theoretical power spectrum $\mathcal{P}$ of
different redshift slices is inevitably described as an unequal-time power
spectrum. This implies that these unequal-time effects and the modes along the
line of sight are folded on top of each other in the observed power spectrum.
We thus first delineate the connection between the observed equal-time power
spectrum and the theoretical 3D unequal-time power spectrum. This connection
is accomplished by relying on the flat-sky approximation of the unequal-time
angular power spectrum $\mathbb{C}_{\ell}$.

In this construction process, we show that one is free to consider also
alternative 3D statistics to the canonically defined observed power spectrum
$P$. We thus construct an observable frequency-angular power spectrum
$\widetilde{\mathbb{C}}$ and show how this newly introduced statistic
naturally includes radial mode contributions and how we can eliminate the need
for a priori distance measure assumptions, usually needed in the wave mode
construction (so-called Alcock-Paczynśki effects.) This enables one to make
measurements independently of the choice of a cosmological model, by
introducing a dimensionless quantity depending only on observable variables
(Fourier counterparts to the angles and redshifts). We also investigate the
properties of this new statistic and verify that, in most current practical
applications, it retains all beneficial properties of the canonical observed
power spectrum $P$ and that the residual contributions to the modes along the
line of sight generated by the redshift dependence can be safely disregarded.
Another powerful aspect of this frequency-angular power spectrum
$\widetilde{\mathbb{C}}$ is that the BAO features shift with cosmological
distance, making it possible to infer distances directly by using the position
of BAO peaks, which is a robust and well-understood measurement.

In the latter part of the paper, we focus on a formulation of the Fourier-
space $P(k)$ that includes corrections due to unequal-time when correlating
sources (or bins of them) at different redshifts. Starting from the observable
angular spectrum, we show how to calculate contributions along the line of
sight and quantify them for some example cases. Starting from the equal-time
standard case, we find an expression for a series expansion to include
unequal-time terms and calculate their amplitude and scale dependencies. Such
corrections generally appear at second order in the radial separation between
sources, $\delta\chi$. Still, there will be a contribution from the first
order when cross-correlating sources with a different bias. These first-order
terms give rise to an imaginary part of the power spectrum, which translates
into odd multipoles when performing the classic Legendre polynomials
expansion. Moreover, unequal-time corrections generate higher-order
multipoles, including the odd ones, even in the RSD case (where typically only
even multipoles appear). This might represent a new cosmological observable
with a yet unexplored potential. We note that such contributions, originating
from observable projection effects, are expected to appear also in the higher
n-point functions, with the consequence of giving rise to contributions that
might be expected to be zero from purely theoretical considerations.

We find that unequal-time corrections give rise to terms typically scaling
with $\mathcal{H}/k$. These contributions are generally small, but they
present some interesting features. First of all, multi-tracer analyses depend
on the difference between the tracer biases but also on their time
derivatives, introducing the exciting possibility of studying the bias
evolution in a new way. In redshift space, this dependence extends to
derivatives of the growth rate, again opening up a new possible avenue for
studying cosmological models.

As a last part, we consider unequal-time effects arising due to the long
displacement field via the IR resummation mechanism. We model these
contributions at the linear level of Lagrangian perturbation theory re-summing
the linear displacements. We show that unequal-time contributions result in
effective smoothing of the original equal-time power spectrum on scales
$k_{*}\sim kH\delta\sigma$ (with $\delta\sigma$ of order few Mpc$/h$). The
cumulative effect is thus far smaller than what can potentially be probed by
current and upcoming experiments.

In summary, we investigated the effects of the unequal-time contributions in
the observed power spectrum and some representative case studies based on
examples of the source biases and their redshift evolution. We defined a new
observable in angular-frequency space that naturally includes transverse and
radial modes and promises to become a more convenient way to analyze galaxy
surveys than the canonical observed power spectrum. In Fourier space, our
calculation of unequal time effects unveiled a deeper understanding of the
behaviour of galaxy clustering along the line of sight, which opens up the
possibility of adding a new tool for cosmological studies with galaxy
clustering measurements.

###### Acknowledgements.

AR acknowledges funding from the Italian Ministry of University and Research
(MIUR) through the “Dipartimenti di eccellenza” project “Science of the
Universe”. Z.V. is partially supported by the Kavli Foundation.

##  Appendix A Angular power spectrum and the choice of the mean distance

Keeping in mind the different options for the choice of the $\bar{\chi}$,
starting from Equation (4) the angular power spectrum can thus be written as

$\displaystyle\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}(\bm{\ell}^{\prime})\right\rangle=(2\pi)^{2}\int\frac{d\chi}{\chi^{2}}\frac{d\chi^{\prime}}{\chi^{\prime
2}}\,W\left(\chi\right)W^{\prime}\left(\chi^{\prime}\right)\delta^{D}\left(\bm{\tilde{\ell}}+\bm{\tilde{\ell}^{\prime}}\right)\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
e^{i\delta\chi
k_{\hat{n}}}\mathcal{P}\big{(}k_{\hat{n}}\bm{\hat{n}},\bm{k}_{\perp},\chi,\chi^{\prime}\big{)}\,.$
(73)

Using the delta function representation in the new variables, we can write

$\delta^{\rm
2D}\left(\bm{\tilde{\ell}}+\bm{\tilde{\ell}^{\prime}}\right)=\delta^{\rm
2D}\left(\frac{\chi^{\prime}\bm{\ell}+\chi\bm{\ell^{\prime}}}{\chi\chi^{\prime}}\right)=\bar{\chi}^{2}\mathcal{A}(\delta)\,\delta^{\rm
2D}\left(\bm{\ell}+\bm{\ell^{\prime}}+\varphi(\delta)\bm{\Delta}\right)\,,$
(74)

where $\bm{\Delta}=\bm{\ell}^{\prime}-\bm{\ell}$, $\bar{\chi}$ in the mean
distance, $\delta=\frac{1}{2}\delta\chi/\bar{\chi}$, and $\varphi(\delta)$ is
an off-diagonal phase of the Dirac delta function. Specifically, for
arithmetic, geometric and harmonic coordinates, respectively, this gives us

$\displaystyle\delta^{\rm
2D}\left(\bm{\tilde{\ell}}+\bm{\tilde{\ell}}^{\prime}\right)$
$\displaystyle=\delta^{\rm
2D}\left(\frac{\bm{\ell}+\bm{\ell}^{\prime}-\delta\bm{\Delta}}{\chi_{\rm
a}(1-\delta^{2})}\right)=\chi^{2}_{\rm a}(1-\delta^{2})^{2}\delta^{\rm
2D}\left(\bm{\ell}+\bm{\ell}^{\prime}-\bm{\Delta}\delta\right)\,,$ (75)
$\displaystyle\delta^{\rm
2D}\left(\bm{\tilde{\ell}}+\bm{\tilde{\ell}}^{\prime}\right)$
$\displaystyle=\delta^{\rm
2D}\left(\frac{(\bm{\ell}+\bm{\ell}^{\prime})\sqrt{1+\delta^{2}}-\delta\bm{\Delta}}{\chi_{\rm
g}}\right)=\frac{\chi^{2}_{\rm g}}{(1+\delta^{2})}\delta^{\rm
2D}\left(\bm{\ell}+\bm{\ell}^{\prime}-\bm{\Delta}\delta/\sqrt{1+\delta^{2}}\right)\,,$
$\displaystyle\delta^{\rm
2D}\left(\bm{\tilde{\ell}}+\bm{\tilde{\ell}}^{\prime}\right)$
$\displaystyle=\delta^{\rm
2D}\left(\frac{(\bm{\ell}+\bm{\ell}^{\prime})\left(1+\sqrt{1+4\delta^{2}}\right)-2\delta\bm{\Delta}}{\chi_{\rm
h}\left(1+\sqrt{1+4\delta^{2}}\right)}\right)=\chi^{2}_{\rm h}\delta^{\rm
2D}\left(\bm{\ell}+\bm{\ell}^{\prime}-2\bm{\Delta}\delta/\big{(}1+\sqrt{1+4\delta^{2}}\big{)}\right)\,,$

where we can identify the factor $\mathcal{A}$ and phase $\varphi$ in each
case. We have

$\displaystyle\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}(\bm{\ell}^{\prime})\right\rangle=(2\pi)^{2}\int
d\chi
d\chi^{\prime}\;W\left(\chi\right)W^{\prime}\left(\chi^{\prime}\right)\frac{\bar{\chi}^{2}}{\chi\chi^{\prime}}\mathcal{A}(\delta)\delta^{\rm
2D}\big{(}\bm{\ell}+\bm{\ell}^{\prime}+\varphi(\delta)\bm{\Delta}\big{)}\mathbb{C}_{\ell}\left(\chi,\chi^{\prime}\right)\,,$
(76)

where we have

$\mathbb{C}_{\ell}\left(\chi,\chi^{\prime}\right)=\frac{1}{\chi\chi^{\prime}}\int\frac{dk_{\hat{n}}}{2\pi}\leavevmode\nobreak\
e^{i\delta\chi
k_{\hat{n}}}\mathcal{P}\big{(}k_{\hat{n}}\bm{\hat{n}},\bm{k}_{\perp},\chi,\chi^{\prime}\big{)}\,.$
(77)

Since we can write

$\delta^{\rm
2D}\big{(}\bm{\ell}+\bm{\ell}^{\prime}+\varphi(\delta)\bm{\Delta}\big{)}=\delta^{\rm
2D}\big{(}\bm{\ell}+\bm{\ell}^{\prime}\big{)}+\Big{(}e^{\varphi(\delta)\bm{\Delta}\cdot\overset{\rightarrow}{\partial}_{\bm{\ell}}}-1\Big{)}\delta^{\rm
2D}\big{(}\bm{\ell}+\bm{\ell}^{\prime}\big{)}\,,$ (78)

we have

$\left\langle\hat{\delta}(\bm{\ell})\hat{\delta}(\bm{\ell}^{\prime})\right\rangle=(2\pi)^{2}\delta^{\rm
2D}\big{(}\bm{\ell}+\bm{\ell}^{\prime}\big{)}\sum_{n=0}^{\infty}\frac{(\overset{\leftarrow}{\partial}_{\bm{\ell}^{\prime}}\cdot\bm{\Delta})^{n}}{n!}C^{(n)}(\ell)\,,$
(79)

and where we have introduced

$C^{(n)}(\ell)=\int d\chi
d\chi^{\prime}\;W\left(\chi\right)W^{\prime}\left(\chi^{\prime}\right)\frac{\bar{\chi}^{2}}{\chi\chi^{\prime}}\mathcal{A}(\delta)\varphi(\delta)^{n}\mathbb{C}(\ell,\chi,\chi^{\prime})\,.$
(80)

##  Appendix B From redshift to comoving distance

Let us also expand $\delta\chi(\bar{z},\delta z)=\chi(\bar{z}+1/2\delta
z)-\chi(\bar{z}-1/2\delta z)$ as a function of $\delta z$, we have:

$\delta\chi=\frac{d\chi(\bar{z})}{d\bar{z}}\delta
z+\frac{1}{3}\frac{d^{3}\chi(\bar{z})}{d\bar{z}^{3}}(\delta
z/2)^{3}+\ldots=\frac{1}{H(\bar{z})}\delta
z+\frac{1}{24}\left(\frac{d^{2}}{d\bar{z}^{2}}\frac{1}{H(\bar{z})}\right)\delta
z^{3}+\ldots\,.$ (81)

Since we can write:

$\frac{d^{2}}{dz^{2}}\frac{1}{H(z)}=-3\frac{\Omega_{m}\left(1-\tfrac{9}{4}\Omega_{m}\right)}{(1+z)^{2}H(z)}\,,$
(82)

this gives us:

$\delta\chi=\frac{1}{H(\bar{z})}\delta
z-\frac{1}{8}\frac{\Omega_{m}\left(1-\tfrac{9}{4}\Omega_{m}\right)}{(1+\bar{z})^{2}H(\bar{z})}\delta
z^{3}+\ldots\simeq\left[1-c_{3}(\delta z)^{2}\right]\frac{\delta
z}{H(\bar{z})}\,,$ (83)

and thus:

$\Omega(\omega,k_{\hat{n}})\equiv\int d\delta z\;e^{i\delta\chi
k_{\hat{n}}-i\omega\delta
z}=e^{-c_{3}\omega\frac{d^{3}}{d\omega^{3}}}(2\pi)\delta^{\rm
D}\left(\frac{k_{\hat{n}}}{H}-\omega\right)\simeq\left(1-c_{3}\omega\frac{d^{3}}{d\omega^{3}}\right)(2\pi)\delta^{\rm
D}\left(\frac{k_{\hat{n}}}{H}-\omega\right)\,.$ (84)

##  Appendix C Unequal time contributions to the Kaiser terms

Here we derive the expansion up to the second order in the unequal time
variable around a mean. We treat the deviation $\delta\chi$ around the mean
comoving distance as the small contribution. We subsequently check the
validity of this expansion on several examples. A redshift-dependent physical
quantity $F$ we can then simply expand as:

$F(z[\chi_{i}(\chi,\delta\chi)])=D(z[\chi_{i}(\chi,0)])+\frac{d}{d\delta\chi}F(z[\chi_{i}(\chi,\delta\chi)])\Big{|}_{\delta\chi=0}\delta\chi+\frac{1}{2}\frac{d^{2}}{d^{2}\delta\chi}F(z[\chi_{i}(\chi,\delta\chi)])\Big{|}_{\delta\chi=0}(\delta\chi)^{2}+\dots\,,$
(85)

where $i\in\\{1,2\\}$ labels the two positions we are concerned with when
considering two-point correlations. For the first derivative, we have:

$\frac{d}{d\delta\chi}F(z[\chi_{i}(\chi,\delta\chi)])=\frac{d\chi_{i}}{d\delta\chi}\frac{dz}{d\chi_{i}}\frac{d}{dz}F(z[\chi_{i}(\chi,\delta\chi)])=HF^{\prime}\frac{d\chi_{i}}{d\delta\chi}\,,$
(86)

where we use the label $F^{\prime}\equiv dF/dz$, and also have
$dz/d\chi_{i}=(d\chi_{i}/dz)^{-1}=H$. For the second derivative, we have:

$\frac{d^{2}}{d^{2}\delta\chi}F(z[\chi_{i}(\chi,\delta\chi)])=\frac{d}{d\delta\chi}\left(HF^{\prime}\frac{d\chi_{i}}{d\delta\chi}\right)=H\left(HF^{\prime}\right)^{\prime}\left(\frac{d\chi_{i}}{d\delta\chi}\right)^{2}+HF^{\prime}\frac{d^{2}\chi_{i}}{d\delta\chi^{2}}\,.$
(87)

Using the arithmetic coordinate setup, i.e. the coordinates defined relative
to the arithmetic mean, we have $d\chi_{1}/d\delta\chi=1/2$,
$d\chi_{2}/d\delta\chi=-1/2$, and thus:

$\frac{d}{d\delta\chi}F(z[\chi_{1/2}(\chi,\delta\chi)])=\pm\frac{1}{2}HF^{\prime}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm and}\leavevmode\nobreak\ \leavevmode\nobreak\
\frac{d^{2}}{d^{2}\delta\chi}F(z[\chi_{i}(\chi,\delta\chi)])=\frac{1}{4}H\left(HF^{\prime}\right)^{\prime}=\frac{1}{4}H^{2}\Big{(}(\ln
H)^{\prime}F^{\prime}+F^{\prime\prime}\Big{)}\,.$ (88)

In the case of redshift-space distortions, we need to evaluate the factor:

$\displaystyle\left(b_{A}+fD\mu^{2}\right)_{z_{1}}\left(b_{B}+fD\mu^{2}\right)_{z_{2}}$
$\displaystyle=\left(b_{A}+fD\mu^{2}\right)\left(b_{B}+fD\mu^{2}\right)+\frac{1}{2}H\left(\Delta^{(0)}_{1}+\Delta^{(1)}_{1}fD\mu^{2}\right)\delta\chi$
(89) $\displaystyle\hskip
130.88284pt+\frac{1}{8}H^{2}\left(\Delta^{(0)}_{2}+\Delta^{(1)}_{2}fD\mu^{2}+\Delta^{(2)}_{2}f^{2}D^{2}\mu^{4}\right)\left(\delta\chi\right)^{2}+\ldots\,,$

where it is convenient to introduce the factors that depend on the mean
redshift:

$\displaystyle\Delta^{(0)}_{1}$
$\displaystyle=b^{\prime}_{A}b_{B}-b^{\prime}_{B}b_{A}\,,\leavevmode\nobreak\
\leavevmode\nobreak\
\Delta^{(1)}_{1}=b^{\prime}_{A}-b^{\prime}_{B}+\gamma_{1}(b_{B}-b_{A})\,,$
(90) $\displaystyle\Delta^{(0)}_{2}$
$\displaystyle=b_{A}\left(\gamma_{0}b_{B}^{\prime}+b_{B}^{\prime\prime}\right)+\left(\gamma_{0}b_{A}^{\prime}+b_{A}^{\prime\prime}\right)b_{B}-2b^{\prime}_{A}b^{\prime}_{B}\,,\leavevmode\nobreak\
\Delta^{(1)}_{2}=\gamma_{2}b_{A}+(\gamma_{0}-2\gamma_{1})b_{A}^{\prime}+b_{A}^{\prime\prime}+A\leftrightarrow
B\,,\leavevmode\nobreak\
\Delta^{(2)}_{2}=2\left(\gamma_{2}-\gamma^{2}_{1}\right)\,.$

Above, we introduced factors $\gamma_{0}$, $\gamma_{1}$ and $\gamma_{2}$ that
are functions of mean redshift. They are introduced by taking the redshift
derivatives of the Hubble parameter, linear growth $D$ and its logarithmic
growth rate $f$. Starting from the Hubble parameter, we introduced
$\gamma_{0}\equiv(\ln H)^{\prime}=\frac{3}{2}\frac{\Omega_{m}}{(1+z)}$, while
the redshift derivative of linear growth is simply $D^{\prime}=-fD/(1+z)$.
Equation of motion for the growth rate
$adf/da=-f(2+f)+(1+f)\frac{3}{2}\Omega_{m}$, gives us:

$(\ln
f)^{\prime}=\frac{1}{1+z}\left(2+f-(1+f)\frac{3}{2}\frac{\Omega_{m}}{f}\right)=\gamma_{1}+f/(1+z),$
(91)

where we introduced the factor
$\gamma_{1}\equiv\left(2-(1+f)\frac{3}{2}\frac{\Omega_{m}}{f}\right)/(1+z)$
and thus $(fD)^{\prime}=\gamma_{1}fD$. We also have:

$\displaystyle\gamma_{2}\equiv\frac{1}{fHD}\left(H(fD)^{\prime}\right)^{\prime}=\gamma^{\prime}_{1}+\gamma_{0}\gamma_{1}+\gamma_{1}^{2}\,,$
(92)

where, using $(\Omega_{m})^{\prime}=3\Omega_{m}\Omega_{\Lambda}/(1+z)$, we can
compute the redshift derivative of $\gamma_{1}$, to obtain:

$\gamma^{\prime}_{1}=\left(\Big{(}1+(1+z)\gamma_{1}/f-3(1+f)\Omega_{\Lambda}/f\Big{)}\gamma_{0}-\gamma_{1}\right)\frac{1}{1+z}\,.$
(93)

For general tracers, their biases and their time evolution can differ from
tracer to tracer and which consequently gives rise to linear corrections in
$\delta\chi$ in Equation (89). We see that $\Delta^{(0)}_{1}$ contributions
can be generated either by different bias values or different bias change
rates, while in the case of the redshift-space related $\Delta^{(1)}_{1}$
term, we have the bias values, and bias change rates contribute additively. On
the other hand, second-order contributions are also present in the auto-
correlations of any tracer.

In the case of dark matter, i.e. $b_{A}=b_{B}=D$, these expressions can be
significantly simplified, as we shall see below. However, in the case of dark
matter, it is useful to identify the specific factors of the powers of
$\mu^{2}$ in the Kaiser formula, as it can serve as the crosscheck of the
validity of our expansion, i.e. if the expansion up to the second order in the
$\delta\chi$ suffices. In this case, as in the case of all autocorrelations,
the first-order contributions vanish ($\Delta^{(0)}_{1}=\Delta^{(1)}_{1}=0$),
while in the case of the second-order contributions, we can identify:

$\displaystyle\frac{D_{1}D_{2}}{D^{2}}-1\approx\frac{\Delta^{(0)}_{2}}{D^{2}}\frac{1}{8}\left(H\delta\chi\right)^{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{\left(f_{1}+f_{2}\right)}{2f}\frac{D_{1}D_{2}}{D^{2}}-1\approx\frac{\Delta^{(1)}_{2}}{2D}\frac{1}{8}\left(H\delta\chi\right)^{2}\,,\leavevmode\nobreak\
\leavevmode\nobreak\
\frac{f_{1}f_{2}}{f^{2}}\frac{D_{1}D_{2}}{D^{2}}-1\approx\Delta^{(2)}_{2}\frac{1}{8}\left(H\delta\chi\right)^{2}\,.$
(94)

To proceed a bit further, let us assume a simple power-law model for the bias
time dependence $b=b_{0}D^{\eta}$. This gives $b^{\prime}=\eta b(\ln
D)^{\prime}$ and $b^{\prime\prime}=\eta b\left((\ln
D)^{\prime\prime}+\eta((\ln D)^{\prime})^{2}\right)$, and thus besides the
first derivative we have stated above, we also need the second derivative
$D^{\prime\prime}=\left(-fD/(1+z)\right)^{\prime}=\big{(}1-(1+z)\gamma_{1}\big{)}\frac{fD}{(1+z)^{2}}$.
Combining and using these in Equation (90), we obtain for the linear terms in
$\delta\chi$:

$\Delta^{(0)}_{1}=\frac{(\eta_{B}-\eta_{A})}{1+z}fb_{A}b_{B}\,,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\Delta^{(1)}_{1}=(\eta_{B}b_{B}-\eta_{A}b_{A})\frac{f}{1+z}+\gamma_{1}(b_{B}-b_{A})\,,$
(95)

both of which vanish in case $b_{A}=b_{B}=D$, as we have stated above. For the
second order $\delta\chi$ contributions we have:

$\displaystyle\Delta^{(0)}_{2}$
$\displaystyle=b_{A}b_{B}\left(-(\eta_{A}+\eta_{B})\left(1+f-\frac{3}{2}\frac{\Omega_{m}}{f}\right)+(\eta_{A}-\eta_{B})^{2}f\right)\frac{f}{(1+z)^{2}}\,,$
(96) $\displaystyle\Delta^{(1)}_{2}$
$\displaystyle=b_{A}\left(\gamma_{2}+(\gamma_{1}-\gamma_{0})\frac{\eta_{A}f}{1+z}+\left(1+(\eta_{A}-1)f\right)\frac{\eta_{A}f}{(1+z)^{2}}\right)+A\leftrightarrow
B\,,$ $\displaystyle\Delta^{(2)}_{2}$
$\displaystyle=2\left(\gamma_{2}-\gamma^{2}_{1}\right)\,.$

In the case of dark matter, when $b_{A}=b_{B}=D$, we have:

$\displaystyle\Delta^{(0)}_{2}$
$\displaystyle=-2D^{2}\left(1+f-\frac{3}{2}\frac{\Omega_{m}}{f}\right)\frac{f}{(1+z)^{2}}\,,$
(97) $\displaystyle\Delta^{(1)}_{2}$
$\displaystyle=2D\left(\gamma_{2}+(\gamma_{1}-\gamma_{0})\frac{f}{1+z}+\frac{f}{(1+z)^{2}}\right)\,,$
$\displaystyle\Delta^{(2)}_{2}$
$\displaystyle=2\left(\gamma^{\prime}_{1}+\gamma_{0}\gamma_{1}\right)\,.$

These expressions can be used to check the ones given in Equations (94). In
Figure 7, we compare these relations. The points are obtained from direct
calculations given on the right-hand side of Equations (94), while solid lines
represent the quadratic approximations whose coefficients are given by
Equations (97). We see that the agreement between the two is excellent in all
three cases, and thus we can conclude that the expansion up to the quadratic
order suffices for estimating the unequal time effects in any current galaxy
survey.

Figure 7:  Comparison of the unequal-time Kaiser terms. Three panels show the
three unequal-time contributions in the linear Kaiser power spectrum
proportional to $\mu^{0}$, $\mu^{2}$ and $\mu^{4}$. These come as different
combinations of the linear growth factor $D(z)$ and growth rate $f(z)$ shown
as two-point unequal-time factors that appear in the power spectrum, given as
a function of comoving distance $\delta\chi$. Points represent the direct
calculations, while solid lines give the values of the coefficients up to the
$\delta\chi^{2}$ order. We see that in all cases, the agreement with the
quadratic approximation is excellent.

##  Appendix D Unequal-time power spectrum in Lagrangian perturbation theory

In the Lagrangian formalism for describing gravitational clustering, the
theoretical power spectrum of a general biased tracer field can be expressed
as

$\mathcal{P}(\bm{k},z,z^{\prime})=\int d^{3}q\leavevmode\nobreak\
e^{-i\bm{k}\cdot\bm{q}}\left(\sum_{\rm a,b}c_{\rm a}c_{\rm b}\left\langle
O_{\rm a}(\bm{q}_{1},z)O_{\rm
b}(\bm{q}_{2},z^{\prime})e^{i\bm{k}\cdot(\psi(\bm{q}_{2},z^{\prime})-\psi(\bm{q}_{1},z))}\right\rangle-1\right)\,,$
(98)

where $\psi(\bm{q})$ is the displacement field, relating the Lagrangian
particle position $\bm{q}$ to the Eulerian position
$\bm{x}=\bm{q}+\psi(\bm{q})$, and $O_{\rm a}(\bm{q},\tau)$ and $c_{\rm a}$ are
the set of operators and corresponding biased coefficients describing specific
biased tracers. Given our interest in the unequal-time effects in the
observable two-point statistics, we are interested in estimating the
corrections around some mean redshift $\bar{z}$. The product of the two bias
operators $OO^{\prime}$ is not of particular interest, given that it also
leads to the $(\mathcal{H}/k)$ type of correction we have investigated in
Appendix C, and the results of which we have estimated in earlier parts of
this section. Here we shall thus focus on the effects of the long displacement
components $\psi(\bm{q})$.

Without going into the details (see [56, 57, 76, 77] for some recent work on
Lagrangian perturbation theory), we can represent the theoretical two-point
function

$\displaystyle\mathcal{P}(k,z,z^{\prime})=\int d^{3}q\
e^{i\bm{k}\cdot\bm{q}}\xi^{(0)}(q,\bar{z})e^{-\frac{1}{2}k_{i}k_{j}A_{ij}(\bm{q},z,z^{\prime})}+\ldots\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ {\rm where}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
A_{ij}(\bm{q},z,z^{\prime})=\left\langle\Delta_{i}\Delta_{j}\right\rangle_{c},$
(99)

$\Delta_{i}=\psi_{i}(\bm{q}_{2},z^{\prime})-\psi_{i}(\bm{q}_{1},z)$ is the
difference of the linear displacements, and the $\ldots$ represents the higher
order perturbative terms, as well as the unequal-time expansion term in powers
of $(H\delta\chi)$, following the procedure given in Appendix C. If, for
example, we consider dark matter dynamics, $\xi^{(0)}(q)$ can be interpreted
as a linear correlation function, and the first $\mathcal{P}(k)$ term above is
simply the IR-resummed linear power spectrum (in the equal-time limit). The
second displacement cumulant $A_{ij}$ can be decomposed as follows:

$\displaystyle A_{ij}(\bm{q},z,z^{\prime})$ $\displaystyle=\delta^{\rm
K}_{ij}X(q,z,z^{\prime})+\hat{q}_{i}\hat{q}_{j}Y(q,z,z^{\prime})$ (100)
$\displaystyle=\frac{1}{3}\delta^{\rm K}_{ij}\left(D^{2}+D^{\prime
2}\right)\Xi_{0}(0)-\frac{2}{3}\delta^{\rm
K}_{ij}DD^{\prime}\Xi_{0}(q)+2\left(\hat{q}_{i}\hat{q}_{j}-\frac{1}{3}\delta^{\rm
K}_{ij}\right)DD^{\prime}\Xi_{2}(q)\,,$

where we have introduced
$\Xi_{0}(q)=\int_{0}^{\infty}\frac{dk}{2\pi^{2}}\leavevmode\nobreak\
\mathcal{P}_{0}(k)j_{0}(kq)$ and
$\Xi_{2}(q)=\int_{0}^{\infty}\frac{dk}{2\pi^{2}}\leavevmode\nobreak\
\mathcal{P}_{0}(k)j_{2}(kq)$, and where the scale-dependent part of the linear
power spectrum $\mathcal{P}_{0}$ is equivalent to the one introduced in
Equation (36). Equivalently, we can write

$\displaystyle X(q,z,z^{\prime})$
$\displaystyle=\frac{1}{3}\left(D^{2}+D^{\prime
2}\right)\Xi_{0}(0)-\frac{2}{3}DD^{\prime}\left(\Xi_{0}(q)+\Xi_{2}(q)\right)\,,$
(101) $\displaystyle Y(q,z,z^{\prime})$ $\displaystyle=2\leavevmode\nobreak\
DD^{\prime}\Xi_{2}(q)\,.$

Figure 8:  Scale dependence of the equal-time displacement correlators $X$ and
$Y$ and corresponding unequal-time contributions $\delta X$ and $\delta Y$ are
shown for two different redshifts, $z=0.5$ and $2.0$. $\delta Y$ contributions
are suppressed relative to the equal-time counterpart $Y$ by at least an order
of magnitude on all scales and for all redshifts. Conversely, $\delta X$
exhibits approximately constant behaviour in $q$, with the grey band
indicating the range between the to limiting regimes $\delta X(q\to 0)$ and
$\delta X(q\to\infty)$.

If we consider only the unequal-time Zeldovich power spectrum limit, these
results are equivalent to the ones obtained in [61]. The unequal-time product
of two growth rates we can expand up to the second order in $\delta\chi$, in a
similar way done in Appendix C, giving us

$\frac{D(z_{1})D(z_{2})}{D(\bar{z})^{2}}\approx
1-\frac{1}{4}\gamma_{\times}(H\delta\chi)^{2}\,,$ (102)

where we introduced the factor
$\gamma_{\times}=f\left(1+f-\frac{3}{2}\frac{\Omega_{m}}{f}\right)/(1+z)^{2}$.
Besides the product, we are also interested in expanding the sum of the two
growth factors. This gives us

$\frac{D(z_{1})^{2}+D(z_{2})^{2}}{2D(\bar{z})^{2}}\approx
1-\frac{1}{4}\gamma_{+}(H\delta\chi)^{2}\,,$ (103)

where we introduced the factor
$\gamma_{+}=f\left(1-f-\frac{3}{2}\frac{\Omega_{m}}{f}\right)/(1+z)^{2}$.
Using this expansion, we can express

$\displaystyle X(q,z,z^{\prime})$ $\displaystyle=X(q,\bar{z})+\delta
X(q,\bar{z})(H\delta\chi)^{2}\,,$ (104) $\displaystyle Y(q,z,z^{\prime})$
$\displaystyle=Y(q,\bar{z})+\delta Y(q,\bar{z})(H\delta\chi)^{2}\,,$

where we introduced the quadratic corrections

$\displaystyle\delta X(q)$
$\displaystyle=\frac{1}{6}D^{2}\bigg{(}\gamma_{\times}\left(\Xi_{0}(q)+\Xi_{2}(q)\right)-\gamma_{+}\Xi_{0}(0)\bigg{)}\,,$
(105) $\displaystyle\delta Y(q)$
$\displaystyle=-\frac{1}{2}D^{2}\gamma_{\times}\Xi_{2}(q)\,.$

Accordingly, we can thus write
$A_{ij}(\bm{q},z,z^{\prime})=A_{ij}(\bm{q},\bar{z})+\delta
A_{ij}(\bm{q},\bar{z})(H\delta\chi)^{2}$. In Figure 8 we show the magnitudes
and scale dependence of the $X$ and $Y$ correlators, as well as their
corresponding unequal-time contributions $\delta X$ and $\delta Y$ for two
different redshifts, $z=0.5$ and $2.0$. We are interested in the relative
behaviour of the $\delta X$ and $\delta Y$ compared to the $X$ and $Y$. First,
we notice that $\delta Y$ contributions are more than an order of magnitude
smaller than $Y$ on all scales at low redshifts, with this difference further
decreasing at higher redshifts, i.e. $\delta Y/Y\lesssim 1/10$ on all scales
and for all redshifts. We can thus neglect $\delta Y$ contributions from our
further considerations. For $\delta X$, similar argumentation is not valid, as
it is the contribution that dominates on small scales at any redshift. The
scale dependence of $\delta X$ is actually bounded for at any $z$ so that we
can write $0.3\lesssim(\delta X(q\to 0)-\delta X(q\to\infty))/\delta X(q_{\rm
BAO})\lesssim 1.3$ for all $z$, where the lower bound is reached for high $z$
and vice versa. In other words, the grey bound shown in Figure 8, which
indicates the deviation of $\delta X$ from some constant value, gets narrow at
higher redshifts. Combining these considerations justifies the following
approximation for the unequal-time contribution of the displacement correlator

$\delta A_{ij}(\bm{q},\bar{z})\simeq\delta X(q,\bar{z})\delta^{\rm
K}_{ij}\simeq\frac{1}{6}D(\bar{z})^{2}\Big{(}\gamma_{\times}(\bar{z})\Xi_{0}(q)-\gamma_{+}(\bar{z})\Xi_{0}(0)\Big{)}\delta^{\rm
K}_{ij}\simeq 2\delta\sigma(\bar{z})^{2}\delta^{\rm K}_{ij}\,,$ (106)

where $\delta\sigma^{2}$ is the scale-independent displacement dispersion due
to the unequal-time effects. As discussed above, the magnitude of
$\delta\sigma^{2}$ is bounded from above and below, and we can thus write

$-\frac{1}{12}\gamma_{+}(\bar{z})\leq\frac{\delta\sigma(\bar{z})^{2}}{D(\bar{z})^{2}\Xi_{0}(0)}\leq\frac{1}{12}\left(\gamma_{\times}(\bar{z})-\gamma_{+}(\bar{z})\right)\,.$
(107)

We rely on this approximation in deriving the estimates of the unequal-time
effects due to the IR-resummation of long displacements discussed in
Subsection III.3.

## References

## References

  * Amendola  _et al._ [2018] L. Amendola _et al._ , Cosmology and fundamental physics with the Euclid satellite, Living Rev. Rel. 21, 2 (2018), arXiv:1606.00180 [astro-ph.CO] .
  * Aghamousa  _et al._ [2016] A. Aghamousa _et al._ (DESI), The DESI Experiment Part I: Science,Targeting, and Survey Design, (2016), arXiv:1611.00036 [astro-ph.IM] .
  * Abate  _et al._ [2012] A. Abate _et al._ (LSST Dark Energy Science), Large Synoptic Survey Telescope: Dark Energy Science Collaboration, (2012), arXiv:1211.0310 [astro-ph.CO] .
  * Spergel  _et al._ [2015] D. Spergel _et al._ , Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report, (2015), arXiv:1503.03757 [astro-ph.IM] .
  * Doré  _et al._ [2014] O. Doré _et al._ , Cosmology with the SPHEREX All-Sky Spectral Survey, (2014), arXiv:1412.4872 [astro-ph.CO] .
  * Bacon  _et al._ [2020] D. J. Bacon _et al._ (SKA), Cosmology with Phase 1 of the Square Kilometre Array: Red Book 2018: Technical specifications and performance forecasts, Publ. Astron. Soc. Austral. 37, e007 (2020), arXiv:1811.02743 [astro-ph.CO] .
  * Schlegel  _et al._ [2022] D. J. Schlegel _et al._ , The MegaMapper: A Stage-5 Spectroscopic Instrument Concept for the Study of Inflation and Dark Energy, (2022), arXiv:2209.04322 [astro-ph.IM] .
  * Spezzati  _et al._ [2023a] F. Spezzati, S. Libanore, E. Vanzan, A. Raccanelli, and Y. Wang, Cosmology with ATLAS Explorer, (2023a), arXiv:23XX.XXXXX [astro-ph.CO] .
  * Weinberg  _et al._ [2013] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess, and E. Rozo, Observational Probes of Cosmic Acceleration, Phys. Rept. 530, 87 (2013), arXiv:1201.2434 [astro-ph.CO] .
  * Mortonson  _et al._ [2013] M. J. Mortonson, D. H. Weinberg, and M. White, Dark Energy: A Short Review, (2013), arXiv:1401.0046 [astro-ph.CO] .
  * Ross  _et al._ [2013] A. J. Ross _et al._ , The Clustering of Galaxies in SDSS-III DR9 Baryon Oscillation Spectroscopic Survey: Constraints on Primordial Non-Gaussianity, Mon. Not. Roy. Astron. Soc. 428, 1116 (2013), arXiv:1208.1491 [astro-ph.CO] .
  * Alvarez  _et al._ [2014] M. Alvarez _et al._ , Testing Inflation with Large Scale Structure: Connecting Hopes with Reality, (2014), arXiv:1412.4671 [astro-ph.CO] .
  * Percival  _et al._ [2004] W. J. Percival _et al._ (2dFGRS), The 2dF Galaxy Redshift Survey: Spherical harmonics analysis of fluctuations in the final catalogue, Mon. Not. Roy. Astron. Soc. 353, 1201 (2004), arXiv:astro-ph/0406513 .
  * Eisenstein  _et al._ [2005] D. J. Eisenstein _et al._ (SDSS), Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies, Astrophys. J. 633, 560 (2005), arXiv:astro-ph/0501171 .
  * Cole  _et al._ [2005] S. Cole _et al._ (2dFGRS), The 2dF Galaxy Redshift Survey: Power-spectrum analysis of the final dataset and cosmological implications, Mon. Not. Roy. Astron. Soc. 362, 505 (2005), arXiv:astro-ph/0501174 .
  * Tegmark  _et al._ [2006] M. Tegmark _et al._ (SDSS), Cosmological Constraints from the SDSS Luminous Red Galaxies, Phys. Rev. D 74, 123507 (2006), arXiv:astro-ph/0608632 .
  * Padmanabhan  _et al._ [2007] N. Padmanabhan _et al._ (SDSS), The Clustering of Luminous Red Galaxies in the Sloan Digital Sky Survey Imaging Data, Mon. Not. Roy. Astron. Soc. 378, 852 (2007), arXiv:astro-ph/0605302 .
  * Guzzo  _et al._ [2008] L. Guzzo _et al._ , A test of the nature of cosmic acceleration using galaxy redshift distortions, Nature 451, 541 (2008), arXiv:0802.1944 [astro-ph] .
  * Percival and White [2009] W. J. Percival and M. White, Testing cosmological structure formation using redshift-space distortions, Mon. Not. Roy. Astron. Soc. 393, 297 (2009), arXiv:0808.0003 [astro-ph] .
  * Ho  _et al._ [2008] S. Ho, C. Hirata, N. Padmanabhan, U. Seljak, and N. Bahcall, Correlation of CMB with large-scale structure: I. ISW Tomography and Cosmological Implications, Phys. Rev. D 78, 043519 (2008), arXiv:0801.0642 [astro-ph] .
  * Reid  _et al._ [2010] B. A. Reid _et al._ , Cosmological Constraints from the Clustering of the Sloan Digital Sky Survey DR7 Luminous Red Galaxies, Mon. Not. Roy. Astron. Soc. 404, 60 (2010), arXiv:0907.1659 [astro-ph.CO] .
  * Kazin  _et al._ [2010] E. A. Kazin _et al._ (SDSS), The Baryonic Acoustic Feature and Large-Scale Clustering in the SDSS LRG Sample, Astrophys. J. 710, 1444 (2010), arXiv:0908.2598 [astro-ph.CO] .
  * Ross  _et al._ [2011] A. J. Ross, W. J. Percival, M. Crocce, A. Cabre, and E. Gaztanaga, Measuring Redshift-Space Distortions using Photometric Surveys, Mon. Not. Roy. Astron. Soc. 415, 2193 (2011), arXiv:1102.0968 [astro-ph.CO] .
  * Samushia  _et al._ [2012] L. Samushia, W. J. Percival, and A. Raccanelli, Interpreting large-scale redshift-space distortion measurements, Mon. Not. Roy. Astron. Soc. 420, 2102 (2012), arXiv:1102.1014 [astro-ph.CO] .
  * Beutler  _et al._ [2014] F. Beutler _et al._ (BOSS), The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Testing gravity with redshift-space distortions using the power spectrum multipoles, Mon. Not. Roy. Astron. Soc. 443, 1065 (2014), arXiv:1312.4611 [astro-ph.CO] .
  * Gil-Marín  _et al._ [2015] H. Gil-Marín, J. Noreña, L. Verde, W. J. Percival, C. Wagner, M. Manera, and D. P. Schneider, The power spectrum and bispectrum of SDSS DR11 BOSS galaxies – I. Bias and gravity, Mon. Not. Roy. Astron. Soc. 451, 539 (2015), arXiv:1407.5668 [astro-ph.CO] .
  * Gil-Marín  _et al._ [2016] H. Gil-Marín _et al._ , The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: RSD measurement from the LOS-dependent power spectrum of DR12 BOSS galaxies, Mon. Not. Roy. Astron. Soc. 460, 4188 (2016), arXiv:1509.06386 [astro-ph.CO] .
  * Beutler  _et al._ [2017] F. Beutler _et al._ (BOSS), The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Fourier space, Mon. Not. Roy. Astron. Soc. 464, 3409 (2017), arXiv:1607.03149 [astro-ph.CO] .
  * Sanchez  _et al._ [2017] A. G. Sanchez _et al._ (BOSS), The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological implications of the configuration-space clustering wedges, Mon. Not. Roy. Astron. Soc. 464, 1640 (2017), arXiv:1607.03147 [astro-ph.CO] .
  * Ivanov  _et al._ [2020] M. M. Ivanov, M. Simonović, and M. Zaldarriaga, Cosmological Parameters from the BOSS Galaxy Power Spectrum, JCAP 05, 042, arXiv:1909.05277 [astro-ph.CO] .
  * D’Amico  _et al._ [2020] G. D’Amico, J. Gleyzes, N. Kokron, K. Markovic, L. Senatore, P. Zhang, F. Beutler, and H. Gil-Marín, The Cosmological Analysis of the SDSS/BOSS data from the Effective Field Theory of Large-Scale Structure, JCAP 05, 005, arXiv:1909.05271 [astro-ph.CO] .
  * Alam  _et al._ [2021] S. Alam _et al._ (eBOSS), Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory, Phys. Rev. D 103, 083533 (2021), arXiv:2007.08991 [astro-ph.CO] .
  * Zhao  _et al._ [2021] G.-B. Zhao _et al._ , The completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: a multitracer analysis in Fourier space for measuring the cosmic structure growth and expansion rate, Mon. Not. Roy. Astron. Soc. 504, 33 (2021), arXiv:2007.09011 [astro-ph.CO] .
  * Chen  _et al._ [2022] S.-F. Chen, Z. Vlah, and M. White, A new analysis of galaxy 2-point functions in the BOSS survey, including full-shape information and post-reconstruction BAO, JCAP 02 (02), 008, arXiv:2110.05530 [astro-ph.CO] .
  * [35] A. Raccanelli and Z. Vlah, Power spectrum in the cave, arXiv:2305.16278 [astro-ph.CO] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Lewis  _et al._ [2000] A. Lewis, A. Challinor, and A. Lasenby, Efficient computation of CMB anisotropies in closed FRW models, Astrophys. J. 538, 473 (2000), arXiv:astro-ph/9911177 .
  * Lesgourgues [2011] J. Lesgourgues, The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview, (2011), arXiv:1104.2932 [astro-ph.IM] .
  * Matsubara [2000] T. Matsubara, The Correlation function in redshift space: General formula with wide angle effects and cosmological distortions, Astrophys. J. 535, 1 (2000), arXiv:astro-ph/9908056 .
  * Yoo [2009] J. Yoo, Complete Treatment of Galaxy Two-Point Statistics: Gravitational Lensing Effects and Redshift-Space Distortions, Phys. Rev. D 79, 023517 (2009), arXiv:0808.3138 [astro-ph] .
  * Yoo  _et al._ [2009] J. Yoo, A. L. Fitzpatrick, and M. Zaldarriaga, A New Perspective on Galaxy Clustering as a Cosmological Probe: General Relativistic Effects, Phys. Rev. D 80, 083514 (2009), arXiv:0907.0707 [astro-ph.CO] .
  * Bonvin and Durrer [2011] C. Bonvin and R. Durrer, What galaxy surveys really measure, Phys. Rev. D 84, 063505 (2011), arXiv:1105.5280 [astro-ph.CO] .
  * Challinor and Lewis [2011] A. Challinor and A. Lewis, The linear power spectrum of observed source number counts, Phys. Rev. D 84, 043516 (2011), arXiv:1105.5292 [astro-ph.CO] .
  * Jeong  _et al._ [2012] D. Jeong, F. Schmidt, and C. M. Hirata, Large-scale clustering of galaxies in general relativity, Phys. Rev. D 85, 023504 (2012), arXiv:1107.5427 [astro-ph.CO] .
  * Bertacca  _et al._ [2012] D. Bertacca, R. Maartens, A. Raccanelli, and C. Clarkson, Beyond the plane-parallel and Newtonian approach: Wide-angle redshift distortions and convergence in general relativity, JCAP 10, 025, arXiv:1205.5221 [astro-ph.CO] .
  * Yoo and Desjacques [2013] J. Yoo and V. Desjacques, All-Sky Analysis of the General Relativistic Galaxy Power Spectrum, Phys. Rev. D 88, 023502 (2013), arXiv:1301.4501 [astro-ph.CO] .
  * Limber [1954] D. N. Limber, The Analysis of Counts of the Extragalactic Nebulae in Terms of a Fluctuating Density Field. II, Astrophys. J. 119, 655 (1954).
  * Kaiser [1987] N. Kaiser, Clustering in real space and in redshift space, Mon. Not. Roy. Astron. Soc. 227, 1 (1987).
  * Alcock and Paczynski [1979] C. Alcock and B. Paczynski, An evolution free test for non-zero cosmological constant, Nature 281, 358 (1979).
  * Szalay  _et al._ [1998] A. S. Szalay, T. Matsubara, and S. D. Landy, Redshift space distortions of the correlation function in wide angle galaxy surveys, Astrophys. J. Lett. 498, L1 (1998), arXiv:astro-ph/9712007 .
  * Papai and Szapudi [2008] P. Papai and I. Szapudi, Non-Perturbative Effects of Geometry in Wide-Angle Redshift Distortions, Mon. Not. Roy. Astron. Soc. 389, 292 (2008), arXiv:0802.2940 [astro-ph] .
  * Raccanelli  _et al._ [2010] A. Raccanelli, L. Samushia, and W. J. Percival, Simulating Redshift-Space Distortions for Galaxy Pairs with Wide Angular Separation, Mon. Not. Roy. Astron. Soc. 409, 1525 (2010), arXiv:1006.1652 [astro-ph.CO] .
  * Castorina and White [2018] E. Castorina and M. White, Beyond the plane-parallel approximation for redshift surveys, Mon. Not. Roy. Astron. Soc. 476, 4403 (2018), arXiv:1709.09730 [astro-ph.CO] .
  * Pryer  _et al._ [2022] D. Pryer, R. E. Smith, R. Booth, C. Blake, A. Eggemeier, and J. Loveday, The galaxy power spectrum on the lightcone: deep, wide-angle redshift surveys and the turnover scale, JCAP 08 (08), 019, arXiv:2111.01811 [astro-ph.CO] .
  * Desjacques  _et al._ [2018] V. Desjacques, D. Jeong, and F. Schmidt, Large-Scale Galaxy Bias, Phys. Rept. 733, 1 (2018), arXiv:1611.09787 [astro-ph.CO] .
  * Vlah  _et al._ [2015a] Z. Vlah, U. Seljak, and T. Baldauf, Lagrangian perturbation theory at one loop order: successes, failures, and improvements, Phys. Rev. D 91, 023508 (2015a), arXiv:1410.1617 [astro-ph.CO] .
  * Vlah  _et al._ [2015b] Z. Vlah, M. White, and A. Aviles, A Lagrangian effective field theory, JCAP 09, 014, arXiv:1506.05264 [astro-ph.CO] .
  * Senatore and Zaldarriaga [2015] L. Senatore and M. Zaldarriaga, The IR-resummed Effective Field Theory of Large Scale Structures, JCAP 02, 013, arXiv:1404.5954 [astro-ph.CO] .
  * Baldauf  _et al._ [2015] T. Baldauf, M. Mirbabayi, M. Simonović, and M. Zaldarriaga, Equivalence Principle and the Baryon Acoustic Peak, Phys. Rev. D 92, 043514 (2015), arXiv:1504.04366 [astro-ph.CO] .
  * Ding  _et al._ [2018] Z. Ding, H.-J. Seo, Z. Vlah, Y. Feng, M. Schmittfull, and F. Beutler, Theoretical Systematics of Future Baryon Acoustic Oscillation Surveys, Mon. Not. Roy. Astron. Soc. 479, 1021 (2018), arXiv:1708.01297 [astro-ph.CO] .
  * Chisari and Pontzen [2019] N. E. Chisari and A. Pontzen, Unequal time correlators and the Zel’dovich approximation, Phys. Rev. D 100, 023543 (2019), arXiv:1905.02078 [astro-ph.CO] .
  * Kitching and Heavens [2017] T. D. Kitching and A. F. Heavens, Unequal-Time Correlators for Cosmology, Phys. Rev. D 95, 063522 (2017), arXiv:1612.00770 [astro-ph.CO] .
  * Raccanelli  _et al._ [2014] A. Raccanelli, D. Bertacca, O. Doré, and R. Maartens, Large-scale 3D galaxy correlation function and non-Gaussianity, JCAP 08, 022, arXiv:1306.6646 [astro-ph.CO] .
  * Raccanelli  _et al._ [2016a] A. Raccanelli, D. Bertacca, R. Maartens, C. Clarkson, and O. Doré, Lensing and time-delay contributions to galaxy correlations, Gen. Rel. Grav. 48, 84 (2016a), arXiv:1311.6813 [astro-ph.CO] .
  * Raccanelli  _et al._ [2016b] A. Raccanelli, F. Montanari, D. Bertacca, O. Doré, and R. Durrer, Cosmological Measurements with General Relativistic Galaxy Correlations, JCAP 05, 009, arXiv:1505.06179 [astro-ph.CO] .
  * Raccanelli  _et al._ [2018] A. Raccanelli, D. Bertacca, D. Jeong, M. C. Neyrinck, and A. S. Szalay, Doppler term in the galaxy two-point correlation function: wide-angle, velocity, Doppler lensing and cosmic acceleration effects, Phys. Dark Univ. 19, 109 (2018), arXiv:1602.03186 [astro-ph.CO] .
  * McDonald and Seljak [2009] P. McDonald and U. Seljak, How to measure redshift-space distortions without sample variance, JCAP 10, 007, arXiv:0810.0323 [astro-ph] .
  * McDonald [2009] P. McDonald, Gravitational redshift and other redshift-space distortions of the imaginary part of the power spectrum, JCAP 11, 026, arXiv:0907.5220 [astro-ph.CO] .
  * Spezzati  _et al._ [2023b] F. Spezzati, A. Raccanelli, and Z. Vlah, in preparation, (2023b), arXiv:23XX.XXXXX [astro-ph.CO] .
  * Hamilton [1997] A. J. S. Hamilton, Linear redshift distortions: A Review, in  _Ringberg Workshop on Large Scale Structure_ (1997) arXiv:astro-ph/9708102 .
  * Chluba  _et al._ [2015] J. Chluba, J. Hamann, and S. P. Patil, Features and New Physical Scales in Primordial Observables: Theory and Observation, Int. J. Mod. Phys. D 24, 1530023 (2015), arXiv:1505.01834 [astro-ph.CO] .
  * Slosar  _et al._ [2019] A. Slosar _et al._ , Scratches from the Past: Inflationary Archaeology through Features in the Power Spectrum of Primordial Fluctuations, Bull. Am. Astron. Soc. 51, 98 (2019), arXiv:1903.09883 [astro-ph.CO] .
  * Achúcarro  _et al._ [2022] A. Achúcarro _et al._ , Inflation: Theory and Observations, (2022), arXiv:2203.08128 [astro-ph.CO] .
  * Chen  _et al._ [2020] S.-F. Chen, Z. Vlah, and M. White, Modeling features in the redshift-space halo power spectrum with perturbation theory, JCAP 11, 035, arXiv:2007.00704 [astro-ph.CO] .
  * Ferraro  _et al._ [2022] S. Ferraro, N. Sailer, A. Slosar, and M. White, Snowmass2021 Cosmic Frontier White Paper: Cosmology and Fundamental Physics from the three-dimensional Large Scale Structure, (2022), arXiv:2203.07506 [astro-ph.CO] .
  * Vlah and White [2019] Z. Vlah and M. White, Exploring redshift-space distortions in large-scale structure, JCAP 03, 007, arXiv:1812.02775 [astro-ph.CO] .
  * Chen  _et al._ [2021] S.-F. Chen, Z. Vlah, E. Castorina, and M. White, Redshift-Space Distortions in Lagrangian Perturbation Theory, JCAP 03, 100, arXiv:2012.04636 [astro-ph.CO] .