# The observed number counts in luminosity distance space

José FonsecaaaaCorresponding author. , Stefano ZazzerabbbCorresponding author.
, Tessa Baker  and Chris Clarkson

###### Abstract

Next generation surveys will provide us with an unprecedented number of
detections of supernovae Type Ia and gravitational wave merger events. Cross-
correlations of such objects offer novel and powerful insights into the large-
scale distribution of matter in the universe. Both of these sources carry
information on their luminosity distance, but remain uninformative about their
redshifts; hence their clustering analyses and cross-correlations need to be
carried out in luminosity distance space, as opposed to redshift space. In
this paper, we calculate the full expression for the number count fluctuation
in terms of a perturbation to the observed luminosity distance. We find the
expression to differ significantly from the one commonly used in redshift
space. Furthermore, we present a comparison of the number count angular power
spectra between luminosity distance and redshift spaces. We see a wide
divergence between the two at large scales, and we note that lensing is the
main contribution to such differences. On such scales and at higher redshifts
the difference between the angular power spectra in luminosity distance and
redshift spaces can be roughly 50$\displaystyle\%$. We also investigate cross-
correlating different redshift bins using different tracers, i.e. one in
luminosity distance space and one in redshift, simulating the cross-
correlation angular power spectrum between background gravitational
waves/supernovae and foreground galaxies. Finally, we show that in a cosmic
variance limited survey, the relativistic corrections to the density-only term
ought to be included.

##  1 Introduction

Forthcoming gravitational wave experiments such as the ground-based Einstein
Telescope (ET) [1, 3, 2], and Cosmic Explorer (CE) [4], as well as the space-
based Laser Interferometer Space Antenna (LISA) [5], will detect rapidly
growing numbers of gravitational wave (GW) merger events. These are detected
when inspiraling compact objects, such as black holes and neutron stars,
merge. Such objects are, in principle, the result of astrophysical processes
in galaxies, and therefore trace the large-scale distribution of matter as
their host galaxies do. However, unless an electromagnetic counterpart
identifies the unique host galaxy of the merger, one can only estimate the
luminosity distance to the GW source; the redshift of the event is not
explicitly contained in gravitational waveform in a measurable way.

But GW mergers are not the only observed transient events for which one
estimates a luminosity distance. Supernovae Type Ia (SNIa) are another
example. Although one can do a spectroscopic follow-up of the host galaxy, the
redshift of the host is more commonly measured photometrically, if the host
galaxy is correctly identified. Thus, the most direct measurement of the
distance to a SNIa is using the distance modulus, which is dependent on the
luminosity distance (and systematic effects).

Hence gravitational waves and supernovae both provide a measure of the
luminosity distance of the source, but they remain agnostic on its redshift.
Photometric Type Ia supernovae (SNIa) have already been established as a
powerful tool for the construction of the cosmological distance ladder via
their property of acting as standard candles [6, 7]. Additionally, the
detection of GWs from inspiraling compact binaries can also be exploited to
measure an event’s luminosity distance, and as GWs can be thought of as
acoustic rather than visual, a more commonly used term for them is then
standard sirens [8, 9, 10, 11, 12, 13, 14].

Further, the Legacy Survey of Space and Time (LSST) of the Vera Rubin
Observatory is predicted to observe around $\displaystyle 10^{6}$ photometric
supernovae during its 10-year observing time [16, 17, 18, 19], and a similar
number of GW events is forecasted to be seen by following third generation,
ground-based GW observatories such as ET [16, 20, 21, 22, 23, 24, 1, 3, 2, 15]
and CE [4, 25, 26, 27]. This unprecedented number of detected objects will
allow us to go beyond using them as standard sirens and standard candles only.
In fact, we will be able to use them to study their clustering across various
scales, a method currently only applicable to galaxies [29, 28, 30, 31, 32].

Cross-correlating gravitational waves with other large-scale structure tracers
has lately been established as a potentially powerful probe of the
distribution of matter in the universe, as well as in understanding the
evolution of astrophysical black holes through cosmic time [16, 20, 21, 22,
23, 24]. However, this method will be possible only with the new generations
of surveys which will provide us with an unparalleled wealth of data. In this
regime, one needs to have a complete description of the observed clustering of
sources for which we measure luminosity distances. One could in principle
assume a cosmology and translate an observed luminosity distance into an
estimate of the redshift. However, doing so imposes a strong prior on any
potential constraints coming from these objects, likely biasing them.

Since in General Relativity (GR) both photons and GWs follow null geodesics,
both will be distorted equally by lensing caused by the foreground matter
distribution [32, 40, 33, 34, 35, 36, 37, 38, 39], which will impact the
inferred luminosity distance to the event [42, 43, 44, 8, 9, 33, 16]. Lensing
is not the only physical effect that alter the observed luminosity distance,
as in any perturbed FLRW universe the observed luminosity distance acquires
linear corrections [42, 46, 47, 41, 48]. One should note that the exact
expression of the perturbation depends on what one considers as the reference
points, as we will discuss later. Therefore, the observed luminosity distance
is an excellent probe of large-scale structure. Analysing the angular power
spectra of these tracers [33, 49, 48], together with computing cross-
correlations between different tracers that live in luminosity distance space
(LDS), can be a powerful tool in constraining cosmological parameters, and
further studying the distance-redshift relationship [33, 50, 51, 52, 53, 16].

Similarly to the more commonly explored counterpart in redshift space (RS)
[54, 55, 56, 57, 58, 59], clustering analysis in luminosity distance space has
to take into account distortion effects caused by the gradient of peculiar
velocities — like the well-studied Redshift Space Distortions (RSDs), radial
velocity effects, Sachs-Wolfe terms and the integrated Sachs-Wolfe (ISW)
effect [45, 29, 33, 28, 30, 31, 32]. While Doppler and RSD terms are dominant
at low redshift, the other effects could provide significant corrections at
higher distances: in particular, lensing effects are integrated over distance,
thus at higher redshifts they will be expected to increase in amplitude. While
LSST will be able to observe SNIa events at $\displaystyle z<4$ [17, 18, 19],
ET/CE are predicted to access the mergers of stellar mass binary black holes
up to $\displaystyle z\sim 20$ [1, 2, 15, 4]. Therefore, clustering analysis
at these previously inaccessible distances would require robust calculations
of the distortion effects caused by lensing, allowing for the possibility of
detecting lensing effects that would otherwise be buried under low-redshift
contributions.

The goal of this paper is to present the full calculation of the observed
number count fluctuations of sources in luminosity distance space (LDS),
including all relativistic corrections. Additionally, we developed a code to
compute the angular power spectrum in LDS. Our general results can be applied
to any tracer in LDS, such as GWs or SNIa. We will then use our results to
assess the relative importance of the various contributions to the angular
power spectrum LDS tracers, including cross-correlations with galaxy surveys.
By this means, we construct a theoretical formalism applicable to data coming
from future surveys like LSST and ET. Forecasts on the observability of these
auto- and cross-correlations, and in particular the detectability of the
various contributions is outside the scope of this paper and will be treated
in future work. Similarly, forecast cosmological constraints using catalogues
of LDS objects are beyond the scope of the current paper.

The paper is structured as follows. In section 2, we compute the number
density contrast in terms of the luminosity distance and volume perturbations.
We also clarify the expression for the total perturbation in the observed
luminosity distance (as opposed to background one), and subsequently compute
the full expression for the number counts fluctuation. Section 3 explores the
difference between the angular power spectra in redshift and luminosity
distance spaces, as well as the relevance of the various terms at different
distances. Finally, section 4 is devoted to summary and conclusions. In
Appendix A we present a thorough calculation on how we computed the transfer
functions implemented in CAMB, and a summary in B. Throughout this paper we
will use natural units by fixing the speed of light to one, i.e.,
$\displaystyle c=1$.

##  2 The observed number counts in luminosity distance space

In this section we compute the number counts fluctuation for tracers in
luminosity distance space, i.e. tracers for which we do not measure position
or redshift but luminosity distance. First, we will compute the over-density
in the number counts as a function of the perturbation in luminosity distance.
We then add the effect of a flux-limited survey (magnification bias) for the
two most relevant luminosity distance tracers, SNIa and GWs. Furthermore, we
will identify the appropriate perturbation in luminosity distance to use in
this derivation, and clearly lay out the computation of the volume
perturbation when using luminosity distance as our observed quantity. Finally,
in section 2.4 we report the general expression including all relativistic
corrections. Readers principally interested in the evaluation of our results
can skip to section 2.4 and then proceed to section 3.

In this paper we will use a perturbed, flat FLRW metric in conformal Newtonian
gauge given by

$ds^{2}=a^{2}(\eta)\left[-(1+2\Psi)d\eta^{2}+(1-2\Phi)\delta_{ab}dx^{a}dx^{b}\right]\,,$
(2.1)

where $\displaystyle\Phi$ and $\displaystyle\Psi$ are the metric potentials
and $\displaystyle\eta$ is conformal time.

###  2.1 Number density perturbation

Let us consider a number of objects seen in a given (observed) direction
$\displaystyle{\bm{n}}$ at a given (observed) luminosity distance
$\displaystyle D_{L}$ and denote it by $\displaystyle
N({\bm{n}},D_{L})d\Omega_{{\bm{n}}}dD_{L}$, where
$\displaystyle\Omega_{{\bm{n}}}$ is an infinitesimal solid angle element. _Per
se_ , this quantity does not carry any relevant cosmological information. It
is the variance of its fluctuation with respect to a cosmological average of
the sources that one can use to relate to the primordial power spectrum.
Therefore, we want to determine the fluctuation of this quantity at first
order in perturbation theory. To compute the number density contrast we then
start from:

$\Delta_{N}=\frac{N-\langle N\rangle}{\langle N\rangle}$ (2.2)

where $\displaystyle\langle N\rangle$ is an average over directions. These
quantities are determined at the observed luminosity distance, although
ultimately we need to relate them to an affine parameter which one can take to
be the background redshift. The number count is simply given by a number
density times a volume, i.e., $\displaystyle N=\rho V$, then

$\Delta({\bm{n}},D_{L})=\frac{\rho({\bm{n}},D_{L})V({\bm{n}},D_{L})-\bar{\rho}(D_{L})\bar{V}(D_{L})}{\bar{\rho}(D_{L})\bar{V}(D_{L})}\,,$
(2.3)

where we used bars for average/background quantities. We now expand the
observed volume and observed (physical, as opposed to comoving) number density
around their background values, i.e.,

$\displaystyle\displaystyle\rho({\bm{n}},D_{L})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bar{\rho}(D_{L})+\delta\rho({\bm{n}},D_{L})\,,$
(2.4a) $\displaystyle\displaystyle V({\bm{n}},D_{L})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\bar{V}(D_{L})+\delta
V({\bm{n}},D_{L})\,.$ (2.4b)

Substituting these into (2.3), one reaches

$\Delta({\bm{n}},D_{L})=\frac{\delta\rho({\bm{n}},D_{L})}{\bar{\rho}(D_{L})}+\frac{\delta
V({\bm{n}},D_{L})}{\bar{V}(D_{L})}\,.$ (2.5)

This expression is similar to the one found in redshift space. However, we
note that the end result will necessarily yield a different number count than
that in redshift space. The size of an arbitrary cell, and thus the number of
e.g. encompassed GW sources/SNIa events, will inherently depend on the
observing quantity used (i.e. $\displaystyle z$ or $\displaystyle D_{L}$),
affecting the first term on the RHS in eq. (2.5). Further, a perturbation in
the latter won’t be identical in both spaces, thus the volume (and the volume
perturbation) of the cell will differ between redshift and luminosity distance
space, altering the last term in eq. (2.5). Ultimately, the way the observed
spaces relate to a background affine parameter is different.

So far we have expanded density and volume in terms of their background
quantities but they are still written in terms of the observed luminosity
distance. Therefore, we need to go from the observed $\displaystyle D_{L}$ to
the background quantity, $\displaystyle\bar{D}_{L}$, through the affine
parameter which we take to be the background redshift, $\displaystyle\bar{z}$;
thus, $\displaystyle\bar{D}_{L}\equiv\bar{D}_{L}(\bar{z})$. Note that in
redshift space we perturb from observed to background quantities, and then
from the observed parameter to the background parameter. Here we should do the
same, i.e.,

$D_{L}=\bar{D}_{L}+\delta D_{L}\,.$ (2.6)

Note that in eq.(2.5) one approximates
$\displaystyle\bar{V}(D_{L})\simeq\bar{V}(\bar{D}_{L})$ as the numerator is
already a perturbed quantity. The same is done for
$\displaystyle\bar{\rho}(D_{L})$.

Next, we expand $\displaystyle\bar{\rho}$ to relate the density perturbation
to a background quantity, i.e.:

$\bar{\rho}(D_{L})=\bar{\rho}(\bar{D}_{L}+\delta
D_{L})=\bar{\rho}(\bar{D}_{L})+\frac{\partial\bar{\rho}}{\partial
D_{L}}\bigg{|}_{D_{L}=\bar{D}_{L}}\delta D_{L}\,.$ (2.7)

Then,

$\delta\rho({\bm{n}},D_{L})=\rho({\bm{n}},D_{L})-\bar{\rho}(D_{L})=\rho({\bm{n}},D_{L})-\bar{\rho}(\bar{D}_{L})-\frac{\partial\bar{\rho}}{\partial
D_{L}}\bigg{|}_{D_{L}=\bar{D}_{L}}\delta D_{L}\,,$ (2.8)

therefore

$\frac{\delta\rho({\bm{n}},D_{L})}{\bar{\rho}(D_{L})}=\delta_{n}-\frac{1}{\bar{\rho}(\bar{D}_{L})}\frac{\partial\bar{\rho}}{\partial
D_{L}}\bigg{|}_{D_{L}=\bar{D}_{L}}\delta D_{L}\,.\ $ (2.9)

where
$\displaystyle\delta_{n}=\left(\rho({\bm{n}},D_{L})-\bar{\rho}(\bar{D}_{L})\right)/\bar{\rho}(\bar{D}_{L})$
is the density contrast in the Newtonian gauge.

Written in terms of the background redshift $\displaystyle\bar{z}$ and noting
that the comoving number density $\displaystyle n$ is related to the physical
one $\displaystyle\rho$ via $\displaystyle\rho=a^{-3}n$, then:

$\displaystyle\displaystyle\frac{1}{\bar{\rho}}\frac{\partial\bar{\rho}}{\partial
D_{L}}\bigg{|}_{D_{L}=\bar{D}_{L}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{a^{-3}n}\frac{\partial(a^{-3}n)}{\partial
D_{L}}=a^{3}n^{-1}\frac{\partial(a^{-3}n)}{\partial a}\left(\frac{\partial
D_{L}}{\partial z}\frac{\partial z}{\partial a}\right)^{-1}$ (2.10)
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{a}\left[-3+\frac{\partial\ln
n}{\partial\ln
a}\right]\bar{r}^{-1}\left(1+\frac{1}{\bar{r}\mathcal{H}}\right)^{-1}(-a^{2})=\frac{\gamma}{\bar{D}_{L}}\left[3-\frac{\partial\ln
n}{\partial\ln a}\right],\ $

where we used

$\displaystyle\displaystyle\frac{{\rm d}\bar{D}_{L}}{{\rm d}z}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\bar{r}+\frac{1}{\cal
H}=\frac{\bar{D}_{L}}{(1+z)\gamma}\,,$ (2.11)
$\displaystyle\displaystyle\frac{{\rm d}a}{{\rm d}z}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle-a^{2}\,,$ (2.12)
$\displaystyle\displaystyle\gamma$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\frac{\bar{r}\cal H}{1+\bar{r}\cal H}\,,$ (2.13)

where $\displaystyle\bar{r}$ is the background comoving distance to the
source. Using these results in eq. (2.5) we have

$\Delta({\bm{n}},D_{L})=\delta_{n}-\left[3-\frac{\partial\ln n}{\partial\ln
a}\right]\gamma\frac{\delta D_{L}}{\bar{D}_{L}}+\frac{\delta
V({\bm{n}},D_{L})}{\bar{V}(D_{L})}\,,$ (2.14)

where we note that the partial derivative of the comoving number density is
the evolution bias [60].

In the case of a luminosity limited survey we need to include a further
perturbation, which arises from lensing magnification. This can be thought of
as the change in the observed number density of sources w.r.t. the change in
the threshold of detection — i.e. a perturbation in the observed distance
might nudge an event above/below the minimum luminosity of the survey. This
term will depend on the type of tracer that populates our number density
contrast, as it inherently depends on the means of observation. For
gravitational waves, where $\displaystyle\sigma$ is the signal-to-noise ratio
(SNR) of the GW in the detector network:

$\displaystyle\displaystyle\Delta_{O}^{GW}(D_{L},\hat{n},\sigma>\sigma_{th})=\Delta_{O}^{GW}(D_{L},\hat{n})+\frac{\partial\ln
n}{\partial\ln\sigma}\bigg{|}_{\sigma=\sigma_{th}}\times\frac{\delta\sigma}{\sigma},$
(2.15)

where the first term on the RHS is the number density contrast for a survey
not limited by luminosity/SNR, while the second is the magnification bias
multiplied by a perturbation in the SNR.

In order to compute $\displaystyle\delta\sigma/\sigma$ we start from:

$\displaystyle\displaystyle\sigma(D_{L})=\sigma(\bar{D}_{L}+\delta
D_{L})=\sigma(\bar{D}_{L})+\frac{\partial\sigma}{\partial
D_{L}}\bigg{|}_{D_{L}=\bar{D}_{L}}\delta D_{L},$ (2.16)

which is equivalent to

$\displaystyle\displaystyle\sigma(D_{L})=\sigma(\bar{D}_{L})+\delta\sigma.$
(2.17)

From [61, 62, 12, 34] we find that the SNR is:

$\sigma=\sqrt{\frac{5}{96\pi^{4/3}}}\frac{1}{D_{L}}(\mathcal{M}_{z})^{5/6}\theta\sqrt{I}\propto\frac{1}{D_{L}},$
(2.18)

where $\displaystyle I$ is an integral over frequency which includes the
sensitivity curve of the detector333In the case of a network of detectors the
signal-to-noise is added in quadrature, retaining the proportionality to
$\displaystyle D_{L}^{-1}$., $\displaystyle\theta$ is a function of angles
that describes the orientation of the binary with respect to the detector and
$\displaystyle\mathcal{M}_{z}$ the redshifted chirp mass. Thus, using eq.
(2.6):

$\displaystyle\displaystyle\frac{\delta\sigma}{\sigma(\bar{D}_{L})}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{\sigma(D_{L})-\sigma(\bar{D}_{L})}{\sigma(\bar{D}_{L})}=\frac{\bar{D}_{L}}{D_{L}}-1=\left(1+\frac{\delta
D_{L}}{\bar{D}_{L}}\right)^{-1}-1\simeq-\frac{\delta D_{L}}{\bar{D}_{L}}\,,$
(2.19)

where we used binomial expansion in the last step. Therefore

$\displaystyle\displaystyle\frac{\delta\sigma}{\sigma}\simeq-\frac{\delta
D_{L}}{\bar{D}_{L}}\,,$ (2.20)

where the relative minus sign implies that an increase in the observed
luminosity distance decreases the corresponding SNR, as it should.

We can now rewrite the number density contrast from eq. (2.15) as:

$\displaystyle\displaystyle\Delta_{O}^{GW}(D_{L},\hat{n},\sigma>\sigma_{th})=\delta_{n}-\left[3-\frac{\partial\ln
n}{\partial\ln a}+\frac{1}{\gamma}\frac{\partial\ln
n}{\partial\ln\sigma}\right]\gamma\frac{\delta
D_{L}}{\bar{D}_{L}}+\frac{\delta V({\bm{n}},D_{L})}{\bar{V}(D_{L})}\,.$ (2.21)

Again, we stress that the partial derivatives w.r.t. the scale factor and the
SNR are, respectively, the evolution and magnification bias for GWs. However,
eq. (2.21) is specific to GWs because of the bias term unique to this case.
Instead, in the case of Supernovae Type Ia, we note that the survey will be
apparent magnitude $\displaystyle m$ limited:

$\Delta_{O}^{SNIa}(D_{L},\hat{n},{m<m_{max}})=\Delta_{O}^{SNIa}(D_{L},\hat{n}){+\frac{\partial\ln
n}{\partial\ln m}\bigg{|}_{m=m_{max}}\times\frac{\delta m}{m}}+\frac{\delta
V({\bm{n}},D_{L})}{\bar{V}(D_{L})}.$ (2.22)

Recalling the definition of absolute magnitude $\displaystyle M$

$\displaystyle\displaystyle M=m-5\log_{10}(D_{L}[\mathrm{Mpc}])+25\,,$ (2.23)

it is then straightforward to calculate the $\displaystyle\delta m$ term, as
$\displaystyle M$ is fixed for all SNIa. Combining it with the evolution bias
term (unchanged) we find:

$\Delta_{O}^{SNIa}(D_{L},\hat{n},{m<m_{max}})=\delta_{n}-\left[3-\frac{\partial\ln
n}{\partial\ln
a}{-}5\frac{1}{\gamma}\frac{\partial\log_{10}n}{\partial{m}}\right]\gamma\frac{\delta
D_{L}}{\bar{D}_{L}}+\frac{\delta V({\bm{n}},D_{L})}{\bar{V}(D_{L})}.$ (2.24)

For simplicity, let us define here the evolution and magnification bias terms:

$\displaystyle\displaystyle b_{e}=\frac{\partial\ln n}{\partial\ln a}$ (2.25a)
$s=\begin{dcases*}{-}\frac{1}{5}\frac{\partial\ln n}{\partial\ln\sigma},&for
GWs;\\\ \frac{\partial\log_{10}n}{\partial{m}},&for SNIa\\\ \end{dcases*}$
(2.25b)

Note that the magnification biases for GWs and SNIa have opposite signs. In
fact, whilst a higher signal-to-noise ratio threshold will reduce the amount
of sources seen, a higher magnitude threshold will increase it, as the two
observables behave oppositely.

Thus we can write the general case:

$\displaystyle\displaystyle\Delta_{O}(D_{L},\hat{n})=\delta_{n}-\left[3-b_{e}{-}\frac{5}{\gamma}s\right]\gamma\frac{\delta
D_{L}}{\bar{D}_{L}}+\frac{\delta V({\bm{n}},D_{L})}{\bar{V}(D_{L})}\,.$ (2.26)

To proceed we therefore need to compute the luminosity distance perturbation
with respect to a background affine parameter, as well as the volume
perturbation.

###  2.2 Luminosity distance perturbation

To evaluate eq. (2.26) we now need to specify the appropriate perturbation in
luminosity distance, $\displaystyle\delta D_{L}/{\bar{D}_{L}}$. Therefore, we
redo the calculation to obtain the expression for the perturbation of the
luminosity distance at a background affine parameter in terms of the perturbed
quantities relevant for cosmology, and which are commonly used in publicly
available codes. Alternative calculations of $\displaystyle\delta D_{L}$ in a
perturbed FLRW universe can be found in [46, 47, 48]. Note that other authors
have computed the luminosity distance perturbation at the observed redshift
[42, 41], leading to a slightly different expression, as expected. For the
sources we consider in this paper we do not have access to an observed
redshift. Our purposes here is to determine the luminosity distance
perturbation with respect to the background luminosity distance at a
background affine parameter. This perturbation will be valid not only for
transients where we directly measure the luminosity distance, but also any
stochastic gravitational background.

We start from the distance-duality relationship between the luminosity
distance and the area distance for an observed source, which is valid as long
as geometric optics holds and photon number is conserved [63, 64], and is
given by:

$D_{L}(z_{s})=(1+z_{s})^{2}D_{A}(z_{s})\,.\ $ (2.27)

From [45], who consider the area distance, we have:

$\displaystyle\displaystyle D_{A}(z_{s})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bar{D}_{A}(z_{s})(1-\kappa(\bm{n}))$ (2.28)

which is the _area distance $\displaystyle D_{A}$ to the source at observed
redshift $\displaystyle z_{s}$_ at direction $\displaystyle\bm{n}$. Note that
barred objects refer to background quantities. The perturbation at the
observed redshift which appears in eq. (2.28) is then found in [42, 45]:

$\displaystyle\displaystyle\kappa(\bm{n})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\left(1-\frac{1}{\bar{r}\mathcal{H}}\right)\bm{v}\cdot\bm{n}+\frac{1}{2}\int_{0}^{\bar{r}}{\rm
d}r\
\frac{\bar{r}-r}{\bar{r}r}\Delta_{\Omega}(\Phi+\Psi)-\frac{\Psi}{\bar{r}\mathcal{H}}+\left(1-\frac{1}{\bar{r}\mathcal{H}}\right)\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi^{\prime}+\Psi^{\prime})$ (2.29)
$\displaystyle\displaystyle+(\Phi+\Psi)-\frac{1}{\bar{r}}\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi+\Psi)$

Note that at the “observed” redshift, multiplying both sides of eq. (2.28) by
$\displaystyle(1+z_{s})^{2}$ implies

$\displaystyle\displaystyle D_{L}(z_{s})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\bar{D}_{L}(z_{s})(1-\kappa(\bm{n}))\,,$ (2.30)

meaning that at the observed redshift the perturbation to the luminosity
distance is the same as for the area distance. This result is widely known and
agrees with the literature [42, 41].

However, to compute the over-density of tracers, we need instead the
perturbation as a function of background affine parameter distance, which we
take to be the background redshift. In this analysis we do not observe a
redshift, only an observed luminosity distance. In order to relate it to
background quantities we exploit the fact that an observed $\displaystyle
D_{L}$ has a corresponding $\displaystyle z_{s}$ which is relatable to the
background $\displaystyle\bar{z}$. The latter is clearly unobservable, but it
is required to describe the background distribution of sources.

Expanding $\displaystyle z_{s}=\bar{z}+\delta z$, the luminosity distance from
eq. (2.27) written in terms of background quantities is then

$\displaystyle\displaystyle D_{L}(z_{s},\bm{n})$
$\displaystyle\displaystyle=(1+\bar{z}+\delta
z)^{2}\,\bar{D}_{A}(\bar{z}+\delta z)\left[1-\kappa(\bm{n})\right]$
$\displaystyle\displaystyle=\left[(1+\bar{z})^{2}+2\delta
z(1+\bar{z})\right]\bar{D}_{A}(\bar{z})\left[1-\kappa(\bm{n})+\frac{1}{\bar{D}_{A}(\bar{z})}\frac{{\rm
d}D_{A}}{{\rm d}z}\bigg{|}_{z_{s}}\delta z\right]$
$\displaystyle\displaystyle=\bar{D}_{L}(\bar{z})\left[1-\kappa(\bm{n})+\frac{2\delta
z}{1+\bar{z}}+\frac{1}{\bar{D}_{A}(\bar{z})}\frac{{\rm d}D_{A}}{{\rm
d}z}\bigg{|}_{z_{s}}\delta z\right]$
$\displaystyle\displaystyle=\bar{D}_{L}(\bar{z})\left[1-\kappa(\bm{n})+\left(1+\frac{1}{\mathcal{H}\bar{r}}\right)\frac{\delta
z}{1+\bar{z}}\right]\,.$ (2.31)

Let us define the luminosity distance perturbation with respect to a
background affine parameter in the same manner as done in eq. (2.30), i.e.

${\delta D}_{L}\equiv D_{L}(z_{s},\bm{n})-\bar{D}_{L}(\bar{z})\,.$ (2.32)

Hence, the luminosity distance fluctuation we need is

$\frac{\delta
D_{L}}{\bar{D}_{L}}(\bar{z})=-\kappa(\bm{n})+\left(1+\frac{1}{{\mathcal{H}}\bar{r}}\right)\frac{\delta
z}{1+\bar{z}}.$ (2.33)

To linear order we have

$\displaystyle\displaystyle\frac{\delta
z}{1+\bar{z}}=\bm{v}\cdot\bm{n}-\Psi-\int_{0}^{\bar{r}}{\rm d}r\
(\Phi^{\prime}+\Psi^{\prime}),$ (2.34)

and combining it with eq. (2.29) in (2.33) gives

$\frac{\delta
D_{L}}{\bar{D}_{L}}=2\bm{v}\cdot\bm{n}-\frac{1}{2}\int_{0}^{\bar{r}}{\rm d}r\
\frac{\bar{r}-r}{\bar{r}r}\Delta_{\Omega}(\Phi+\Psi)-\Phi-2\Psi-2\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi^{\prime}+\Psi^{\prime})+\frac{1}{\bar{r}}\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi+\Psi)\,,$ (2.35)

which agrees with the perturbation identified in [33], and already computed in
[46, 47, 48]. Note that our derivation of the luminosity distance perturbation
at the background affine parameter is different from that found in [33], but
yields the same result. Our derivation is in fact similar to [47, 48], using
as starting point the distance-duality relationship.

###  2.3 Volume perturbation

The final part of our calculation of the number count over-density is to
determine the volume perturbation in eq. (2.26). To do so we follow closely
[29]. Note that we will use similar notation but we write quantities in terms
of luminosity distance. First let us start with the volume element:

$\displaystyle\displaystyle{\rm d}V$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sqrt{-g}\epsilon_{\mu\nu\alpha\beta}\,u^{\mu}{\rm
d}x^{\nu}\,{\rm d}x^{\alpha}\,{\rm
d}x^{\beta}=\sqrt{-g}\;\epsilon_{\mu\nu\alpha\beta}\,u^{\mu}\\!\frac{\partial
x^{\nu}}{\partial D_{L}}\\!\frac{\partial
x^{\alpha}}{\partial\theta_{s}}\\!\frac{\partial
x^{\beta}}{\partial\varphi_{s}}\\!\left|\frac{\partial(\theta_{s},\varphi_{s})}{\partial(\theta_{o},\varphi_{o})}\right|\,{\rm
d}D_{L}\,{\rm d}\theta_{o}\,{\rm d}\varphi_{o}$ (2.36)
$\displaystyle\displaystyle\equiv$ $\displaystyle\displaystyle
v(D_{L},\theta_{o},\varphi_{o})\,{\rm d}D_{L}\,{\rm d}\Omega_{o}~{},$

where $\displaystyle{\rm d}\Omega_{o}=\sin\theta_{o}{\rm d}\theta_{o}{\rm
d}\varphi_{o}$, and $\displaystyle\epsilon_{\mu\nu\alpha\beta}$ is the Levi-
Civita symbol. Given that the volume density in terms of the observer’s
coordinates is defined as

$v\equiv\sqrt{-g}\;\epsilon_{\mu\nu\alpha\beta}u^{\mu}\\!\frac{\partial
x^{\nu}}{\partial D_{L}}\\!\frac{\partial
x^{\alpha}}{\partial\theta_{s}}\\!\frac{\partial
x^{\beta}}{\partial\varphi_{s}}\\!\left|\frac{\partial(\theta_{s},\varphi_{s})}{\partial(\theta_{o},\varphi_{o})}\right|\,,$
(2.37)

we can compute the volume perturbation as

$\frac{\delta V}{V}=\frac{v-\bar{v}}{\bar{v}}=\frac{\delta v}{\bar{v}}\,.$
(2.38)

Recalling that we need to relate all quantities to background parameters (e.q.
formally defining $\displaystyle\bar{r}$ via $\displaystyle\bar{D_{L}}$), one
should note that the background volume density in luminosity distance space is
given by

$\bar{v}(\bar{D}_{L})=\frac{a^{4}\bar{r}^{2}(\bar{D}_{L})}{1+\mathcal{H}\bar{r}(\bar{D}_{L})}\,.$
(2.39)

The angles at the source and the observed angles are related via a small
perturbation:

$\displaystyle\displaystyle\theta_{s}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\theta_{0}+\delta\theta\,,$ (2.40)
$\displaystyle\displaystyle\varphi_{s}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\varphi_{0}+\delta\varphi\,.$ (2.41)

Then, to first order the determinant of the Jacobian of the transformation of
the angular position is

$\left|\frac{\partial(\theta_{s},\varphi_{s})}{\partial(\theta_{o},\varphi_{o})}\right|=1+\frac{\partial\delta\theta}{\partial\theta}+\frac{\partial\delta\varphi}{\partial\varphi}\,.$
(2.42)

Together with the fact that the determinant of the metric is
$\displaystyle\sqrt{-g}=a^{4}(1+\Psi-3\Phi)$, and that the 4-velocity of the
source is $\displaystyle u=(1-\Psi,v^{i})$, one finds that the perturbed
volume density can be expressed as

$v(D_{L})=a^{3}(1+\Psi-3\Phi)\Bigg{[}\frac{{\rm d}r}{{\rm
d}D_{L}}r^{2}\frac{\sin\theta_{s}}{\sin\theta_{o}}\left(1+\frac{\partial\delta\theta}{\partial\theta}+\frac{\partial\delta\varphi}{\partial\varphi}\right)-\left(\Psi\frac{{\rm
d}\bar{r}}{{\rm d}\bar{D}_{L}}+\bm{v}\cdot\bm{n}\frac{{\rm d}\bar{\eta}}{{\rm
d}\bar{D}_{L}}\right)\bar{r}^{2}\Bigg{]}.$ (2.43)

Using the angle perturbation in eq. (2.40) we have

$\sin{\theta_{s}}\simeq\sin{\theta_{0}}+\cos{\theta_{0}}\
\delta\theta=\sin{\theta_{0}}\left(1+\cot{\theta_{0}}\ \delta\theta\right)\,.$
(2.44)

One should also note that the comoving distance is a perturbed quantity, i.e,
$\displaystyle r^{2}=(\bar{r}+\delta r)^{2}\simeq\bar{r}^{2}+2\bar{r}\delta
r$. We then have

$\frac{{\rm d}\bar{r}}{{\rm
d}\bar{D}_{L}}=\left(1+\bar{z}+H\bar{r}\right)^{-1}=\frac{1}{(1+\bar{z})(1+\bar{r}\cal
H)}\,,$ (2.45)

and

$\frac{{\rm d}\bar{\eta}}{{\rm d}\bar{D}_{L}}=-\frac{{\rm d}\bar{r}}{{\rm
d}\bar{D}_{L}}=-\frac{1}{(1+\bar{z})(1+\bar{r}\cal H)}\,.$ (2.46)

We can write at linear order

$\frac{{\rm d}r}{{\rm d}D_{L}}=\frac{{\rm d}\bar{r}}{{\rm
d}\bar{D}_{L}}+\frac{{\rm d}\delta r}{{\rm d}\bar{D}_{L}}-\frac{{\rm d}\delta
D_{L}}{{\rm d}\bar{D}_{L}}\frac{{\rm d}\bar{r}}{{\rm
d}\bar{D}_{L}}=\left(\frac{{\rm d}\bar{r}}{{\rm d}\eta}+\frac{{\rm d}\delta
r}{{\rm d}\lambda}-\frac{{\rm d}\delta D_{L}}{{\rm d}\lambda}\frac{{\rm
d}\bar{r}}{{\rm d}\bar{D}_{L}}\right)\frac{{\rm d}\eta}{{\rm d}\bar{D}_{L}},$
(2.47)

where we take derivatives along the photon geodesic and set $\displaystyle{\rm
d}\eta={\rm d}\lambda$. Note as well that $\displaystyle{\rm d}\bar{r}/{\rm
d}\eta=-1$. Then, expanding eq. (2.47) with eqs. (2.45) and (2.46), and
substituting it into eq. (2.43), together with the expansion described in eq.
(2.44), we obtain:

$v(D_{L})=\frac{a^{4}\bar{r}^{2}}{1+\bar{r}\cal H}\left[1-3\Phi-\frac{{\rm
d}\delta r}{{\rm d}\lambda}+\frac{a}{1+\bar{r}\cal H}\frac{{\rm d}\delta
D_{L}}{{\rm d}\lambda}+\frac{2\delta
r}{\bar{r}}+\left(\cot\theta+\frac{\partial}{\partial\theta}\right)\delta\theta+\frac{\partial\delta\varphi}{\partial\varphi}+\bm{v}\cdot\bm{n}\right]\,.$
(2.48)

Finally we need to perturb $\displaystyle v$ around the background
$\displaystyle D_{L}$, i.e.

$\bar{v}(D_{L})=\bar{v}(\bar{D}_{L})+\frac{d\bar{v}}{d\bar{D}_{L}}\delta
D_{L}\,.$ (2.49)

where

$\frac{d\bar{v}}{d\bar{D}_{L}}=\frac{a^{4}\bar{r}^{2}}{1+\bar{r}\cal
H}\left[\frac{2}{\bar{r}\cal
H}+\frac{1}{1+\bar{r}\mathcal{H}}\left(\bar{r}\mathcal{H}\frac{{\cal
H}^{\prime}}{{\cal H}^{2}}-1\right)-4\right]\frac{\cal
H}{(1+z)(1+\bar{r}\mathcal{H})}\,.$ (2.50)

Then the volume perturbation is given by

$\displaystyle\displaystyle\frac{\delta v}{\bar{v}}({\bm{n}},D_{L})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{v(D_{L},\bm{n})-\bar{v}(D_{L})}{\bar{v}(D_{L})}$
(2.51) $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-3\Phi-\frac{{\rm d}\delta r}{{\rm
d}\lambda}+\frac{a}{1+\bar{r}\cal H}\frac{{\rm d}\delta D_{L}}{{\rm
d}\lambda}+\frac{2\delta
r}{\bar{r}}+\left(\cot\theta+\frac{\partial}{\partial\theta}\right)\delta\theta+\frac{\partial\delta\varphi}{\partial\varphi}+\bm{v}\cdot\bm{n}$
$\displaystyle\displaystyle-\left[\frac{2}{\bar{r}\cal
H}+\gamma\left(\frac{\cal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{r\cal{H}}\right)-4\right]\frac{{\cal
H}\delta D_{L}}{(1+z)(1+\bar{r}\mathcal{H})}\,.$

Using previous results shown in [29] one can start expressing the volume
perturbation in terms of perturbed quantities such as the potentials and
peculiar velocity. The perturbation in position is given by

$\delta r=\int_{0}^{\bar{r}}{\rm d}r(\Phi+\Psi)\,.$ (2.52)

Then

$\frac{{\rm d}\delta r}{{\rm d}\lambda}=-(\Phi+\Psi)\,,$ (2.53)

and

$2\frac{\delta r}{\bar{r}}=\frac{2}{\bar{r}}\int_{0}^{\bar{r}}{\rm d}r\
(\Phi+\Psi)\,,$ (2.54)

is the standard Sachs-Wolfe term. The angular perturbations give rise to the
traditional kappa term caused by lensing

$\left(\cot\theta+\frac{\partial}{\partial\theta}\right)\delta\theta+\frac{\partial\delta\varphi}{\partial\varphi}=-2\kappa_{g}=-\int_{0}^{\bar{r}}{\rm
d}r\ \frac{\bar{r}-r}{\bar{r}r}\Delta_{\Omega}(\Phi+\Psi)\,.$ (2.55)

Further, expressing $\displaystyle\delta D_{L}=\frac{\delta
D_{L}}{\bar{D}_{L}}\,\bar{D}_{L}$, we can write

$\displaystyle\displaystyle\frac{{\rm d}\delta D_{L}}{{\rm
d}\lambda}=\bar{D}_{L}\frac{{\rm d}}{{\rm d}\lambda}\left(\frac{\delta
D_{L}}{\bar{D}_{L}}\right)+\frac{\delta D_{L}}{\bar{D}_{L}}\frac{{\rm
d}D_{L}}{{\rm d}\lambda}\,.$ (2.56)

Finally, substituting eqs. (2.53),(2.54),(2.55) and (2.56) into eq. (2.51)
yields:

$\displaystyle\displaystyle\frac{\delta v}{\bar{v}}({\bm{n}},D_{L})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-2\Phi+\Psi+\frac{\gamma}{\cal H}\frac{{\rm
d}}{{\rm d}\lambda}\left(\frac{\delta
D_{L}}{\bar{D}_{L}}\right)+\bm{v}\cdot\bm{n}-2\kappa_{g}+\frac{2}{\bar{r}}\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi+\Psi)$ (2.57) $\displaystyle\displaystyle+\frac{\delta
D_{L}}{\bar{D}_{L}}\left(1-\gamma\left[\frac{2}{\bar{r}\cal
H}+\gamma\left(\frac{\cal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\cal{H}}\right)-4\right]\right)\,.$

We need to calculate the derivative with respect to the affine parameter along
the photon geodesic, which we take to be the conformal time when considering
the total derivative. We also take

$\frac{{\rm d}}{{\rm d}\lambda}=\frac{{\rm d}}{{\rm
d}\eta}=\frac{\partial}{\partial\eta}-\bm{n}^{i}\frac{\partial}{\partial\bm{x}^{i}}\,.$
(2.58)

Then

$\displaystyle\displaystyle\frac{{\rm d}}{{\rm d}\lambda}\frac{\delta
D_{L}}{\bar{D}_{L}}=-2(\mathcal{H}+\partial_{r})\bm{v}\cdot\bm{n}+\Phi^{\prime}+\partial_{r}\Phi-\frac{\Phi+\Psi}{r_{s}}+\frac{1}{2\bar{r}^{2}}\int_{0}^{\bar{r}}{\rm
d}r\ \left[2+\Delta_{\Omega}\right](\Phi+\Psi)\,,\ $ (2.59)

where we also used the Euler Equation, i.e, $\displaystyle
v_{r}^{\prime}+{\cal H}v_{r}+\partial_{r}\Psi=0$.

Finally, we obtain the expression for the volume perturbation in luminosity
distance space:

$\displaystyle\displaystyle\frac{\delta v}{\bar{v}}({\bm{n}},D_{L})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle-2\Phi+\Psi+\frac{\gamma}{\cal
H}\left[-2(\mathcal{H}+\partial_{r})\bm{v}\cdot\bm{n}+\Phi^{\prime}+\partial_{r}\Phi-\frac{\Phi+\Psi}{r_{s}}+\frac{1}{2\bar{r}^{2}}\int_{0}^{\bar{r}}{\rm
d}r\ \left[2+\Delta_{\Omega}\right](\Phi+\Psi)\right]$ (2.60)
$\displaystyle\displaystyle+\bm{v}\cdot\bm{n}-2\kappa_{g}+\frac{2}{\bar{r}}\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi+\Psi)+\frac{\delta
D_{L}}{\bar{D}_{L}}\left(1-\gamma\left[\frac{2}{\bar{r}\cal
H}+\gamma\left(\frac{\cal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\cal{H}}\right)-4\right]\right)\,.\
$

###  2.4 The observed density contrast in luminosity distance space

Now that we have all the components, we can expand eq. (2.26) using eqs.
(2.35), (2.60) to find:

$\displaystyle\displaystyle\Delta({\bm{n}},D_{L})$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\delta_{n}-2\Phi+\Psi+\frac{\gamma}{\mathcal{H}}\frac{{\rm
d}}{{\rm d}\lambda}\frac{\delta
D_{L}}{\bar{D}_{L}}+v_{r}-2\kappa_{g}+\frac{2}{\bar{r}}\int_{0}^{\bar{r}}{\rm
d}r\ (\Phi+\Psi)-\beta\frac{\delta D_{L}}{\bar{D}_{L}}\,$ (2.62)
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\delta_{n}+A_{D}(\bm{v}\cdot\bm{n})+A_{LSD}\partial_{r}(\bm{v}\cdot\bm{n})+A_{\Psi}\Psi+A_{\Phi}\Phi+A_{\Phi^{\prime}}\Phi^{\prime}+A_{\nabla\Phi}\partial_{r}\Phi$
$\displaystyle\displaystyle+\frac{1}{\bar{r}}\int_{0}^{\bar{r}}{\rm d}r\
(A_{TD}+A_{L}\Delta_{\Omega})(\Phi+\Psi)+A_{ISW}\int_{0}^{\bar{r}}{\rm d}r\
(\Phi^{\prime}+\Psi^{\prime})\,,\ $

recalling that
$\displaystyle\gamma\equiv\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}$,
and where the coefficient of the perturbation in $\displaystyle D_{L}$-space
is:

$\displaystyle\displaystyle\beta$ $\displaystyle\displaystyle\equiv$
$\displaystyle\displaystyle\gamma\left[\frac{2}{\bar{r}\mathcal{H}}+\gamma\left(\frac{\cal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\cal{H}}\right)-1-b_{e}\right]{-}5s-1\,,$
(2.63)

and where we grouped the terms of the various distortion effects into the
following coefficients:

$\displaystyle\displaystyle A_{D}$
$\displaystyle\displaystyle=1-2(\gamma+\beta)\,,$ (2.64a)
$\displaystyle\displaystyle A_{LSD}$ $\displaystyle\displaystyle=-2\
\frac{\gamma}{\mathcal{H}}\,,$ (2.64b) $\displaystyle\displaystyle A_{\Psi}$
$\displaystyle\displaystyle=1-\frac{1}{1+\bar{r}\mathcal{H}}+2\beta\,,$
(2.64c) $\displaystyle\displaystyle A_{\Phi}$
$\displaystyle\displaystyle=-2-\frac{1}{1+\bar{r}\mathcal{H}}+\beta\,,$
(2.64d) $\displaystyle\displaystyle A_{\Phi^{\prime}}$
$\displaystyle\displaystyle=\frac{\gamma}{\mathcal{H}}\,,$ (2.64e)
$\displaystyle\displaystyle A_{\nabla\Phi}$
$\displaystyle\displaystyle=\frac{\gamma}{\mathcal{H}}\,,$ (2.64f)
$\displaystyle\displaystyle A_{TD}$
$\displaystyle\displaystyle=2-\beta+\frac{1}{1+\bar{r}\mathcal{H}}\,,$ (2.64g)
$\displaystyle\displaystyle A_{L}$
$\displaystyle\displaystyle=\frac{1}{2}\left[\left(\frac{\bar{r}-r}{r}\right)(\beta-2)+\frac{1}{1+\bar{r}\mathcal{H}}\right]\,,$
(2.64h) $\displaystyle\displaystyle A_{ISW}$
$\displaystyle\displaystyle=2\beta\,.$ (2.64i)

These coefficients belong to, respectively, density, gradient of velocity
(luminosity distance space distortions — LSD), potentials and derivatives of
potentials, time-delay (Sachs-Wolfe), lensing and ISW. The expression is more
complex than the corresponding one in redshift space, and a comparison of the
terms can be found in Appendix D. In particular, the latter is missing the
term proportional to the gradient of the potentials. The magnification and
evolution biases enter in most terms (with the exception of the derivatives of
potentials and the LSD term); however, for the remainder of the paper these
are neglected, as their modelling is left for future work. One should note
that $\displaystyle A_{LSD}$ had already been identified and included in the
calculation of the angular power spectrum [20, 16], and the 3D power spectrum
and its multipoles [33, 49]. The lensing term $\displaystyle A_{L}$, with
kappa $\displaystyle\kappa_{g}$ and its line-of-sight gradient, is included in
[16, 33], although their equivalent $\displaystyle\beta$ coefficient has no
magnification bias included.

The number density contrast $\displaystyle\delta_{n}$ is in the Newtonian
gauge, but the bias is implicitly defined in the synchronous gauge with
respect to the matter density contrast. If the number density of sources
evolves with time then the density contrast in the two gauges is related via
the expression [28, 65]

$\delta_{n}=b\delta_{M}^{syn}+\left[b_{e}-3\right]\frac{\mathcal{H}v}{k}\,,\ $
(2.65)

where $\displaystyle\delta_{M}^{syn}$ is the matter density contrast in the
synchronous gauge and $\displaystyle b$ is the tracer bias.

Note that hereafter we use redshift and luminosity distance interchangeably as
a measure of distance. Although we will compute the angular power spectrum in
luminosity distance, redshift is the conventional measure used to locate the
line-of-sight distance to a bin of tracers and its width. One has a better
intuition of distance using $\displaystyle z$ rather than $\displaystyle
D_{L}$. At the background level one can convert between the two if we fix the
cosmology. However, at the perturbed level, the observed and background
quantities are related differently with the metric and energy-momentum tensor
perturbations, as we have shown here and is part of the point of our paper.

Figure 1: Comparison between kernels in luminosity distance space (blue) and
redshift space (orange) as a function of redshift. _Left_ : LSD/RSD; _Right:_
Doppler. Dashed lines indicate negative values. We remind the reader that
these are kernel amplitudes, and therefore dimensionless. Further, we set the
magnification and evolution biases, $\displaystyle s$ and $\displaystyle
b_{e}$ to zero to purely investigate the kernel.

The underlying matter density contrast in Newtonian gauge
$\displaystyle\delta_{n}$ will be the same in both redshift and luminosity
distance space, as expected. However, the correction terms from (2.62) differ
significantly from the RS case. Whilst we investigate the effects of some
important terms in this section and in section 3, an interested reader can
find the full terms comparison in appendix D (see as well [28, 66]). Figure 1
and figure 2 show the amplitudes of the terms that are most likely to be
detectable, namely LSD/RSD, Doppler and lensing. On the left of figure 1 we
show the amplitude of both LSDs (2.64b) and redshift space distortions (RSD).
We can see that the redshift dependence of both amplitudes is quite different.
Although the redshift and luminosity density perturbations are linear in the
peculiar velocity when written in terms of background quantities (a step
needed to compute the number counts fluctuation), their effect on the volume
perturbation has a different amplitude and redshift dependence. We see an
inversion of the strength of the LSD/RSD; LSDs are stronger than RSDs after
$\displaystyle z\approx 1.6$ (which agrees with results shown in [20]).

Notably, the Doppler term on the right of figure 1 is different altogether:
while in RS it contributes strongly near the observer and then tends to a
constant value close to zero, in LDS it is roughly constant, tending to 7 at
$\displaystyle z=0$ and 5 at high redshifts (see eq. (2.64a)). Note that we
fixed the biases $\displaystyle b_{e}=s=0$ but this does not change the
overall trend, only the limits at low and high-z. For this particular choice
of evolution and magnification biases, the amplitudes in the different spaces
also have opposite signs. Additionally the LDS Doppler term does not have an
$\displaystyle 1/r$ dependence near the observer. This is an effect of
expanding the observed luminosity distance in terms of the background
redshift, which brings an extra factor of $\displaystyle\delta z$. In fact,
this carries an extra velocity term which eliminates the dependency on
$\displaystyle 1/r$, and that will enter in the expression for the luminosity
distance perturbation, resulting in a non-zero Doppler term at high redshifts.

Figure 2:  Comparison between $\displaystyle A_{L}$ in luminosity distance
space (blue) and redshift space (orange). _Left_ : as a function of
$\displaystyle r(z)$; _Right:_ as a function of $\displaystyle\bar{r}(z)$
(i.e. source distance) at a fixed $\displaystyle r=0.5\bar{r}$. Dashed lines
indicate negative values. The magnification and evolution biases,
$\displaystyle s$ and $\displaystyle b_{e}$, were set to zero.

In figure 2 we plot the lensing coefficient in eq. (2.64h). On the left, we
fix the source at redshift $\displaystyle 5$ and look at $\displaystyle A_{L}$
as a function of $\displaystyle r(z)$ which is the integrand in eq. (2.62).
This implies that the overall trend of the redshift space curve won’t change
depending on the source redshift. However, the amplitude will depend on the
value of $\displaystyle s$, which we have set to zero. In the case of LDS, the
curve’s shape and the zero crossing point will depend on the actual values of
$\displaystyle b_{e}$ and $\displaystyle s$. This means that for some tracers
in LDS one may have a change from magnification near the source to de-
magnification further away, while in RS there is only a magnification effect.
In the right-hand panel, we evaluate the lensing integrand at the halfway
distance between source and observer, $\displaystyle r=0.5\bar{r}$, and
examine how it changes with $\displaystyle\bar{r}(z)$ (i.e the
source/background distance). While in redshift space we have a constant value,
in LDS it depends on the location of the source.

Figure 3: Comparison of the normalised lensing kernels from eq. (2.67) in
luminosity distance space (blue) and redshift space (orange).

This difference can be better seen when using the lensing kernel commonly used
for weak lensing. Weak gravitational lensing is encoded in the convergence
field $\displaystyle\kappa_{c}$, which can be expressed as a weighted
projection of the matter overdensity [67, 68]. In redshift space:

$\displaystyle\displaystyle\kappa_{c}(\bm{\hat{n}})=\int_{0}^{\infty}{\rm d}z\
W_{L}(z)\delta(r(z),\bm{\hat{n}},z)\,,\ $ (2.66)

where, assuming a flat Universe, the lensing kernel $\displaystyle W_{L}$
depends on the amplitude $\displaystyle A_{L}$ from eq. (2.64h) via:

$\displaystyle\displaystyle
W_{L}(z)=\frac{3}{2}\Omega_{m}\frac{H_{0}^{2}}{\mathcal{H}(z)}\frac{r(z)}{c}\int_{z}^{\infty}{\rm
d}z_{s}\ A_{L}(r(z_{s}))\ p(z_{s})\,,\ $ (2.67)

with $\displaystyle p(z_{s})$ being a normalised distribution of tracers. In
figure 3, we plot a comparison of the lensing kernel in luminosity distance
space and in redshift space. We adopt a fitting function from [68]
parameterising the redshift distribution of the galaxy sample from the Canada-
France-Hawaii Telescope Stripe 82 Survey [69, 70]:

$\displaystyle\displaystyle p(z_{s})=A\frac{z^{a}+z^{ab}}{z^{b}+c}\,,\ $
(2.68)

with $\displaystyle A=0.688$, $\displaystyle a=0.531$, $\displaystyle b=7.810$
and $\displaystyle c=0.517$. This is simply a toy model to compare the lensing
kernel in luminosity distance to previous work in redshift space. In LDS the
kernel is broader and thus lensing is sensitive to a wider range of redshifts.
The particular choice of distribution of sources will also affect this,
together with adding the impact of magnification and evolution biases. In
particular, for a specific choice of distribution the resulting kernel can
change sign.

##  3 The angular power spectrum in luminosity distance space

###  3.1 A comparison with power spectra in redshift space

In this section we use the full expression shown in eq. (2.62) to evaluate
numerically the angular power spectrum, and the relevance of the relativistic
corrections. We modified the publicly available code
CAMB444https://github.com/cmbant/CAMB together with its python wrapper pyCAMB,
and implemented the option to calculate the auto and cross-correlation angular
power spectra in luminosity distance space, including correlations between
distinct bins in distance, using the expression in (2.26).555We will make the
code publicly available shortly. In order to compute the angular power
spectrum of sources within a luminosity distance range $\displaystyle
D_{L}\in[D_{L,i}-\Delta D_{L,i}/2,D_{L,i}+\Delta D_{L,i}/2]$, where
$\displaystyle\Delta D_{L,i}$ is the size of the _ith_ bin, we expand the
number density contrast in luminosity distance space into spherical harmonics,
i.e.,

$\Delta({\bm{n}},D_{L,i})=\sum^{\infty}_{\ell=0}\sum^{\ell}_{m=-\ell}a_{\ell
m}(D_{{L,i}})Y_{\ell m}(\hat{\bm{n}})\,.$ (3.1)

The angular power spectrum $\displaystyle C_{\ell}$ is the two-point function,
equivalent to the variance of the $\displaystyle a_{\ell m}$ coefficients, and
is related with the primordial power spectrum $\displaystyle{\cal P}_{\cal R}$
via

$C^{ij}_{\ell}=\langle a_{\ell
m}(D_{L,i})a^{\ast}_{\ell^{\prime}m^{\prime}}(D_{L,j})\rangle=4\pi\int{\rm
d}\ln k\ \Delta^{i}_{\ell}(k,D_{L,i})\Delta^{j}_{\ell}(k,D_{L,j})\ {\cal
P}_{\cal R}(k)~{}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,,$ (3.2)

where $\displaystyle\Delta^{i}_{\ell}$ is the transfer function of the _ith_
bin and combines the distribution of sources, the window function and the
transfer functions of each term in eq. (2.62). Here $\displaystyle i$ and
$\displaystyle j$ stand for sample bins in luminosity distance and/or
redshift, which can be distinct. A comprehensive derivation of the transfer
functions in the bin in harmonic space is found in Appendix A.

Figure 4: Percentage difference between the angular power spectrum in
luminosity distance space (LDS) and redshift space (RS) for two different
binning sizes (top: $\displaystyle\sigma=0.05$, bottom:
$\displaystyle\sigma=0.2$). At low $\displaystyle\ell$ the two can differ
quite significantly, particularly at high redshift. However, at smaller scales
the difference tends to a constant depending on the redshift examined. The
overall difference is also related to the binning used, thus on the
uncertainty on the localisation of the tracer, as better accuracy on the
distance will reduce the difference between the angular power spectra in the
two spaces.

We start from the expression of the overdensity (eq. (2.62)) and the
definition of $\displaystyle C_{\ell}$ to construct
$\displaystyle\Delta^{i}_{\ell}$, as well as the theoretical harmonic space
transfer functions for each contribution in eq. (2.62). These are then
implemented in CAMB. The code also allows the input of the values of both
magnification and evolution bias, although for the following examples we
choose to set them to zero. Note that [16] also changed CAMB to provide a code
to compute the angular power spectra in luminosity distance space. We go
beyond their work and include all contributions present in eq. (2.62) without
the Limber approximation. We also include the magnification bias in our
calculation and code. The latter also allows for cross-correlations between
luminosity distance and redshift spaces.

Before looking at the individual contributions, we compare the angular power
spectrum in the two different spaces. For simplicity we will consider generic
tracers throughout this paper that live in either space with idealised values
for its biases, i.e., $\displaystyle b=1,s=b_{e}=0$. Therefore, one can
directly compare the expressions in luminosity distance space (LDS) and
redshift space (RS). In figure 4 we plot the relative difference between the
angular power spectrum in LDS and the one in RS at multiple redshifts. We use
broad Gaussian windows in the top plot and narrow bins in the bottom plot.
Whilst broad windows are motivated by the large uncertainty in the estimation
of the luminosity distance for GWs [71, 72], narrower bins are used in galaxy
clustering and thus are appropriate for a comprehensive comparison. Further,
SNIa distance uncertainties for LSST will lie in between these two regimes
[73, 74].

Interestingly, using large bins (bottom figure 4), the large scales of the
angular power spectrum can experience a $\displaystyle 50\%$ difference at
$\displaystyle z=2.5$. But at smaller scales all angular power spectra tend to
a constant which depends on redshift. At low redshift, the difference is
roughly $\displaystyle\sim 10\%$, implying a relatively small difference
between the two cases when looking at smaller scales. These differences are
reduced significantly when looking at thinner bins (top figure 4), however, we
remind the reader that the more realistic scenario when using GWs/SNIa would
need larger uncertainties, and thus would be more akin to the bottom plot. As
we will see in figures 6 and 7, the bulk of the difference at low
$\displaystyle\ell$ is due to the lensing contribution. Instead, at higher
$\displaystyle\ell$ the two main contributions are the density auto-
correlation (which is identical in the two spaces), and LSD/RSD terms; in
particular, the latter gives a constant difference between LDS and RS, which
is then visible at smaller scales when the differences in all other
contributions become negligible.

We note that this constant offset at high $\displaystyle\ell$ decreases
towards $\displaystyle z\sim 1.5$ and increases afterwards. Looking back at
figure 1, the amplitude of the LSDs becomes stronger than that of the RSDs
around this same value of redshift. In $\displaystyle\Lambda$CDM,
$\displaystyle z_{t}\simeq 1.5$ is the turnover point for the angular diameter
distance, and assuming no shear in a preferential direction, to a safe
approximation, the angular diameter distance is the same as the area distance
[64] used to identify the perturbation in luminosity distance (in eq. (2.27)).
The imprint of the turnover is then visible in how the difference in the
angular power spectra between LDS and RS evolves with redshift. However, this
behaviour only appears when considering displacement-related effects, i.e.
LSD/RSD, Doppler and ISW. We suggest this is the effect of the extra
$\displaystyle\delta z$ term in the last line of eq. (2.27), as discussed at
the end of section 2.4.

Figure 5: Angular power spectrum for the generic tracers (see text) at
$\displaystyle z=0.5$ (top) and at $\displaystyle z=1.5$ (bottom) and the
auto-correlations of different contributions. Left: in luminosity distance
space. Right: in redshift space.

More fundamentally, what figure 4 shows is that the number counts expression
in redshift space is not a good approximation to estimate clustering of GW
mergers, which are intrinsically observed in LDS. One should note that the
poor angular resolution of such experiments imposes in effect a beam, which
when multiplied by the $\displaystyle C_{\ell}$ dampens the signal at high
$\displaystyle\ell$. Here we did not model this effect in detail, as it is
beyond the scope of this paper.

###  3.2 The relevance of relativistic corrections

We have seen that the angular power spectrum has a different amplitude in the
two spaces, but we have not identified the origin of such a difference. In
figure 5 we compare the angular power spectrum in luminosity distance space,
including the auto-correlations of each contribution in the density contrast
in equation (2.62), with the one in redshift case. We plot the $\displaystyle
C_{\ell}$ up to $\displaystyle\ell_{max}=100$ considering the error in
localisation of a GW event for 3G detectors like ET or CE [75, 3, 21, 22, 23,
24] and that $\displaystyle\ell_{max}=180^{\circ}/\theta$, where
$\displaystyle\theta$ is the angular uncertainty [36]. At first glance, one
can see that lensing has a stronger contribution in LDS as opposed to RS,
together with other terms such as the potentials. However the other
contributions are not so easily compared, especially because of the log scale
of the plot. Nonetheless, we can identify the most relevant terms: lensing
(green line) and velocity gradient (orange line).

Figure 6: Comparison of the auto- and cross-terms contributions to the total
angular power spectrum in luminosity distance space for a generic tracer (see
text) at different redshifts. The higher the redshift and the more the lensing
term contributes to the total.

Figure 7: Comparison of the auto- and cross-terms contributions to the total
angular power spectrum in redshift space for a generic tracer (see text) at
different redshifts. Looking at figure 6 we note that the lensing term in
z-space is weaker at higher redshift compared to the LDS case.

To better gauge the relative importance of the main terms, we then plot in
figure 6 their contributions to the total angular power spectrum in
$\displaystyle D_{L}$-space including the main auto- and cross-term
correlations at several redshifts. As expected, the lensing auto-correlation
contribution becomes more important as redshift increases. However, comparing
with redshift space (figure 7) we see that at high redshifts lensing has a
stronger contribution in LDS as opposed to RS. In particular, we note that
both lensing x lensing and density x lensing in LDS are greater than the RS
case by roughly a factor of 2 when looking at $\displaystyle z=2.5$. This
clearly shows this contribution to be the main difference between the two
spaces, which then drives the percentage difference show in figure 4. In fact,
the factor of 2 between the main contribution, i.e. lensing, is then reflected
in a $\displaystyle 50\%$ difference at low $\displaystyle\ell$ between the
total angular power spectra in the two cases. Additionally the correlation
between the density and lensing is always negative in the RS case, while in
LDS it can change sign due to the extra term in $\displaystyle A_{L}$.

Figure 8: Comparison of the most important contributions (both auto- and
cross-terms) when cross-correlating different redshift bins. We fix a redshift
tracer, e.g. a generic galaxy survey, in the foreground at $\displaystyle
z=0.5$, and a tracer in LDS (left) or in RS (right) in the background. The
latter is set to $\displaystyle z=1$ for the top plots and $\displaystyle
z=1.5$ for the bottom ones.

Finally, we show an example of cross-correlating two different redshift bins
(see figure 8). We first fix a tracer in LDS at a higher bin and cross-
correlate with a lower bin in redshift space. This is what is shown on the
left column of figure 8 together with the contributions to the cross-bin
angular power spectrum in $\displaystyle D_{L}$-space, including the main
auto- and cross-term correlations. The foreground survey is assumed to be a
redshift space galaxy survey, as these are the most widely available. We then
repeat this by setting both bins to redshift space, which is shown on the
right column of figure 8. The former could represent GW/SNIa signals in the
background distorted by a galaxy in front, while the latter is the
conventional case with a galaxy in the background and another one in front.
Whilst we fix the foreground tracer at $\displaystyle z=0.5$, we show plots
with the background one at $\displaystyle z=1$ (top plots) and $\displaystyle
z=1.5$ (bottom plots). We include the three main terms and their correlations
with the density term. Interestingly, whilst the relevant contributions to the
angular power spectra on the top left are all almost completely positive, the
ones on the other plots are not, suppressing the density term on very large
scales and therefore the total angular power spectrum. In all the plots we see
that the density x lensing term is a key contributor to the total power,
especially on very large scales. In addition the lensing-lensing correlation
becomes ever more important as the redshift of the background survey
increases. Although this is both true for the RS x LDS and RS x RS, the sign
of these two contributions is not the same in the two cases. This is reflected
in the amplitude of the cross-bin angular power which can be more than an
order of magnitude different. Therefore, this reinforces the notion that when
using LSS tracers, we cannot model the angular power spectrum in redshift
space as a proxy to the one in luminosity distance space. One should also note
that the shape and amplitude of the lines depends on the right choice of
evolution and magnification biases, which we fix to zero in all cases.

###  3.3 Importance of the relativistic corrections in a cosmic variance
limited survey

Figure 9: Error in measuring a LDS $\displaystyle C_{\ell}$ at different
redshifts. In black we plot the density auto-correlation as reference.

We have computed the full expression for the number density contrast in
luminosity distance space including all relativistic effects. However, we have
yet to show the relevance of this in the observed signal. Our goal in this
subsection is not to provide any forecast on the detectability of the angular
power spectrum of a source survey in luminosity distance. Neither do we intend
to compare future surveys to determine which should include GR corrections, as
we will focus on this in upcoming work. We therefore look at the best case
scenario, i.e., an unrealistic cosmic variance limited survey.

The error in measuring the angular power spectrum $\displaystyle C_{\ell}$ is
simply given by

$\Delta C_{\ell}=\sqrt{\frac{2}{{(2\ell+1)}f_{\rm
sky}}}\left(C_{\ell}+\mathcal{N}_{\ell}\right)\,,$ (3.3)

where $\displaystyle f_{\rm sky}$ is the fraction of the observed sky and
$\displaystyle\mathcal{N}_{\ell}$ is the noise. For now, let us assume the
best case scenario, even if unrealistic, and take the cosmic variance limit,
where shot-noise is much smaller than the signal, and a full sky survey as
well ($\displaystyle f_{\rm sky}=1$). We plot the angular power spectrum and
its variance for this idealised case in figure 9 at different redshifts. We
also plot the density term only in solid black. Although we are working within
an idealised scenario, one can see that at low redshifts all the correction
terms would be well within the uncertainty, i.e., we may never be able to
distinguish the different contributions at low z. In this case, one can
neglect the relativistic corrections, as any realistic sample will have shot-
noise, which increases the uncertainty on the angular power spectrum.
Furthermore, any realistic experiment will introduce sample selection
uncertainties which further increase the error budget. However, as redshift
increases this is no longer true, and the signal cannot be explained by the
density term only. For redshifts $\displaystyle z\gtrsim 1$ the difference
between the density term and the GR corrections becomes ever bigger.
Therefore, as long as the survey is not shot-noise dominated, one is required
to model the signal properly. Which terms have to be considered and the impact
of neglecting them on the estimation of parameters can only be assessed for a
specific experiment, including the target sample and any selection effects. It
is beyond the scope of this paper as we only wanted to create a bottom line of
the relevance of the GR corrections in LDS. One should not that neglecting
relativistic correction at low redshifts in the cosmic variance limit would
only be possible in a single tracer case, since one can go beyond cosmic
variance using multiple large-scale structure tracers [76, 77].

##  4 Summary and conclusions

With the advent of future gravitational wave detectors [15, 27] and optical
surveys which will detect many more supernovae [19], we can start to envisage
using such transient objects as tracers of large-scale structure.
Observationally one can only determine the luminosity distance to these
objects. Therefore, in this paper, we explored clustering in luminosity
distance space. We started by computing the number density contrast in
luminosity distance space in terms of the volume and luminosity distance
perturbations. We also defined the magnification and evolution biases for the
typical astrophysical objects which live in luminosity distance space. We then
computed the luminosity distance perturbation with respect to the background
redshift as well as the volume perturbation. We note that our result agrees
with [46, 47, 48]. We finally computed the full expression for the number
counts fluctuation including all relativistic effects in LDS presented in eq.
(2.62), which is the main result of this paper. This result agrees with a
previous calculation from [33], and includes more terms than previously
considered [33, 49, 20, 16]. We therefore provide an alternative full
derivation for the number density contrast in luminosity distance space. We
only considered transient events in luminosity distance, but our calculation
is in principle valid for the stochastic gravitational background with an
appropriate magnification bias.

In deriving the density contrast we also identified how the effect of the
evolution of the sources across redshift (the evolution bias) and
perturbations in the number of detectable sources (the magnification bias)
should be included in the calculation of the number density contrast. We
derived the effect of the latter for both SNIa and GWs, considering magnitude-
and signal-to-noise-limited surveys, respectively. Detailed modelling of these
biases for realistic GWs and SNIa surveys, including their effects on the
number counts are left for upcoming work [78].

The full expression in LDS is significantly more complex than that in RS, with
contributions such as lensing including additional terms. A comparison between
the amplitude of each contribution is shown explicitly in 1 in appendix D.
Some terms behave substantially differently, such as the Doppler and the
lensing ones. In particular, the Doppler term does not tend to zero far from
the observer, rather it becomes constant. One surprising aspect of clustering
in luminosity distance space is the sensitivity to the Doppler effect which
comes via the redshift perturbation. The area distance perturbation is a
purely geometric effect giving rise to the normal lensing contribution in the
Jacobi map [45]. When moving to luminosity distance the reciprocity relation
brings in the redshift, and therefore the Doppler contribution to the redshift
perturbation.

We then explored the angular power spectrum in LDS. To do so, we changed the
publicly available cosmological solver CAMB, implementing the luminosity
distance number counts transfer functions. The code not only computes the
angular power for luminosity distance sources, but also their correlation with
other tracers of dark matter such as galaxy surveys. The code will be made
available in the future. With this tool, we found a significant percentage
difference with the one in redshift space at large scales. At higher
redshifts, this difference increases and can reach $\displaystyle 50\%$ at
$\displaystyle z=2.5$. This is extremely important as future gravitational
wave detectors have large effective beams, meaning that only low
$\displaystyle\ell$ multipoles would be accessible; these are the same
multipoles where the number density contrast in redshift space is not a good
approximation of the one in luminosity distance space. This should have a
significant impact on the detectability and best-fit values of the
astrophysical biases of gravitational wave mergers. These in turn can be used
to characterise the distribution and evolution of black holes with cosmic
time.

A notable difference between the two spaces is the importance of lensing,
especially in cross-correlations. Cross-correlating different redshift bins
shows different behaviours of the lensing terms, namely the density x lensing
contribution. In the case examined, whilst the angular power spectrum in LDS
is fully positive at higher redshifts, in redshift space it crosses zero at
large scales.

Therefore, mismodelling the angular power spectra when cross-correlating a
background GWs survey with a foreground lensing object, e.g. a galaxy survey,
will lead to a misunderstanding of the gravitational potentials in the line-
of-sight. In fact, if we attain a cosmic variance limited sample in the
future, not properly including all GR effects will lead to a total mismatch
between the observed and the density-only contribution (as we have shown in
figure 9).

Several works in the literature have studied how transient events, such as GW
mergers and SNIa, can be used as tracers of the dark matter distribution in
the universe. Our contribution to this debate was centered in providing a full
calculation of the number density contrast, including all GR effects, in the
space where distances are observed. In contrast with galaxy surveys, here we
estimate luminosity distances rather than redshifts. We have shown that this
difference is substantial and should be included in the modelling and
forecasts.

## Acknowledgments

We are pleased to thank Charles Dalang, Roy Maartens and Stefano Camera for
useful discussions. JF thanks the support of Fundação para a Ciência e a
Tecnologia (FCT) through the research grants UIDB/04434/2020 and
UIDP/04434/2020 and through the Investigador FCT Contract No.
2020.02633.CEECIND/CP1631/CT0002. JF also thanks the hospitality of Astronomy
Unit of QMUL and the University of the Western Cape where part of this work
was developed. We thank the support of FCT and the Portuguese Association of
Researcher and Students in the UK (PARSUK) under the Portugal - United Kingdom
exchange program Bilateral Research Fund (BRF). T.B. is supported by ERC
Starting Grant SHADE (grant no. StG 949572) and a Royal Society University
Research Fellowship (grant no.
URF$\displaystyle\backslash$R1$\displaystyle\backslash$180009). CC is
supported by the UK Science & Technology Facilities Council Consolidated Grant
ST/P000592/1. S.Z. acknowledges support by the Parent Fund University of
London.

##  Appendix A The luminosity distance density contrast in harmonic space

In section 2.4 we presented the number density contrast in luminosity distance
space (see eq. 2.62). In practice one looks at the angular distribution of
sources and the statistical structure of its distribution to relate it with
cosmological parameters and the dark matter density contrast. In this appendix
we will revise the multipole expansion in harmonic space and compute the
angular power spectrum $\displaystyle C_{\ell}$. We will also show how we
derive the transfer functions in harmonic space and summarise our
implementation in CAMB.

###  A.1 Decomposition into spherical harmonics

By definition the number density contrast is a perturbed quantity with spatial
average $\displaystyle\langle\Delta\rangle=0$. Here we work in shells in the
celestial sphere and, thus, do a spherical harmonic decomposition of the field
$\displaystyle\Delta({\bm{n}},D_{L})$ into

$\Delta({\bm{n}},D_{L})=\sum^{\infty}_{\ell=0}\sum^{\ell}_{m=-\ell}a_{\ell
m}(D_{L})Y_{\ell m}(\hat{\bm{n}})\,.$ (A.1)

The only dependence on the distance is encoded in the $\displaystyle a_{\ell
m}$, which are the projection of the field onto the spherical harmonic basis.
It follows that the $\displaystyle a_{\ell m}$ can then be written in terms of
the field as (see appendix C)

$a_{\ell m}(D_{L})=\int{\rm d}\Omega_{{\bm{n}}}\
\Delta({\bm{n}},D_{L})Y^{*}_{\ell m}(\hat{\bm{n}})\,.$ (A.2)

The $\displaystyle a_{\ell m}$ will keep the same statistical properties of
$\displaystyle\Delta$, i.e., $\displaystyle\langle a_{\ell m}\rangle=0$. It is
in effect a random variable with null average where the 2-point function
describes its statistical distribution. It is this 2-point function of the
$\displaystyle a_{\ell m}$ which we call the angular power spectra
$\displaystyle C_{\ell}$, and is defined by

$\langle a_{\ell m}a^{*}_{\ell^{\prime}m^{\prime}}\rangle\equiv C_{\ell}\
\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$ (A.3)

So far we have assumed an infinitesimal shell in $\displaystyle D_{L}$, which
in practice will never happen. In reality, data is binned into intervals of
distance and the “observed” $\displaystyle a_{\ell m}^{i}$ in the i _th_ -bin
is then a weighted quantity, i.e.,

$a_{\ell m}^{i}(\bar{D}_{L})=\int{\rm d}D_{L}\ w_{i}(D_{L})\ a_{\ell
m}(D_{L})\,,$ (A.4)

where $\displaystyle w_{i}(D_{L})$ is the normalized distribution function and
$\displaystyle\bar{D}_{L}$ is the central distance of the bin. The weighting
is usually given both by the distribution of sources $\displaystyle\bar{n}$
along the line-of-sight and by the window function $\displaystyle
W_{i}(D_{L})$ we have chosen, i.e. how we bin the tracers considered. The
window function can be e.g. a top-hat, a gaussian or an error function and
will depend on the type of survey. In all our analysis we opted for broad
gaussian windows. In general one can take the weighting function to be

$w(D_{L})=\frac{1}{\int{\rm d}D_{L}^{\prime}\
W_{i}(D_{L}^{\prime})\bar{n}(D_{L}^{\prime})}W_{i}(D_{L})\bar{n}(D_{L})\,.$
(A.5)

###  A.2 The angular power spectra

Let us consider for now the density term $\displaystyle\delta_{M}$. Note that
the bias is in this case an overall amplitude so we can neglect it for now. On
what follows the comoving distance is implicitly defined as a function of the
luminosity distance, i.e., $\displaystyle r\equiv r(D_{L})$. We can write the
density $\displaystyle a_{\ell m}$ as

$\displaystyle\displaystyle a_{\ell m}^{\delta}(D_{L})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\int\ {\rm
d}\Omega_{{\bm{n}}}\ \delta_{M}({\bm{n}},r)Y^{*}_{\ell m}(\hat{n})$ (A.6)
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(2\pi)^{3}}\int\int{\rm
d}\Omega_{{\bm{n}}}\ {\rm d}^{3}k\ \delta_{\vec{k}}(D_{L})\ Y^{*}_{\ell
m}(\hat{n})\ e^{ikr\hat{k}.{\bm{n}}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{1}{(2\pi)^{3}}\int\int{\rm
d}\Omega_{{\bm{n}}}\ {\rm d}^{3}k\ \delta_{\vec{k}}(D_{L})\ Y^{*}_{\ell
m}(\hat{n})\
\sum_{\ell^{\prime}=0}^{\infty}(2\ell^{\prime}+1)i^{\ell^{\prime}}{\cal
P}_{\ell^{\prime}}(\hat{k}.{\bm{n}})j_{\ell^{\prime}}(kr)$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{i^{\ell}}{2\pi^{2}}\int{\rm d}^{3}k\
\delta_{\vec{k}}(D_{L})\ Y^{*}_{\ell m}(\hat{k})\ j_{\ell}(kr)\,.$

In the second line we Fourier transform $\displaystyle\delta_{M}$, while in
the third and fourth lines we used properties of Legendre polynomials and
spherical harmonics (see appendix C). Then, using Eq. A.4, the two point
function between two bins is

$\displaystyle\displaystyle\left\langle a_{\ell
m}^{\delta}(D_{L,i})a^{\delta*}_{\ell^{\prime}m^{\prime}}(D_{L,j})\right\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Bigg{\langle}\left[\int{\rm d}D_{L}\
w^{i}(D_{L})\frac{i^{\ell}}{2\pi^{2}}\int{\rm d}^{3}k\
\delta_{\vec{k}}(D_{L})\ Y^{*}_{\ell m}(\hat{k})\ j_{\ell}(kr)\right]\times$
(A.7) $\displaystyle\displaystyle\left[\int{\rm d}D_{L}^{\prime}\
w^{j}(D_{L}^{\prime})\frac{(-i)^{\ell^{\prime}}}{2\pi^{2}}\int{\rm
d}^{3}k^{\prime}\ \delta^{*}_{\vec{k}^{\prime}}(D_{L}^{\prime})\
Y_{l^{\prime}m^{\prime}}(\hat{k}^{\prime})\
j_{\ell^{\prime}}(k^{\prime}r^{\prime}))\right]\Bigg{\rangle}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{(-1)^{\ell^{\prime}}i^{\ell+\ell^{\prime}}}{4\pi^{4}}\int{\rm
d}D_{L}\ w^{i}(D_{L})\int{\rm d}^{3}k\ Y^{*}_{\ell m}(\hat{k})\
j_{\ell}(kr)\times$ $\displaystyle\displaystyle\int{\rm d}D_{L}^{\prime}\
w^{j}(D_{L}^{\prime})\int{\rm d}^{3}k^{\prime}\
Y_{l^{\prime}m^{\prime}}(\hat{k}^{\prime})\
j_{\ell^{\prime}}(k^{\prime}r^{\prime})\times$
$\displaystyle\displaystyle\left\langle\delta_{\vec{k}}(D_{L})\delta^{*}_{\vec{k}^{\prime}}(D_{L}^{\prime})\right\rangle$

Here it is useful to introduce transfer function that split the redshift
evolution from the primordial inflationary perturbation. We refer to these as

${\cal T}_{\cal Q}(D_{L},k)\equiv\frac{{\cal Q}(D_{L},k)}{{\cal
R}(D_{L,initial},k)}\,,$ (A.8)

where

$\langle{\cal R}(k){\cal
R}^{*}(k^{\prime})\rangle=(2\pi)^{3}\delta^{3}(\vec{k}-\vec{k}^{\prime})P_{\cal
R}(k)\,,$ (A.9)

and $\displaystyle P_{\cal R}(k)$ is the power spectrum primordial
perturbations and $\displaystyle\cal R$ is the curvature perturbation. This
way

$\langle{\cal Q}(D_{L},k){\cal
Q}^{*}(D_{L}^{\prime},k^{\prime})\rangle=(2\pi)^{3}\delta^{3}(\vec{k}-\vec{k}^{\prime}){\cal
T}_{\cal Q}(D_{L},k){\cal T}_{\cal Q}(D_{L}^{\prime},k)P_{\cal R}(k)\,.$
(A.10)

The primordial power spectrum is given by

${\cal P}_{\cal R}(k)\equiv\frac{k^{3}}{2\pi^{2}}P_{\cal R}(k)=A_{\cal
S}\left(\frac{k}{k_{p}}\right)^{n_{\cal S}-1}\,,$ (A.11)

where $\displaystyle k_{p}$ is a pivot scale and $\displaystyle A_{\cal S}$ is
the primordial amplitude of the power spectrum and $\displaystyle n_{\cal S}$
its spectral index. With this in mind we can then write

$\left\langle\delta_{\vec{k}}(D_{L})\delta^{*}_{\vec{k}^{\prime}}(D_{L}^{\prime})\right\rangle=(2\pi)^{3}\delta^{3}(\vec{k}-\vec{k}^{\prime}){\cal
T}_{\cal\delta}(D_{L},k){\cal T}_{\cal\delta}(D_{L}^{\prime},k)P_{\cal
R}(k)\,.$ (A.12)

We can then simplify eq. A.7 as

$\displaystyle\displaystyle\left\langle a_{\ell
m}^{\delta}(\bar{D}_{L,i})a^{\delta*}_{\ell^{\prime}m^{\prime}}(\bar{D}_{L,j})\right\rangle$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{2(-1)^{\ell^{\prime}}i^{\ell+\ell^{\prime}}}{\pi}\left(\int{\rm
d}\Omega_{\hat{k}}\ Y^{*}_{\ell
m}(\hat{k})Y_{l^{\prime}m^{\prime}}(\hat{k})\right)\int k^{2}{\rm d}k\ P_{\cal
R}(k)$ (A.13) $\displaystyle\displaystyle\left[\int{\rm d}D_{L}\
w^{i}(D_{L}){\cal T}_{\cal\delta}(D_{L},k)\ j_{\ell}(kr)\right]\left[\int{\rm
d}D_{L}^{\prime}\ wj(D_{L}^{\prime}){\cal T}_{\cal\delta}(D_{L}^{\prime},k)\
j_{\ell^{\prime}}(kr^{\prime})\right]$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle 4\pi\int{\rm d}\ln k\
\Delta^{\delta}_{\ell}(k,\bar{D}_{L,i})\Delta^{\delta}_{\ell}(k,\bar{D}_{L,j}){\cal
P}_{\cal R}(k)~{}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,,$

where we implicitly define the effective transfer function
$\displaystyle\Delta^{\delta}$ as

$\Delta^{\delta}_{\ell}(k,\bar{D}_{L,i})=\int_{0}^{\infty}{\rm d}D_{L}\
w^{i}(D_{L})b(D_{L}){\cal T}_{\cal\delta}(D_{L},k)\ j_{\ell}(kr)\,,$ (A.14)

where we reintroduced the bias $\displaystyle b$. Note that this derivation
holds for the potentials as well, but not for their gradient or the velocity
terms. These include a derivative term with respect to $\displaystyle k$ (in
Fourier space), which affects the integrand of the transfer function (see e.g.
$\displaystyle\Delta_{\ell}^{\nabla\Phi}$ and
$\displaystyle\Delta_{\ell}^{LSD}$ in appendix B). As CAMB is written in
conformal time $\displaystyle\eta$ it is more convenient to present
$\displaystyle\Delta^{\delta}$ as

$\Delta^{\delta}_{\ell}(k,D_{L,i})=-\int^{\eta_{0}}_{0}{\rm d}\eta\ \frac{{\rm
d}D_{L}}{{\rm d}\eta}w^{i}(\eta)b(\eta){\cal
T}_{\cal\delta}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,,$ (A.15)

where in units of $\displaystyle c=1$, $\displaystyle r=\eta_{0}-\eta$, and

$\frac{{\rm d}D_{L}}{{\rm d}\eta}=-\frac{1+(\eta_{0}-\eta)\cal H}{a}\,.$
(A.16)

The angular power spectrum is then given by

$C^{ij}_{\ell}=4\pi\int{\rm d}\ln k\
\Delta^{i}_{\ell}(k,D_{L,i})\Delta^{j}_{\ell}(k,D_{L,j})\ {\cal P}_{\cal
R}(k)\,,$ (A.17)

with

$\Delta_{\ell}=\Delta^{\delta}_{\ell}+\Delta^{LSD}_{\ell}+\Delta^{L}_{\ell}+\Delta^{D}_{\ell}+\Delta^{\Phi}_{\ell}+\Delta^{\Psi}_{\ell}+\Delta^{\Phi^{\prime}}_{\ell}+\Delta^{\nabla\Phi}_{\ell}+\Delta^{TD}_{\ell}+\Delta^{ISW}_{\ell}+\Delta^{\delta_{v}}_{\ell}\,.$
(A.18)

##  Appendix B Transfer functions implemented in CAMB

The transfer functions implemented are given by

$\displaystyle\displaystyle\Delta_{\ell}^{\delta}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \left|\frac{{\rm
d}D_{L}}{{\rm d}\eta}\right|w(\eta)b(\eta){\cal
T}_{\delta}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.1)
$\displaystyle\displaystyle\Delta_{\ell}^{LSD}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle-\int_{0}^{\eta_{0}}{\rm d}\eta\ \frac{{\rm
d}^{2}}{{\rm d}^{2}\eta}\left[\left|\frac{{\rm d}D_{L}}{{\rm
d}\eta}\right|w(\eta)\frac{A^{LSD}(\eta)}{k}{\cal
T}_{v}(\eta,k)\right]j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.2)
$\displaystyle\displaystyle\Delta_{\ell}^{L}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\ell(\ell+1)\int^{\eta_{0}}_{0}{\rm
d}\eta^{\prime}\ \left({\cal T}_{\Phi}(\eta^{\prime},k)+{\cal
T}_{\Psi}(\eta^{\prime},k)\right)j_{\ell}(k(\eta_{0}-\eta^{\prime}))\times$
(B.3) $\displaystyle\displaystyle\int^{\eta^{\prime}}_{0}{\rm d}\eta\
\left|\frac{{\rm d}D_{L}}{{\rm
d}\eta}\right|w_{i}(\eta)A_{L}(\eta,\eta^{\prime})\,,$
$\displaystyle\displaystyle\Delta_{\ell}^{D}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \frac{{\rm d}}{{\rm
d}\eta}\left[\left|\frac{{\rm d}D_{L}}{{\rm
d}\eta}\right|w(\eta)\frac{A^{D}(\eta)}{k}{\cal
T}_{v}(\eta,k)\right]j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.4)
$\displaystyle\displaystyle\Delta_{\ell}^{\Phi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \left|\frac{{\rm
d}D_{L}}{{\rm d}\eta}\right|w(\eta)A^{\Phi}(\eta){\cal
T}_{\Phi}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.5)
$\displaystyle\displaystyle\Delta_{\ell}^{\Psi}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \left|\frac{{\rm
d}D_{L}}{{\rm d}\eta}\right|w(\eta)A^{\Psi}(\eta){\cal
T}_{\Psi}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.6)
$\displaystyle\displaystyle\Delta_{\ell}^{\Phi^{\prime}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \left|\frac{{\rm
d}D_{L}}{{\rm d}\eta}\right|w(\eta)A^{\Phi^{\prime}}(\eta){\cal
T}_{\Phi^{\prime}}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.7)
$\displaystyle\displaystyle\Delta_{\ell}^{\nabla\Phi}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \frac{{\rm d}}{{\rm
d}\eta}\left[\left|\frac{{\rm d}D_{L}}{{\rm
d}\eta}\right|w(\eta)A^{\nabla\Phi}(\eta){\cal
T}_{\Phi}(\eta,k)\right]j_{\ell}(k(\eta_{0}-\eta))\,,$ (B.8)
$\displaystyle\displaystyle\Delta_{\ell}^{TD}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int^{\eta_{0}}_{0}{\rm d}\eta^{\prime}\
\left({\cal T}_{\Phi}(\eta^{\prime},k)+{\cal
T}_{\Psi}(\eta^{\prime},k)\right)j_{\ell}(k(\eta_{0}-\eta^{\prime}))\times$
(B.9) $\displaystyle\displaystyle\int^{\eta^{\prime}}_{0}{\rm d}\eta\
\left|\frac{{\rm d}D_{L}}{{\rm
d}\eta}\right|w_{i}(\eta)A_{TD}(\eta,\eta^{\prime})\,,$
$\displaystyle\displaystyle\Delta_{\ell}^{ISW}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int^{\eta_{0}}_{0}{\rm d}\eta^{\prime}\
\left({\cal T}_{\Phi^{\prime}}(\eta^{\prime},k)+{\cal
T}_{\Psi^{\prime}}(\eta^{\prime},k)\right)j_{\ell}(k(\eta_{0}-\eta^{\prime}))\times$
(B.10) $\displaystyle\displaystyle\int^{\eta^{\prime}}_{0}{\rm d}\eta\
\left|\frac{{\rm d}D_{L}}{{\rm d}\eta}\right|w_{i}(\eta)A_{\rm ISW}(\eta)\,,$
$\displaystyle\displaystyle\Delta_{\ell}^{\delta_{v}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{0}^{\eta_{0}}{\rm d}\eta\ \left|\frac{{\rm
d}D_{L}}{{\rm
d}\eta}\right|w(\eta)\left[b_{e}-3\right]\frac{\mathcal{H}}{k}{\cal
T}_{v}(\eta,k)j_{\ell}(k(\eta_{0}-\eta))\,.$ (B.11)

##  Appendix C Properties of spherical harmonics

The spherical harmonics $\displaystyle Y_{\ell m}(\hat{n})$ obey the
differential equation

$\Delta_{{\bm{n}}}Y_{\ell m}(\hat{\bm{n}})=-\ell(\ell+1)Y_{\ell
m}(\hat{\bm{n}})\,,$ (C.1)

where

$\Delta_{{\bm{n}}}=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}\,.$
(C.2)

The spherical harmonics are a natural set of base functions for mode
decomposition on the sphere as they are orthonormal, i.e.

$\int{\rm d}\Omega_{\bm{n}}\ Y_{\ell
m}(\hat{\bm{n}})Y^{*}_{\ell^{\prime}m^{\prime}}({\bm{n}})=\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\,.$
(C.3)

They are also related to the Legendre polynomials by

${\cal
P}_{\ell}({\bm{n}}\cdot{\bm{n}}^{\prime})=\frac{4\pi}{2\ell+1}\sum^{m=\ell}_{m=-\ell}Y_{\ell
m}(\hat{\bm{n}})Y^{*}_{\ell m}(\hat{\bm{n}}^{\prime})\,.$ (C.4)

It follows from eqs. (C.3) and (C.4) the useful expression

$\int{\rm d}\Omega_{{\bm{n}}}\ {\cal
P}_{\ell^{\prime}}({\bm{n}}\cdot{\bm{n}}^{\prime})\ Y^{*}_{\ell
m}(\hat{\bm{n}})\ =\frac{4\pi}{2\ell+1}Y^{*}_{\ell
m}(\hat{\bm{n}}^{\prime})\delta_{\ell\ell^{\prime}}\,,$ (C.5)

which we will use regularly, and which is valid for the complex conjugate.
Another useful property is

$e^{ir{\bm{n}}\cdot{\bm{n}}^{\prime}}=\sum_{\ell=0}^{\infty}(2\ell+1)\
i^{\ell}\ {\cal P}_{\ell}({\bm{n}}\cdot{\bm{n}}^{\prime})\ j_{\ell}(r)\,,$
(C.6)

where $\displaystyle j_{\ell}$ are the spherical Bessel functions. Then

$\displaystyle\displaystyle\int{\rm d}\Omega_{{\bm{n}}}\
e^{ir{\bm{n}}\cdot{\bm{n}}^{\prime}}Y^{*}_{\ell m}({\bm{n}})$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\int{\rm
d}\Omega_{{\bm{n}}}\ \sum_{\ell^{\prime}=0}^{\infty}(2\ell^{\prime}+1)\
i^{\ell^{\prime}}\ {\cal P}_{\ell^{\prime}}({\bm{n}}\cdot{\bm{n}}^{\prime})\
j_{\ell^{\prime}}(r)Y^{*}_{\ell m}({\bm{n}})$ (C.7)
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\sum_{\ell^{\prime}=0}^{\infty}(2\ell^{\prime}+1)\
i^{\ell^{\prime}}\ j_{\ell^{\prime}}(r)\frac{4\pi}{2\ell+1}Y^{*}_{\ell
m}(\hat{\bm{n}}^{\prime})\delta_{\ell\ell^{\prime}}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle 4\pi\ i^{\ell}\
j_{\ell}(r)Y^{*}_{\ell m}(\hat{\bm{n}}^{\prime})$

Finally, a further property that will be useful follows from eqs. (C.1) and
(C.4), i.e.

$\Delta_{{\bm{n}}}{\cal
P}_{\ell}({\bm{n}}\cdot{\bm{n}}^{\prime})=-\ell(\ell+1){\cal
P}_{\ell}({\bm{n}}\cdot{\bm{n}}^{\prime})\,.$ (C.8)

##  Appendix D Comparison of the amplitudes

Term | LDS | RS
---|---|---
$\displaystyle A_{LSD,RSD}$ | $\displaystyle-\frac{2\bar{r}}{1+\bar{r}\mathcal{H}}$ | $\displaystyle-\frac{1}{\mathcal{H}}$
$\displaystyle A_{D}$ | $\displaystyle 3-10s+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(\frac{4}{\bar{r}\mathcal{H}}+2b_{e}\right)-2\left(\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\right)^{2}\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\mathcal{H}}\right)$ | $\displaystyle\frac{5s-2}{\bar{r}\mathcal{H}}-5s+b_{e}-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}$
$\displaystyle A_{\Psi}$ | $\displaystyle-1+10s+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(\frac{3}{\bar{r}\mathcal{H}}-2-2b_{e}\right)+2\left(\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\right)^{2}\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\mathcal{H}}\right)$ | $\displaystyle 1+\frac{5s-2}{\bar{r}\mathcal{H}}-5s+b_{e}-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}$
$\displaystyle A_{\Phi}$ | $\displaystyle-3+5s+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(\frac{1}{\bar{r}\mathcal{H}}-1-b_{e}\right)+\left(\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\right)^{2}\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\mathcal{H}}\right)$ | $\displaystyle 5s-2$
$\displaystyle A_{\Phi^{\prime}}$ | $\displaystyle\frac{\bar{r}}{1+\bar{r}\mathcal{H}}$ | $\displaystyle\frac{1}{\mathcal{H}}$
$\displaystyle A_{\nabla\Phi}$ | $\displaystyle\frac{\bar{r}}{1+\bar{r}\mathcal{H}}$ | $\displaystyle 0$
$\displaystyle A_{TD}$ | $\displaystyle 3-5s+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(-\frac{1}{\bar{r}\mathcal{H}}+1+b_{e}\right)-\left(\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\right)^{2}\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\mathcal{H}}\right)$ | $\displaystyle 5s-2$
$\displaystyle A_{L}$ | $\displaystyle\frac{1}{2}\left[\left(\frac{\bar{r}-r}{r}\right)\left(5s-3+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(\frac{2}{\bar{r}\mathcal{H}}-1-b_{e}\right)+\left(\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\right)^{2}\left(\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}-\frac{1}{\bar{r}\mathcal{H}}\right)\right)+\frac{1}{1+\bar{r}\mathcal{H}}\right]$ | $\displaystyle\frac{1}{2}(5s-2)\frac{\bar{r}-r}{r}$
$\displaystyle A_{ISW}$ | $\displaystyle-1+10s+\frac{\bar{r}\mathcal{H}}{1+\bar{r}\mathcal{H}}\left(\frac{4}{\bar{r}\mathcal{H}-2-2b_{e}}\right)$ | $\displaystyle\frac{2-5s}{\bar{r}\mathcal{H}}+5s-b_{e}+\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}$
Table 1: Comparison of the amplitudes of the corrections to the number counts
fluctuation between luminosity distance space and redshift space.

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