# A fractal LTB model cannot explain Dark Energy

Erick Pastén  erick.contreras[at]postgrado.uv.cl Víctor H. Cárdenas
victor.cardenas[at]uv.cl Instituto de Física y Astronomía, Universidad de
Valparaíso, Gran Bretaña 1111, Valparaíso, Chile

###### Abstract

We revisited the problem of describing, on average, a fractal distribution of
matter using a Lemaitre-Tolman-Bondi (LTB) solution. Here we study the fractal
structure of our local universe having a fractal dimension and a scale
transition, ensuring an homogeneous bang-time function. We test our model with
the latest type Ia supernova data, the Pantheon compilation, and discuss
problems and possible improvements for it, concluding that a fractal
transition in LTB cosmology cannot be used to explain the effects of dark
energy without requiring an inhomogeneous big-bang, but it can be useful to
study structures at low scales.

##  I Introduction

Dark Energy is the usual explanation for the apparent acceleration implied by
the type Ia supernova (SNIA) data et al. and Project (1999), et al. (1998).
However, the suggestion for the existence of dark energy is ultimately based
on the cosmological principle, that of assuming global homogeneity and
isotropy. The requirement of an extra parameter $\Omega_{\Lambda}$ is then
necessary to explain the dimming of supernovae magnitude at large redshift.

Although at large scales the universe appears being homogeneous and isotropic
in agreement to the CMB observations, at small scales it is far from being
like that due to the presence of complex structures that produce
underdensities Keenan et al. (2013), fractal-like structures Labini
(2011),Labini et al. (1998) and bulk flows that are not at rest with respect
to the Hubble flow Feindt (2013) Hudson et al. (1999) Magoulas et al. (2016).
There have been many works claiming that some of these effects can mimic an
apparent acceleration. Possibly the combination of many (even all) of these
contributions may have a strong effect than we previously thought Celerier
(2006) Enqvist (2007) Cosmai et al. (2019) Tsagas (2011) Asvesta et al.
(2022). As a step towards the understanding of the influence of the local
structure in cosmology, in this work we explore a simple approach by studying
the local fractal structure of the universe with the luminosity distance
relations from SNIA data.

In a recent paper Cosmai et al. (2019, 2023), the authors stated that a
fractal-like $M(r)$ function can fit the SNIA data using a non-dark energy LTB
model written as:

$M(r)\propto r^{D},$ (1)

where $D$ is the fractal dimension. Although interesting, the analysis is
based on very special assumptions that make the model an extremely exceptional
case. Here, we propose an analysis based also on the LTB metric but using both
a fractal dimension and also a length-scale that seems to work without any
theoretical problems, ensuring also an homogeneous big bang.

##  II The LTB model

Lemaitre Lemaitre (1933), Tolman Tolman (1934) and Bondi Bondi (1947) were the
first to studied isotropic and radial inhomogeneous universes, known as LTB
models. Here we restrict ourselves to the study of pressureless matter
universes. Assuming for generality the existence of a cosmological constant
with equation of state $\rho_{\Lambda}=-p_{\Lambda}$, the energy-momentum
tensor is given by:

$T_{\mu}^{\nu}=-\rho_{M}(r,t)\delta^{\mu}_{0}\delta^{0}_{\nu}-\rho_{\Lambda}\delta^{\mu}_{\nu},$
(2)

in which the matter density $\rho_{M}$ varies in time and radially in space,
meanwhile $\rho_{\Lambda}$ remains fixed. The metric of a LTB universe can be
written as:

$ds^{2}=-dt^{2}+\frac{R^{\prime 2}}{1+2E(r)}dr^{2}+R^{2}d\Omega^{2},$ (3)

where $R=R(r,t)$ is a generalization of the scale factor and $E(r)$ is a
function associated with the total energy. Here a prime indicates a derivative
respect to $r$ and a dot a derivative respect to time $t$. From Einstein’s
equation we get the evolution equation:

$\frac{\dot{R}^{2}}{R^{2}}=\frac{2GM(r)}{R^{3}}+\frac{8\pi
G\rho_{\Lambda}}{3}+\frac{2E}{R^{2}},$ (4)

where $M(r)$ is an integration constant (in time).

Notice that by writing $R(r,t)=a(t)r$ where $a(t)$ is the scale factor and
$2E(r)=-Kr^{2}$ with $K$ being the curvature constant factor, the metric
reduces to the FLRW metric and (4) reduces to the usual Friedmann equation.

The $M(r)$ function can be interpreted as the mass contained in a sphere of
radius $R=ar$ in the FLRW limit. The matter density is defined through:

$\displaystyle\frac{M^{\prime}}{R^{\prime}R^{2}}=4\pi\rho_{M}.$ (5)

So the mass function $M(r)$ is defined as the integral of $\rho_{M}$ in the
LTB space as:

$M(r)=\int_{0}^{r}\rho_{M}(4\pi R^{\prime}R^{2}dr)$ (6)

Defining the local Hubble rate by

$H(r,t)=\frac{\dot{R}(r,t)}{R(r,t)},$ (7)

we can write (4) as:

$H^{2}=\frac{2GM(r)}{R^{3}}+\frac{8\pi G\rho_{\Lambda}}{3}+\frac{2E}{R^{2}}$
(8)

By defining the local matter density parameter $\Omega_{m}$ as

$2GM(r)=H_{0}^{2}(r)\Omega_{m}(r)R_{0}^{3}(r),$ (9)

where $H_{0}(r)=H(r,t_{0})$ is the present Hubble local rate,
$R_{0}=R(r,t_{0})$ being an initial condition that can be gauged, and for the
curvature term

$2E(r)=H_{0}^{2}(r)\Omega_{k}(r)R_{0}^{2}(r),$ (10)

then, the local Friedman equation takes the form

$H^{2}(r,t)=H_{0}^{2}(r)(\Omega_{m}(r)A^{3}+(1-\Omega_{m}(r))A^{2}),$ (11)

where $A(r,t)=R_{0}/R$. This means that we can choose any $R_{0}(r)$ function
and the equations remains the same. This freedom is usually fixed by taking
$R_{0}(r)=r$ Enqvist (2007). The Friedman equation (11) defines completely the
relation between the functions $H_{0}(r)$, $\Omega_{m}(r)$, $R(r,t)$ and
$H(r,t)$.

One of the most used constraint to LTB models is based on the time that has
passed from the big bang until now Garcia-Bellido and Haugbølle (2008), known
as the bang time $t_{BT}(r)$ defined by:

$\begin{split}H_{0}(r)(t-&t_{BT}(r))=\int_{0}^{R(r,t)/R_{0}(t)}\frac{dx}{\sqrt{\Omega_{m}/x+\Omega_{k}}}\\\
&=\frac{R(r,t)}{R_{0}(r)\sqrt{\Omega_{k}(r)}}\sqrt{1+\frac{R_{0}(r)\Omega_{m}(r)}{R(r,t)\Omega_{k}(r)}}\\\
&-\frac{\Omega_{m}(r)}{\sqrt{\Omega_{k}(r)^{3}}}\sinh^{-1}{\sqrt{\frac{\Omega_{m}(r)R(r,t)}{\Omega_{k}(r)R_{0}(r)}}}.\end{split}$
(12)

This function means that not all locations in the universe were created at the
same time. If we set $t=t_{0}$ we can obtain the current age of the universe
$t_{BB}(r)$:

$\begin{split}&H_{0}(r)(t_{0}-t_{BB}(r))=H_{0}(r)t_{u}(r)=\int_{0}^{1}\frac{dx}{\sqrt{\Omega_{m}/x+\Omega_{k}}}\\\
&=\frac{1}{\sqrt{\Omega_{k}(r)}}\sqrt{1+\frac{\Omega_{m}(r)}{\Omega_{k}(r)}}-\frac{\Omega_{m}(r)}{\sqrt{\Omega_{k}(r)^{3}}}\sinh^{-1}{\sqrt{\frac{\Omega_{m}(r)}{\Omega_{k}(r)}}},\end{split}$
(13)

where we have defined $t_{u}(r)$ as the local age of the universe. It is
interesting to note that we need to fix two functions here, chosen among
$\Omega_{m}(r)$, $H_{0}(r)$ and $t_{BT}(r)$, and also fix a gauge (usually
$R_{0}(r,t)=r$) to define completely the scale factor function $R(r,t)$. We
obtain a similar result if we use instead the functions $M(r)$ and $E(r)$:

$\begin{split}\sqrt{2}(t-&t_{BT}(r))=\int_{0}^{R(r,t)}\frac{dx}{\sqrt{2GM(r)/x+2E(r)}}\\\
&=\frac{R(r,t)}{\sqrt{E}}\sqrt{1+\frac{GM(r)}{R(r,t)E(r)}}\\\
&-\frac{GM(r)}{\sqrt{E(r)^{3}}}\sinh^{-1}{\sqrt{\frac{GM(r)R(r,t)}{E(r)}}}\end{split}$
(14)

In this case, it seems we need to set three functions, $t_{BT}(r)$, $M(r)$ and
$E(r)$, to define $R(r,t)$, but in fact one of them is a pure gauge. Usually
$M(r)\propto r^{3}$ is assumed as this gauge (as we mentioned in the
introduction, in some fractal-like models it has been proposed to use Eq.(1)
as the gauge, with $D$ being the fractal dimension Cosmai et al. (2019)), so
we just need to define two functions as we will see.

Another way to define LTB model is thinking in a density contrast profile
$\delta(r,t)$ defined as:

$\delta(r,t)=\frac{\rho(r,t)-\rho_{0}^{\infty}}{\rho_{0}^{\infty}},$ (15)

where $\rho_{0}^{\infty}=\rho(\infty,t_{0})$ is the mean density constant very
far outside the local inhomogeneity at the present time. This function is
useful because this allows us to describe the mass profile using the data
directly from the observational photometric local surveys. We have for
$t=t_{0}$:

$\rho_{0}(r)=\rho_{0}^{\infty}(\delta_{0}(r)+1)$ (16)

Although the functions $\Omega_{m}(r)$, $M(r)$ and $E(r)$ have a clear
meaning, it is difficult to relate them directly to the observed quantity
$\delta_{0}(r)$. So instead of using the complexities of (13), we can
approximate a solution by assuming the universe being described by FLRW and
define a integrated density function as:

$\rho_{0}^{H}(r)\propto\frac{M(r)}{r^{3}}\propto(\delta^{H}(r)+1),$ (17)

which is exactly true for an homogeneous universe. Using the gauge
$R_{0}(r)=r$ we can write:

$\delta^{H}(r)+1\propto\frac{M(r)}{r^{3}}\propto\Omega_{m}(r)H_{0}(r)^{2},$
(18)

where $\delta^{H}(r)$ is the integrated density contrast. Then, constraining
$t_{BT}(r)$ and $\delta^{H}(r)$ we can define completely $R(r,t)$. It is
important to note that this is not exactly the integrated density contrast,
but is a useful approximation that allows us to define the universe in terms
of density real parameters.

##  III Luminosity Distance

The luminosity distance in LTB models can be obtained from:

$d_{L}=(1+z)^{2}R(r,t),$ (19)

where $r=r(z)$ and $t=t(z)$ are defined by the geodesic equations.

$\displaystyle\frac{dr}{dz}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{1-k}}{(1+z)\dot{R^{\prime}}}$ (20)
$\displaystyle\frac{dt}{dz}$ $\displaystyle=$
$\displaystyle-\frac{R^{\prime}}{(1+z)\dot{R^{\prime}}}$ (21)

These equations are necessary to make contact with the observations. In our
particular case, we use the Pantheon sample et al. (2018) that release the
redshift $z$, distance modulus $\mu(z)$ and the error $\sigma_{\mu}(z)$ for
$1048$ type Ia supernova that can be compared to the theoretical expectation

$\mu(z)=5\log_{10}\frac{d_{L}(z)}{10\text{pc}}.$ (22)

##  IV Fractals in LTB Model

In this section we follow the treatment of Ref. Cosmai et al. (2019, 2023). In
that work, the authors argue that from the statistical analysis of the 3D
galaxy distribution, the average conditional density can be described locally
as

$\langle n(r)\rangle\sim r^{-\gamma}$ (23)

where $\gamma$ is a phenomenological parameter that can take values from $0.2$
to $0.9$ in a scale range from $1$ Mpc/$h$ until $100$ Mpc/$h$. Beyond that
scale, a transition to homogeneity ($\gamma=0$) is expected whose behavior is
not completely clear. Then for the structure of a real galaxy we can
approximate the density of matter as:

$\rho_{M}(r)\sim\langle n(r)\rangle$ (24)

a behaviour that can be described using a LTB model. In this case, the
function $M(r)$ is related to this fractal density function due to equation
(6):

$M(r)=\int_{0}^{r}\langle n(r)\rangle(4\pi R^{\prime}R^{2}dr),$ (25)

where $M(r)$ can be understood as the matter inside a volume of radius $r$. In
Cosmai et al. (2019), it was stated that a fractal-like $M(r)$ function can
fit the SNIA data in a pressureless LTB model given by

$M(r)=\Phi r^{D},$ (26)

where $D=3-\gamma$ is the fractal dimension and $\Phi$ is the mass-scale. Let
us study this model at some detail.

First of all, we have to emphasize a subtlety in the notation. The authors of
Cosmai et al. (2019) perform the integration of the equation

$\dot{R}^{2}+2R\ddot{R}+k=0,$ (27)

assuming that at $t=0$, the $R$ function takes the value $R_{0}$ from which
leads the definition:

$R(r,0)=R_{0}(r).$ (28)

In the LTB literature usually $R_{0}(r)$ is the present scale factor (which is
usually a gauge in LTB cosmology when $\Omega_{m}(r)$ is used instead of
$M(r)$), while $R(r,0)=0$ is the initial scale factor in the homogeneous big
bang itself (usually setted to $0$). This leads to a confusion with the common
definitions in LTB cosmology literature and should be pointed out.

Second, following the previous redefinition for the $R_{0}$ function, an
inhomogeneous negative bang time function $t_{BT}(r)<0$, that notably is
dependent on the fractal dimension of matter $D$ Cosmai et al. (2023), is
obtained :

$t_{BT}=-\frac{2}{3}\sqrt{\frac{R_{0}^{3}(r)}{2GM(r)}}.$ (29)

However, an inhomogeneous bang time is something physically undesirable if we
are trying to understand the influence of the fractal distribution of matter
in cosmology. A similar problem was pointed out by Kenworthy et al. (2019)
discussing the work made in Hoscheit and Barger (2018) about the influence of
a large void in LTB cosmology, as an inhomogenous big bang results in
different ages of the universe at difference places of it. This introduces a
new fundamental inhomogeneity that requires new physics in the same sense than
the addition of dark energy. If we perform the integration of the equation for
a purely flat universe, with an homogeneous bang time function and the
function $M(r)$ with the usual limits ($t\in[0,t],\>R\in[0,R(r,t)]$ we just
get the EdS results:

$R(r,t)=\left(\frac{9M(r)}{2}\right)^{1/3}t^{2/3}.$ (30)

We can see here that the election of the function $M(r)$, with the assumption
of global flatness and homogeneous big-bang is just a gauge related to the
scale of the radial coordinate. Also, the Hubble function can be written as

$\displaystyle H(t)=\frac{\dot{R}}{R}=\frac{2}{3t},$ (31)

independently of the $M(r)$ function used. We have no divergences for $H_{0}$
at $r=0$, but also the fractal structure described with $M(r)$ cannot be used
to explain any $H_{0}$ tension.

The question now emerges, can we use LTB models with $E(r)=0$ to describe
fractal distribution of matter and assuming an homogeneous big bang? In the
following we describe our proposal to do that.

###  IV.1 Sharp transition model

First, we propose to define a function $M(r)$ that can describe both, the
internal fractal statistical behaviour of the density and also a transition to
a FLRW universe (a transition between fractality and non-fractality). The
function $M(r)$ could be described as

$\displaystyle M_{in}(r)$ $\displaystyle=\Phi_{in}r^{D},$ $\displaystyle r<L$
(32) $\displaystyle M_{out}(r)$ $\displaystyle=\Phi_{out}r^{3},$
$\displaystyle r>L,$ (33)

where $M\sim r^{3}$ is the mass for a FLRW universe. At this point we have
four free parameters in the model: the mass scale $\Phi_{in}$ and
$\Phi_{out}$, the fractal dimension $D$ and the scale of the fractal structure
$L$. If we demand for continuity at $r=L$ we can reduce to three free
parameters as:

$\displaystyle\Phi_{in}L^{D}=\Phi_{out}L^{3},$ (34)

then the model becomes

$\displaystyle M_{in}(r)$
$\displaystyle=\Phi_{out}L^{3}\left(\frac{r}{L}\right)^{D},$ $\displaystyle
r<L$ (35) $\displaystyle M_{out}(r)$ $\displaystyle=\Phi_{out}r^{3},$
$\displaystyle r>L.$ (36)

Defining the function in this way, the effects between the internal fractal
structure and the external EdS universe should be noticeable. A similar
approach was developed in Ruffini and Stahl (2017), but the difference is that
they fix the scale of transition about $r=2300$ Mpc meanwhile for us this is a
free parameter to test.

As a first approach, we assume $\Omega_{\Lambda}=0$ and the universe being
just EdS beyond the transition. We use low redshift type Ia supernova data to
estimate the transition scale for fractality. We also use the full sample
111We can understand this approach as being a tool to estimate the fractal
transition in low redshift data, but also as a possible explanation for dark
energy with fractal structure using the full sample. However, this last
approach is questionable as it does not exist a clear evidence for fractal
structure for scales $r>100$ Mpc, in which case we have to add another
parameter to be fixed. Following Alexander et al. (2009) and considering an
EdS universe for $r>L$, we choose for $\Phi_{out}$ to be in agreement with the
age of the universe in an EdS universe:

$\displaystyle\Phi_{out}=\frac{4\pi}{3}\rho_{0}=\frac{H_{0}^{2}}{2},$ (37)

where $H_{0}\sim h_{out}/3000$ Mpc is the Hubble parameter outside the fractal
behaviour in the FLRW limit. With this scale we have the radial coordinate in
Mpc and just two free parameters: $D$ and $L$. The $R(r,t)$ can then be
written as:

$\displaystyle R_{in}(r,t)$ $\displaystyle=$
$\displaystyle\left(\frac{9\Phi_{in}}{2}\right)^{\frac{1}{3}}L\left(\frac{r}{L}\right)^{D/3}t^{2/3},\>\>r<L$
(38) $\displaystyle R_{out}(r,t)$ $\displaystyle=$
$\displaystyle\left(\frac{9\Phi_{out}}{2}\right)^{\frac{1}{3}}rt^{2/3},\>\>r>L,$
(39)

It is important to note that in this model, the density is surprisingly
homogeneous. Using equation (11) we can obtain the same density profile than
an EdS universe:

$\displaystyle\rho_{0}(r)=\frac{3\Phi_{out}}{4\pi},$ (40)

but the integrated density at $t_{0}$ is:

$\displaystyle\rho_{0}^{H}(r)\propto r^{D-3}=r^{-\gamma},\>\>r<L$ (41)

which is constant for $r>L$. Using the continuity condition at $L$ we obtain:

$\rho_{0}^{H}(r)=\left\\{\begin{array}[]{lr}\frac{3\Phi_{out}}{4\pi}\left(\frac{r}{L}\right)^{-\gamma}&,r<L\\\
\frac{3\Phi_{out}}{4\pi}&,r>L\end{array}\right.$ (42)

The behaviour of $M(r)$ and the densities are displayed in Fig. 1 and Fig. 2.
It is interesting to note that the fractal behaviour can be also be defined
as:

$\displaystyle\rho_{0}^{H}(r)$ $\displaystyle=$
$\displaystyle\rho_{0}\left(\frac{r}{L}\right)^{-\gamma},\>\>r<L$ (43)
$\displaystyle\rho_{0}^{H}(r)$ $\displaystyle=$
$\displaystyle\rho_{0},\>\>r>L$ (44)

As simple it is, this model assumes a sharp transition which is not very
realistic. Purely Fractal and Fractal-EdS universe are plotted in Fig. 3.

Figure 1: $M(r)$ profiles. In black appears the fractal model with a sharp
transition to EdS; in red pointed line EdS and in yellow pointed line a purely
fractal universe. We use $D=3.36$, $L=100Mpc$ and $h_{out}\sim 0.73$. The
vertical gray line is the lenght $L$ of the transition. The functions
$M_{1}(r)$ and $M_{2}(r)$ that describes a smooth transition in equations (45)
and (46) are not plotted as those are very similar to the black line (see,
e.g., fig. 2 of Calcagni (2017)).  Figure 2: Integrated density profiles for
different values of $D$. We use $L=100Mpc$ and $h_{out}\sim 0.73$. The
vertical gray line is the length $L$ of the transition. Note that profiles
with $D>3$ are similar to an over-dense region and $D<3$ to an under-dense
region. $D=3$ correspond to the common constant integrated density profile in
EdS universe.  Figure 3: Luminosity Distance for the different universes. Same
description as in FIG. 1. Note that we use the purely fractal universe with
the scale $\Phi_{in}$ to illustrate.

###  IV.2 Smooth transition model

A smooth slope transition around $r=L$ could be introduced considering a
multi-power mass function. Motivated by the work developed in Calcagni (2017),
a binomial “average” mass profile could be useful for this case. If $D<3$ we
can write:

$\displaystyle M_{1}(r)$
$\displaystyle=\Phi_{out}\left(L^{3}\left(\frac{r}{L}\right)^{D}+r^{3}\right).$
(45)

Note that as $D<3$, for $r\gg L$ the term $r^{3}$ dominates the mass function,
meanwhile for $r\ll L$ the $r^{D}$ term is stronger. In the opposite case when
$D>3$ we can propose:

$\displaystyle M_{2}(r)$
$\displaystyle=\frac{\Phi_{out}}{\frac{1}{L^{3}}\left(\frac{L}{r}\right)^{D}+\frac{1}{r^{3}}},$
(46)

which follows a similar asymptotic behaviour. We choose the same $\Phi_{out}$
value described in the last section to allow for consistent convergences at
$r\gg L$. This approach has the advantage of being numerically friendlier as
we have a unique mass profile for all $r$ space.

##  V Statistical analysis with SNIa data

To test our models, we use the Pantheon sample comprised by 1048 SNIa in the
redshift range $z\in(0.01,2.3)$. As we are interested in studying a possible
transition between fractal behaviour and a FLRW universe, we consider two
approaches. First, using only low redshift SNIa, using a cutoff of $z<0.3$
corresponding approximately to $\sim 800$Mpc. Second, using the full sample.
We select $h_{out}\sim 0.73$ in the first case to match local Hubble parameter
scale measures Dainotti et al. (2021) and $h_{out}\sim 0.45$ in the second
case, in order to converge for low scales of the Hubble parameter that fits
CMB with an EdS universe Alnes et al. (2005). The full Tripp Formulae for
distance modulus is:

$\mu_{obs}=m_{b}^{*}-M=m_{b}+\alpha x-\beta c+\Delta_{M}-M,$ (47)

here, $m_{b}$ corresponds to the peak apparent magnitude in the B-band, $M$ is
the absolute B-band magnitude of a fiducial SNIa, $x$ and $c$ are light curve
shape and color parameters, $\alpha$ is a coefficient of the relation between
luminosity and stretch and $\beta$ is a coefficient of the relation between
luminosity and colour, and $\Delta_{M}$ is a corrections based on the mass of
the host galaxy. All of these parameters, usually called nuisance parameters,
are needed to standardize the SNIa data leading to the corrected magnitude
$m_{b}^{*}$. Those correction are provided in Pantheon catalogue, and the
parameter $M$ can be marginalized. This relation can be used to perform
statistical analysis with the distance theoretical modulus for any
cosmological model given by Eq.(22).

##  VI Results

We use the code EMCEE Foreman-Mackey et al. (2013) to test the models against
the SNIa data. This is a pure python implementation of the affine invariant
ensemble sampler for Markov chain Monte Carlo proposed by Goodman and Weare
Goodman and Weare (2010). From this analysis we obtain the best fit values for
$D$ and $L$ for each model. We used directly the corrected magnitudes from the
catalogue, but also another approach allowing the nuisance parameters $\alpha$
and $\beta$ to vary along the cosmological parameters. In this last approach,
we ignore the step mass correction as this does not impact significantly the
cosmological fit.

###  VI.1 Sharp transition

The results for the sharp transition model and details are shown in Table 1 .
We note that for the low redshift supernovae we can get a relatively good fit,
but not for the full sample. We also note high errors in the parameter
estimation when we do not use tight priors,possible attributed to the sharp
transition at $r=L$. Best fit for the full sample is plotted in Figure 4.

To look for insights into the value of the fractal dimension $D$, we also
consider priors on the parameter $L$, which is our fractal transition scale.
Instead of fixing it at a convenient value, as previous works have done, here
we introduce Gaussian priors for $L$ using different values coming from large
scale research of the transition scale for homogeneity. For example, in Yadav
et al. (2010) the authors concluded that the scale of homogeneity should be
less than $260h^{-1}$ Mpc. Also in Yadav et al. (2005) and Hogg et al. (2005)
the authors have suggested even a smaller scale of $L=60h^{-1}$Mpc. Let us
consider four values for $L$: $L=60h^{-1}$, $L=100h^{-1}$, $L=150h^{-1}$ and
$L=200h^{-1}$. Considering Gaussian priors on this values with a 5% of error,
we find the results showed in the last panel in Table 1 for low redshift
supernovae at $z<0.3$. Again, for sharp transition model the error on $D$ is
very high. This means that a flat LTB fractal model with a sharp transition,
homogeneous big bang and no cosmological constant does not perform well to fit
SN IA data.

Figure 4: Best fit function for corrected magnitudes of the full Pantheon
sample using a fractal sharp transition. We get a notably worst fit than
previous studies in fractal LTB models that use an inhomogeneous big bang.

Figure 5: Best fit functions for corrected magnitudes of the full Pantheon
sample using the two $M(r)$ functions of the fractal smooth transition. The
errors in the parameter estimation are much more lower than the sharp case,
but still the fit does not perform well.

###  VI.2 Smooth transition

The results for the best fit for the sharp transition model and details are
shown in Table 2. We note that the errors in the parameter estimation are
notably lower when the model is tested against the corrected magnitudes of the
Pantheon sample, which support the idea that a smooth function performs much
better in MCMC sampling. However, the $\chi^{2}_{red}$ values are even worse
than the sharp transition model for all cases. We conclude that a flat LTB
fractal model with a smooth transition, homogeneous big bang and no
cosmological constant neither perform well to fit SN IA data. Best fits are
plotted in Figure 5.

Table 1:  Best fit values for $D$ and $L$ using a sharp transition, fitting
for corrected magnitudes as are given in Pantheon catalogue. Also we perform a
fit with a tight prior for $L$ with the corrected magnitudes to look for
insights into the value of $D$.

Corrected magnitudes
$h_{out}$ $z_{cut}$ $D$ $L(Mpc)$ $\chi^{2}_{red}$ $0.73$ $z<0.3$
$3.16^{+0.36}_{-0.63}$ $696.83^{+113.01}_{-298.50}$ $1.15$ $0.45$ Full Sample
$3.47^{1.04}_{-1.38}$ $1354.17^{232.08}_{-313.73}$ 1.43

Tight priors for $L$ and corrected magnitudes
$L$ Prior $D$ $L(Mpc)$ $\chi^{2}_{red}$ $60h^{-1}$
$2.97\begin{subarray}{c}+1.26\\\ -1.07\end{subarray}$ 85.02
$\begin{subarray}{c}+0.53\\\ -0.52\end{subarray}$ $1.15$ $100h^{-1}$
$2.74\pm\begin{subarray}{c}+1.24\\\ -1.09\end{subarray}$
$137.02\begin{subarray}{c}+0.54\\\ -0.51\end{subarray}$ $1.15$ $150h^{-1}$
$2.73\pm\begin{subarray}{c}+1.2\\\ -1.1\end{subarray}$
$205.02\begin{subarray}{c}+0.54\\\ -0.51\end{subarray}$ $1.16$ $200h^{-1}$
$2.6\pm\begin{subarray}{c}+1.15\\\ -1.13\end{subarray}$
$273.00\begin{subarray}{c}+0.51\\\ -0.5\end{subarray}$ $1.16$

Table 2:  Best fit values for $D$ and $L$ using a smooth transition, fitting
for corrected magnitudes. Due to the low errors in the parameter estimation we
did not extent further this approach.

Corrected magnitudes $M_{1}(r)$ function
$h_{out}$ $z_{cut}$ $D$ $L(Mpc)$ $\chi^{2}_{red}$ $0.73$ $z<0.3$
$3.16^{+0.018}_{-0.018}$ $1020.86^{+0.015}_{-0.021}$ $1.37$ $0.45$ Full Sample
$3.15^{0.017}_{-0.017}$ $1020.84^{+0.026}_{-0.014}$ 1.46 Corrected magnitudes
$M_{2}(r)$ function
$h_{out}$ $z_{cut}$ $D$ $L(Mpc)$ $\chi^{2}_{red}$ $0.73$ $z<0.3$
$3.14^{+0.019}_{-0.018}$ $1020.86^{+0.022}_{-0.020}$ $1.33$ $0.45$ Full Sample
$3.15^{+0.014}_{-0.017}$ $1020.85^{+0.021}_{-0.018}$ $1.46$

##  VII Discussion and conclusions

We have demonstrated that a purely fractal model following $M(r)\sim r^{D}$
with an homogeneous big bang is not different from an usual EdS universe with
another scaling relation. As an alternative to the fractal model proposed in
Cosmai et al. (2019), a more consistent model was developed, allowing for a
transition between a fractal scale with $M\sim r^{D}$ to an homogeneous
universe $M\sim r^{3}$ and not requiring the idea of a inhomogeneous big bang.
Two analyses to put in stress our model with the Pantheon data were performed.
In the case of a sharp transition, we got a good fit for low redshift
supernovae, but the model fails to describe the full behaviour of the Pantheon
data sample. We conclude that fractal LTB models cannot explain the effects of
dark energy. Overall, we looked for insights in fractal dimension value
performing a different analysis for low redshift supernovae using tight priors
for the transition scale. It was observed that the errors are still high at
$z<0.3$. The results are not very different from the smooth transition case,
giving a slightly worst fit than the sharp case, although the errors were much
lower. The physical validity of our models could be studied more deeply if we
summarize the problems that those have. First, we did not take into account a
cosmological constant. This constant could be added to the models in order to
fit the full SN IA data looking for effects of the local fractal structure in
the cosmological constant value. Also, we assumed that the $M$ scale should
converge to a fixed $H_{0}$ experimental value. Two possibilities accounting
for convergence to CMB $h\sim 0.45$ and another to SNIA estimation to $h\sim
0.73$ were used. Those values could be modified allowing for other definitions
like presented in Camarena and Marra (2020) Valkenburg et al. (2013). In
future works, we propose to study how a fractal structure in LTB models can
impact $\Lambda$CDM cosmology. We reinforce the importance of take into
account the full information about our local large scale structure to the
development of a better cosmological description that lead us to stringer
constraints on the $\Lambda$CDM model.

## ACKNOWLEDGMENTS

EP acknowledge the support through a graduate scholarship ANID-Subdirección de
Capital Humano/Doctorado Nacional/2021-21210824.

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# Multi-wavelength spectroscopic probes: biases from neglecting light-cone
effects

Jan-Albert Viljoen  José Fonseca  Roy Maartens

###### Abstract

Next-generation cosmological surveys will observe larger cosmic volumes than
ever before, enabling us to access information on the primordial Universe, as
well as on relativistic effects. In a companion paper, we applied a Fisher
analysis to forecast the expected precision on $f_{\rm NL}$ and the
detectability of the lensing magnification and Doppler contributions to the
power spectrum. Here we assess the bias on the best-fit values of $f_{\rm NL}$
and other parameters, from neglecting these light-cone effects. We consider
forthcoming 21cm intensity mapping surveys (SKAO) and optical galaxy surveys
(DESI and Euclid), both individually and combined together. We conclude that
lensing magnification at higher redshifts must be included in the modelling of
spectroscopic surveys. If lensing is neglected in the analysis, this produces
a bias of more than 1$\sigma$ – not only on $f_{\rm NL}$, but also on the
standard cosmological parameters.

##  1 Introduction

An important scientific goal of forthcoming galaxy surveys is to tighten
constraints on primordial non-Gaussianities with large-scale structure. In a
companion paper [1], we used a Fisher forecast to estimate the constraints on
the local-type primordial non-Gaussianity parameter $f_{\rm NL}$, from its
effect on the clustering bias of future surveys with high redshift resolution:
a bright galaxy sample (similar to the BGS survey with the Dark Energy
Spectroscopic Instrument (DESI) [2]), an H$\alpha$ survey (similar to that of
Euclid [3]), and 21cm intensity mapping surveys like those planned for the
SKAO-MID telescope, in lower- and higher-frequency bands, denoted IM1 and IM2
[4]. This multi-wavelength choice of four surveys is motivated by: high
redshift resolution in order to detect the Doppler effect; good coverage of
redshifts in the range $0<z<2$; a negligible cross-shot noise between optical
and 21cm intensity samples; and the very different systematics affecting
optical and 21cm radio surveys, which are suppressed in cross-correlations. We
showed in [1] that the full combination of four surveys could achieve a
constraint of $\sigma(f_{\rm NL})\sim 1.5$, detect the Doppler effect with a
signal-to-noise ratio $\sim$8, and isolate the contribution from lensing
magnification with $\sim$2% precision. Basic properties of the surveys and the
uncertainties on $f_{\rm NL}$ and relativistic effects are shown in Table 1.

Table 1: Survey volumes and uncertainties on $f_{\rm NL}$ and relativistic
terms in (1.1), marginalising over the standard cosmological parameters
(excluding priors). $\sigma(\varepsilon_{\rm P})$ is always $\gg 1$ and is not
shown. Uncertainties when marginalising also over the Gaussian clustering
biases in each bin, $b_{A}(z_{i})$, are in brackets. $\otimes$ is a multi-
tracer combination and $\oplus$ is a sum of Fisher information. (From [1].)

Redshift Survey $\Omega_{\rm sky}/{\rm deg}^{2}$ $\sigma(f_{\rm NL})$
$\sigma(\varepsilon_{\rm D})$ $\sigma(\varepsilon_{\rm L})$ $0.1-0.58$ BGS
(DESI-like) 15,000 26.36 (26.38) 7.57 (7.57) 0.32 (0.39) IM2 (SKAO-like)
20,000 35.33 (35.74) 18.04 (18.07) $-$ IM2$\otimes$BGS 10,000 2.10 (2.12) 0.14
(0.14) 0.12 (0.13) $0.9-1.8$ H$\alpha$ (Euclid-like) 15,000 9.32 (9.34) 9.08
(9.08) 0.04 (0.04) $0.35-3.05$ IM1 (SKAO-like) 20,000 4.65 (4.72) 6.28 (6.29)
$-$ $0.60-3.05$ IM1$\otimes$H$\alpha$ 10,000 3.05 (3.06) 0.37 (0.37) 0.03
(0.03) $0.1-3.05$ (IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 10,000 1.70
(1.70) 0.13 (0.13) 0.03 (0.03) IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$
10,000 1.55 (1.55) 0.13 (0.13) 0.02 (0.02)

Table 1 shows that marginalisation over the clustering biases degrades the
single-tracer constraints slightly, but has a negligible effect on multi-
tracer constraints. This follows from the very large number of redshift bins
and consequent cross-bin correlations.

Relativistic light-cone effects are usually neglected in the modelling of
galaxy and 21cm intensity mapping surveys. This has been mainly a reasonable
approximation up to now, but next-generation surveys will make higher demands
on theoretical accuracy. In previous galaxy surveys with smaller survey sizes,
or at lower redshifts, or with shot-noise-dominated samples, it was safe not
to consider the light-cone corrections in the number density contrast. But as
future surveys probe larger volumes, go to higher redshifts, and reduce shot-
noise, it is important to understand whether neglecting such effects in the
modelling can bias our measurements of parameters.

The observed number density/ temperature contrast $\Delta_{A}$ is given by the
contrast at the source $\delta_{A}$, modulated not only by redshift-space
distortions, but also by lensing magnification and other relativistic effects
(see [1] for more details):

$\displaystyle\Delta_{A}=\delta_{A}+\Delta^{\rm RSD}+\varepsilon_{\rm
D}\Delta^{\rm Dopp}+\varepsilon_{\rm L}\Delta^{\rm Lens}+\varepsilon_{\rm
P}\Delta^{\rm Pot}\,.$ (1.1)

The $\varepsilon_{\rm I}$ parameters have fiducial value 1, corresponding to
the correct theoretical expression. The errors $\sigma(\varepsilon_{\rm I})$
give an estimate of the detectability of these contributions. The Doppler
term, $\Delta^{\rm Dopp}={[b_{\rm e}-5s+(5s-2)/{\mathcal{H}}\chi+{\rm
d}\ln{\mathcal{H}}/{\rm d}\ln(1+z)]}\bm{v}\cdot\bm{n}$, is sourced by radial
peculiar velocities; the lensing term, $\Delta^{\rm Lens}={(5s-2)}\kappa$, is
sourced by the lensing convergence, and the potential term is sourced by
Sachs-Wolfe, integrated Sachs-Wolfe, and time-delay effects. Here $s$ is the
magnification bias (with $5s-2=0$ for intensity mapping) and $b_{\rm e}$ is
the evolution bias.

There is an important distinction between the lensing term and the other two
relativistic terms:

  * •

The lensing convergence $\kappa$ receives contributions from matter
fluctuations $\delta_{\rm m}$ all along the line-of-sight and therefore mixes
different scales [5]. This means that the scale dependence of $\kappa$ differs
from that of $\delta_{\rm m}$. As a result, the lensing term $\Delta^{\rm
Lens}$ is not fully degenerate, but partially degenerate with the clustering
term $\delta_{A}=b_{A}\,\delta_{\rm m}$ [6, 7, 8].

  * •

The Doppler term scales in Fourier space as $\Delta^{\rm Dopp}\sim\delta_{\rm
m}\,H_{0}/k$. It is only non-negligible on ultra-large scales.

  * •

The potential term scales in Fourier space as $\Delta^{\rm
Pot}\sim\Phi\sim\delta_{\rm m}\,H_{0}^{2}/k^{2}$, where $\Phi$ is the Bardeen
potential. This is only non-negligible on even larger scales than the Doppler
term.

  * •

In correlations, the leading Doppler contribution is $\delta_{\rm
m}\times\mbox{Doppler}\sim(\delta_{\rm m})^{2}\,H_{0}/k$, which dominates over
the leading potential contribution, $\delta_{\rm m}\times\Phi\sim(\delta_{\rm
m})^{2}\,H_{0}^{2}/k^{2}$.

  * •

The ultra-large scale relativistic effects are partially degenerate with the
contribution of scale-dependent clustering bias from $f_{\rm NL}$. A simple
model of scale-dependent bias is given by (see [9] for improved models):

$\delta_{\rm g}(z,\bm{k})=\left[b(z)+3{f_{\rm NL}}\,\frac{\delta_{\rm
crit}\big{[}b(z)-1\big{]}{D(z_{\rm d})(1+z_{\rm d})\,\Omega_{\rm
m0}H_{0}^{2}}}{D(z)\,T(k)\,k^{2}}\right]\delta_{\rm m}(z,\bm{k})\,,$ (1.2)

where $\delta_{\rm crit}=1.686$ is the threshold density contrast for
spherical collapse, $T$ is the matter transfer function (normalised to 1 on
ultra-large scales), $D$ is the growth factor (normalised to 1 at $z=0$), and
$z_{\rm d}$ is the redshift at decoupling.

The Doppler term dominates over the potential contributions on ultra-large
scales. Since it has $H_{0}/k$ and $(H_{0}/k)^{2}$ contributions, it is
partially degenerate with the effect of $f_{\rm NL}$, and it follows that
neglecting the Doppler contribution could bias future measurements of $f_{\rm
NL}$, as pointed out in [10, 11, 12, 13, 7, 14, 15, 16] (see also [17, 18, 19,
20] for the galaxy bispectrum). The degree of bias is determined by the
amplitudes of the Doppler and $f_{\rm NL}$ contributions, which depend on the
surveys considered. Neglecting the lensing contribution can also lead to a
biased measurement of $f_{\rm NL}$ – by biasing the estimate of clustering
bias that affects the amplitude of the $f_{\rm NL}$ contribution [6, 7]. In
general, the conclusion is that one needs to include the relativistic effects
in the modeling to avoid a biased estimation of the $f_{\rm NL}$ best-fit
value. However, the degree of bias depends on the survey specifications. One
should note that $f_{\rm NL}$ is also sensitive to bias from observational
systematics such as stellar contamination [21] and foreground contamination of
21cm intensity maps [22, 23]. The analysis in our companion paper [1] takes
into account these systematics, and shows that their effect on $\sigma(f_{\rm
NL})$ is considerably reduced in multi-tracer constraints.

In addition to primordial non-Gaussianity, we can ask whether neglecting the
relativistic effects will also bias the measurements of standard cosmological
parameters [24, 25, 5, 7, 26, 8]. The overall conclusion of previous work is
that the neglect of lensing can produce a significant bias, while the neglect
of Doppler and potential contributions has negligible impact. Our results are
consistent with this.

To fully include all relativistic effects, all radial correlations of redshift
bins, and all wide-angle effects, we use the angular power spectra

$C^{AB}_{\ell}(z^{A}_{i},z^{B}_{j})=4\pi\int{\rm d}\ln
k\,\Delta_{A\ell}(z^{A}_{i},k)\,\Delta_{B\ell}(z^{B}_{j},k)\,\mathcal{P}_{\zeta}(k)\,,$
(1.3)

which also do not require any Alcock-Paczynski correction. Here
$\mathcal{P}_{\zeta}(k)$ is the primordial power spectrum, and $A,B$ denote
tracers. We study the importance of including relativistic effects in single-
and multi-tracer cases, and at low and high redshifts. It turns out that for
the surveys we consider, lensing magnification (at higher redshifts) must be
included for $f_{\rm NL}$ and standard cosmological parameters. Omitting the
lensing contribution produces a signficant bias ($>1\sigma$) on the best-fit
value of $f_{\rm NL}$ and standard cosmological parameters. The bias induced
by neglecting the Doppler effect is below $1\sigma$ for all parameters, but
since this effect has signal-to-noise $\sim 8$, it should in principle be
included.

In section 2 we review how to estimate the bias on best-fit values of
parameters. Results are given in section 3 and we conclude in section 4. Our
fiducial cosmology is given by: $A_{\rm s}=2.142\times 10^{-9}$, $n_{\rm
s}=0.967$, ${\Omega_{\rm cdm0}}=0.26$, ${\Omega_{\rm b0}}=0.05$, $w=-1$,
$H_{0}=67.74\,$ km/s/Mpc, $f_{\rm NL}=0$, where $\Omega_{\rm cdm0}+\Omega_{\rm
b0}=\Omega_{\rm m0}$. We use redshift bins of width $\Delta z=0.03$; a minimum
multipole $\ell_{\rm min}=5$ to avoid large-scale systematics in the galaxy
surveys (e.g. stellar contamination) and the intensity surveys (foreground
contamination); and a maximum multipole $\ell_{\rm
max}(z)=0.2h(1+z)^{2/(2+n_{s})}\chi(z)$, to stay in the linear-perturbation
regime.

##  2 Estimating the bias on parameter measurements

One can estimate biases on the best-fit values of parameters from unaccounted
systematics or incomplete theoretical modelling in broadly two ways. In one
approach, one can use the ‘correct’ mock data and do inference on this data
using the ‘wrong’ model (see [16] for a recent example). This approach
provides a precise evaluation of the biasing introduced but is time-consuming
and computationally expensive. We follow the alternative approach, which uses
the Fisher matrix formalism in the case of nested model selection [27].

In the Fisher formalism we assume that the posterior distribution of a set of
$m$ parameters $\bm{\vartheta}$, given some $n$ dimensional data vector
$\bm{X}$, is Gaussian:

$\mathcal{P}({\bm{\vartheta}}|{\bm{X}})={\left[(2\pi)^{m}\det{(\bm{F}^{-1})}\right]^{-1/2}}\,\exp\left[-\frac{1}{2}({\bm{\vartheta}}-\bar{\bm{\vartheta}})^{T}{\bm{F}}\,({\bm{\vartheta}}-\bar{\bm{\vartheta}})\right]\,.$
(2.1)

The best-fit values $\bar{\bm{\vartheta}}$ maximise the posterior. The Fisher
matrix $\bm{F}$ is the inverse of the covariance of the parameters and the
marginal error of a parameter is given by

$\sigma^{2}(\vartheta_{\alpha})=\left[{\bm{F}}^{-1}\right]_{\vartheta_{\alpha}\vartheta_{\alpha}}\,.$
(2.2)

Using Bayes theorem, we relate the posterior with the likelihood of the data
${\cal L}$ via

${\cal P}({\bm{\vartheta}}|{\bm{X}})\propto{\cal
L}({\bm{X}}|\bm{\vartheta})\,{\Pi}(\bm{\vartheta})\,,$ (2.3)

where $\Pi$ is the prior. For simplicity we do not consider any priors on the
parameters. Assuming the likelihood to be Gaussian,

${\cal
L}({\bm{X}}|\bm{\vartheta})=\left[(2\pi)^{n}\det{\bm{\Gamma}}\right]^{-1/2}\,\exp\left[-\frac{1}{2}({\bm{X}}-\bar{\bm{X}})^{T}{\bm{\Gamma}}^{-1}({\bm{X}}-\bar{\bm{X}})\right]\,,$
(2.4)

where $\bm{\Gamma}$ is the covariance of the data, and $\bar{\bm{X}}$ is the
value that maximises the likelihood. Since both $\bar{\bm{X}}$ and
$\bm{\Gamma}$ can depend on $\bm{\vartheta}$, one sees that the modelling
between the data and the parameters we want to fit influences our estimates.

Making incorrect model assumptions will therefore lead to a shift in the best-
fit value of the parameters considered. In the example of nested models, there
is a sub-model with parameters $\psi_{i}$, so that
$\vartheta_{\alpha}=\\{\psi_{i},\varphi_{I}\\}$. Suppose that we make an
incorrect assumption, and fix the values of $\varphi_{I}$ to incorrect values
$\hat{\varphi}_{I}$, instead of the true values $\bar{\varphi}_{I}$. This
shifts $\varphi_{I}$ by the amount,

$\delta\varphi_{I}=\hat{\varphi}_{I}-\bar{\varphi}_{I}\,.$ (2.5)

As a consequence, we will estimate the remaining parameter values to be
$\hat{\psi}_{i}$, instead of their true values $\bar{\psi}_{i}$. The bias on
the best-fit values is:

$\delta\psi_{i}=\hat{\psi}_{i}-\bar{\psi}_{i}\,.$ (2.6)

We relate (2.6) to (2.5) through the covariance of the parameters, determined
by the amount of information we expect to extract from the survey [28, 23]:

$\delta\psi_{i}=-\sum_{I,j}\,\delta\varphi_{I}\,F_{\varphi_{I}\psi_{j}}\,\left[{\bm{H}}^{-1}\right]_{\psi_{j}\psi_{i}}\,,$
(2.7)

where $\bm{F}$ is the Fisher matrix of the total set $\vartheta_{\alpha}$, and
$\bm{H}$ is the Fisher matrix of the sub-model with the parameters $\psi_{i}$
that we wish to fit.

The angular power spectra are given by the covariance of the maps
${\bm{X}}=[a_{\ell m}^{A}(z^{A}_{i})]$, where $A$ ranges over the different
tracers and $i$ over the redshift bins. The Fisher matrix is (see [1]):

$\displaystyle F_{{\vartheta}_{\alpha}{\vartheta}_{\beta}}=\sum_{\ell_{\rm
min}}^{\ell_{\rm max}}\frac{(2\ell+1)}{2}f_{\rm sky}\,{\rm
Tr}\Big{[}\big{(}\partial_{{\vartheta}_{\alpha}}\bm{C}_{\ell}\big{)}\,\bm{\Gamma}_{\ell}^{-1}\big{(}\partial_{{\vartheta}_{\beta}}\bm{C}_{\ell}\big{)}\,\bm{\Gamma}_{\ell}^{-1}\Big{]}\,,$
(2.8)

where

$\bm{\Gamma}_{\ell}=\bm{C}_{\ell}+\bm{\mathcal{N}}_{\ell}\,.$ (2.9)

The noise $\bm{\mathcal{N}}$ depends on the survey: shot-noise in the case of
galaxy surveys, instrumental noise in the case of intensity mapping. We also
need to incorporate the effects of the telescope beam on intensity mapping.
(More details are given in [1]).

For the forecasts we use the parameters

${\vartheta}_{\alpha}=\Big{\\{}f_{\rm NL},\,\varepsilon_{\rm
D},\,\varepsilon_{\rm L},\,\varepsilon_{\rm P};\,\Omega_{\rm m0},\,w,\,n_{\rm
s},\,H_{0},\,A_{\rm s};\,{b_{A}(z_{i})}\Big{\\}}\,,$ (2.10)

where $b_{A}(z_{i})$ are the Gaussian clustering bias values for each tracer
in each redshift bin. We are interested in the effect of neglecting the
Doppler, lensing and potential effects, so that

$\displaystyle\varphi_{I}=\big{\\{}\varepsilon_{\rm D},\,\varepsilon_{\rm
L},\,\varepsilon_{\rm P}\big{\\}}\,.$ (2.11)

Neglecting the relativistic effects means setting $\hat{\varepsilon}_{\rm
I}=0$, which means a shift from the true values of

$\delta\varepsilon_{\rm D}=\delta\varepsilon_{\rm L}=\delta\varepsilon_{\rm
P}=-1\,.$ (2.12)

We then determine the bias on the best-fit values of the remaining parameters:

${\psi}_{i}=\big{\\{}f_{\rm NL};\,\Omega_{\rm m0},\,w,\,n_{\rm
s},\,H_{0},\,A_{\rm s}\big{\\}}\,.$ (2.13)

This bias is best expressed as normalised by the errors $\sigma(\psi_{i})$. We
thus define the normalised biases from neglecting individual relativistic
effect as

$\delta^{I}\psi_{i}\equiv\frac{\delta\psi_{i}(\varphi_{I})}{\sigma(\psi_{i})}\,,$
(2.14)

where $\delta\psi_{i}(\varphi_{I})$ denotes the case of (2.7) when $I$ is
fixed at one value and only $j$ is summed over. It follows that the normalised
bias from all relativistic effects combined is

$\delta^{\rm
rel}\psi_{i}=\frac{\delta\psi_{i}}{\sigma(\psi_{i})}\quad\mbox{where}\quad\delta\psi_{i}=\sum_{I}\delta\psi_{i}(\varphi_{I})\,.$
(2.15)

In general, if $|\delta^{I}\psi_{i}|<1$, then any induced bias is smaller than
the error bars and can be safely neglected. On the other hand, if
$|\delta^{I}\psi_{i}|>1$ then the effect $I$ should not be neglected in the
model. Note however that the approximation made in deriving (2.7) breaks down
when $|\delta^{I}\psi_{i}|>1$. For biases above $1\sigma$, the value of
$\delta^{I}\psi_{i}$ is not reliable – it does not quantify the bias, but
qualitatively it confirms that the bias is larger than the error bars. This is
sufficient for our purposes.

##  3 Results

Table 2: Marginal errors (in percentage) on standard cosmological parameters,
for individual surveys and in multi-tracer ($\otimes$) combination, including
all relativistic effects. $\oplus$ denotes the sum of independent multi-tracer
pairs (excluding the IM2 band overlap in band IM1). Results exclude priors.

Redshift Survey $\Omega_{\rm{m0}}$ $n_{\rm s}$ $H_{0}$ $w$ $A_{\rm s}$
$0.10-0.58$ BGS (DESI-like) 2.34 5.95 12.43 5.25 19.41 IM2 (SKAO-like) 3.57
8.89 18.57 4.85 28.03 IM2$\otimes$BGS 1.14 2.04 3.78 3.30 5.96 $0.90-1.80$
H$\alpha$ (Euclid-like) 1.18 3.57 6.79 3.19 10.37 $0.35-3.05$ IM1 (SKAO-like)
2.27 5.64 11.23 2.53 16.57 $0.60-3.05$ IM1$\otimes$H$\alpha$ 1.13 2.85 5.60
2.29 8.32 $0.10-3.05$ (IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$ ) 0.69
1.58 3.08 1.35 4.61 IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 0.68 1.55
3.02 1.34 4.51

###  3.1 Constraints on standard cosmological parameters

Before estimating biases on best-fit values, we determine how well the
standard cosmological parameters can be constrained. Table 2 presents the
marginal errors as a fraction of the fiducial values, expressed in
percentages. Generally the high-redshift surveys give better constraints –
since they observe a larger volume and hence sample more scales, which reduces
cosmic variance. However, in the multi-tracer combination, the low-$z$ surveys
provide better constraints than the high-$z$ surveys. The reason is that
cosmic variance is effectively cancelled, thereby removing the advantage of
bigger volume at high $z$. Furthermore, the low $z$ surveys have the advantage
of lower noise. The exception is the dark energy equation of state $w$, which
benefits from measurements before and after dark energy domination, that are
available only in IM1. When all surveys are combined, the best constraints are
achieved. In particular, the error on $\Omega_{m0}$ is sub-percent, while all
others are a few percent. The full multi-tracer combination produces a
significant improvement in precision.

Figure 1: Contour plots of the standard cosmological parameters and $f_{\rm
NL}$: full multi-tracer combination of all 4 surveys (black); low-$z$ multi-
tracer pair (blue); high-$z$ multi-tracer pair (red); sum of low-$z$ and high
$z$ Fisher information (green). Fiducial values are indicated by dotted lines.
(Constraints exclude priors.)

Figure 1 displays the contour plots for the standard cosmological parameters,
together with $f_{\rm NL}$. Fiducial values are indicated by the dotted lines,
and black contours indicate the multi-tracer correlation of all the surveys.
The low- and high-$z$ multi-tracer pairs are in blue and red respectively. The
sum of their Fisher information is in green. A strong degeneracy is apparent
between $n_{\rm s}$, $H_{0}$, and $A_{\rm s}$, which is reduced as more data
sets are added. By contrast, $w$ and $\Omega_{\rm m0}$, are differently
degenerate with the other parameters at low and high redshifts. Except for
$w$, all cosmological parameters appear to be uncorrelated with $f_{\rm NL}$,
which is not unexpected.

Another feature of Figure 1 and Table 2 is that the constraints and contours
do not improve significantly when the sum of multi-tracer pairs is replaced by
the full multi-tracer. This indicates that taking them as uncorrelated is a
good approximation, since little information is added from low-$z$ $\otimes$
high-$z$ cross-correlations. The approximation considerably decreases the
computation time needed.

Table 3: Bias on the best-fit value of each standard cosmological parameter
(normalised by the standard deviation) that follows from neglecting
relativistic effects.

Redshift Survey $\delta^{\rm rel}\Omega_{\rm m0}$ $\delta^{\rm rel}n_{\rm s}$
$\delta^{\rm rel}H_{0}$ $\delta^{\rm rel}w$ $\delta^{\rm rel}A_{\rm s}$
$0.10-0.58$ BGS 0.17 -0.17 0.18 -0.1 -0.18 IM2 0.01 -0.01 0.01 0.0 -0.02
IM2$\otimes$BGS 0.25 -0.30 0.44 -0.67 -0.51 $0.90-1.80$ H$\alpha$ 5.46 -7.23
6.98 -7.81 -7.14 $0.35-3.05$ IM1 0.01 -0.01 0.01 -0.02 -0.01 $0.60-3.05$
IM1$\otimes$H$\alpha$ 6.12 -8.71 8.04 -9.14 -8.14 $0.10-3.05$
(IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 3.42 -4.66 4.28 -4.81 -4.17
IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 6.07 -7.53 7.25 -7.34 -7.05

###  3.2 Bias from neglecting relativistic effects

We now consider the bias on the best-fit value from neglecting all
relativistic effects, beginning with the standard cosmological parameters.
Table 3 shows the biases on the best-fit of $\psi_{i}$, normalised to
$\sigma(\psi_{i})$, that follows from neglecting all relativistic effects in
the modeling – i.e., $\delta^{\rm rel}\psi_{i}$, defined in (2.15). At low
$z$, neglecting the relativistic effects is justified, even for the multi-
tracer pair IM2$\otimes$BGS. The same is true for the high-$z$ IM1 on its own.
By contrast, neglecting the relativistic effects in the H$\alpha$ survey on
its own leads to significant bias for all parameters. This bias is then passed
on to any multi-tracer that includes the H$\alpha$ survey.

Table 4: Marginal error and normalised best-fit biases $\delta^{I}f_{\rm NL}$
on $f_{\rm NL}$, from neglecting the Doppler, lensing and potential effects,
and their combination, $\delta^{\rm rel}f_{\rm NL}$.

Redshift Survey $\sigma(f_{\rm NL})$ $\delta^{\rm D}f_{\rm NL}$ $\delta^{\rm
L}f_{\rm NL}$ $\delta^{\rm P}f_{\rm NL}$ $\delta^{\rm rel}f_{\rm NL}$
$0.10-0.58$ BGS 26.38 -0.05 0.17 0.02 0.14 IM2 35.74 -0.03 0.0 0.0 -0.03
IM2$\otimes$BGS 2.12 0.04 0.49 0.09 0.62 $0.90-1.80$ H$\alpha$ 9.34 0.10 6.03
-0.08 6.06 $0.35-3.05$ IM1 4.72 0.04 0.0 -0.06 -0.02 $0.60-3.05$
IM1$\otimes$H$\alpha$ 3.06 0.08 3.11 -0.15 3.04 $0.10-3.05$
(IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 1.70 0.06 1.84 -0.01 1.89
IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 1.55 0.10 2.60 -0.01 2.69

Figure 2 shows the contours corresponding to Table 3. The low-$z$ multi-tracer
pair and the high-$z$ multi-tracer pair disagree on the best-fit value for all
parameters when relativistic effects are neglected. In other words, there is a
tension between low-$z$ and high-$z$ results, which is not eased by combining
them. This clearly exemplifies the problem of theoretical systematics.

Figure 2: As in Figure 1, but now neglecting relativistic effects, i.e.,
lensing, Doppler and potential effects. The best-fit values from the full
multi-tracer combination (black) are given above the distributions, showing
the consequent bias on the true values (marked by dotted lines).

In Table 4, we present the marginal error on $f_{\rm NL}$ and the bias on its
best-fit value, arising from neglecting the Doppler, lensing and potential
effects, (2.14), and their combination, (2.15). The first column reproduces
the results already presented in [1], and serves to normalise the biases on
the true value $\bar{f}_{\rm NL}=0$. The last column is the equivalent of
Table 3 for $f_{\rm NL}$. The columns in between break down the bias into the
three components of the relativistic effects. Note that intensity mapping is
unaffected by lensing magnification.

As in the case of the standard cosmological parameters, neglecting the
relativistic effects at low redshift does not significantly bias the best-fit
value of $f_{\rm NL}$. Similarly, the high redshift IM1 survey does not show
significant bias in $f_{\rm NL}$, and it is again only the H$\alpha$ survey
that suffers a significant bias on the best-fit. This $>1\sigma$ bias
propagates into all multi-tracer combinations with H$\alpha$. It is apparent
that the $>1\sigma$ bias is mainly due to the neglect of lensing
magnification, and it follows that lensing must be included in the analysis.

The Doppler and potential effects can lead to a bias up to $13\%$ of the error
bars, in the case of IM2$\otimes$BGS. This is a significant fraction of the
total 63% bias, with lensing contributing 50%. If we are only interested in
biases above $1\sigma$, then we can neglect the Doppler and potential
contribution. However, there may be other survey combinations for which the
Doppler and potential effects, when added to the lensing effect, push the bias
above $1\sigma$ (or pull it below $1\sigma$).

Figure 3: As in Figure 2, but now neglecting the Doppler and potential
effects, while including the lensing effect.

Figure 2 presents the contours in the case where all relativistic effects are
neglected. This incorrect model will introduce theoretical systematics in the
form of a bias on the best-fit values of the parameters. For the low-$z$
multi-tracer pair, the bias is small enough that the fiducials are still
contained within the $1\sigma$ contours. On the other hand, the high-$z$
multi-tracer pair shows strong biases in all best-fit values, in tension with
the low-$z$ pair. In particular, the wrong theoretical model in the high $z$
case lead us to detect a spurious $f_{\rm NL}\sim 10$ at 3$\sigma$, whereas
the true value implies Gaussian initial conditions. It is also apparent that
when combining the low- and high-$z$ data sets, the spurious $f_{\rm NL}$
detection remains ($\sim 4$ at $\sim 2\sigma$).

Table 3, Table 4 and Figure 2 show that the biases in the best-fit parameters
come overwhelmingly from the Euclid-like H$\alpha$ survey. It is clear that
relativistic effects must be included in the modeling for theoretical
accuracy. Table 4 confirms that for the surveys considered, we can safely omit
the Doppler and potential effects.

To confirm this, we compute the bias on the best-fit value from neglecting the
Doppler and potential but keeping the lensing effect. Figure 3 shows that
including lensing in the modelling is sufficient to de-bias all parameters.
Although some residual bias remains, it is within the 1$\sigma$ contours.
Therefore we conclude that for these surveys, and their combinations, it is
safe to neglect Doppler and potential effects, if the goal is to measure
$f_{\rm NL}$ and the standard cosmological parameters.

We emphasise that the Doppler contribution itself is detectable, with signal-
to-noise of $\sim 8$, and for the purpose of detection, it should be included
in the modeling. We do not expect any significant bias on the standard
cosmological parameters from neglecting the Doppler term, since this term is
only non-negligible on ultra-large scales, which contribute little to standard
constraints. This is confirmed in Table 5 to Table 9 below. The $f_{\rm NL}$
constraints do rely critically on ultra-large scales, and neglecting the
Doppler effect biases the $f_{\rm NL}$ best-fit by 10% for the full multi-
tracer combination of the surveys considered here (Table 4). There may be
other combinations of surveys for which the neglect of the Doppler term
produces a more significant bias on the $f_{\rm NL}$ best-fit.

###  3.3 Gaussian universe

In a Gaussian model of the Universe, with $f_{\rm NL}=0$, do we need to
include relativistic effects to avoid any bias in measurements of the standard
cosmological parameters? We summarise the results for single- and multi-tracer
cases in Table 5 to Table 9 for the parameters $\Omega_{\rm m0}$, $n_{\rm s}$,
$H_{0}$, $w$ and $A_{\rm s}$.

Table 5: Normalised best-fit bias in the matter density $\delta^{I}\Omega_{\rm
m0}$, from neglecting Doppler, lensing and potential effects, and their
combination $\delta^{\rm rel}\Omega_{\rm m0}$.

Redshift Survey $\delta^{\rm D}\Omega_{\rm m0}$ $\delta^{\rm L}\Omega_{\rm
m0}$ $\delta^{\rm P}\Omega_{\rm m0}$ $\delta^{\rm rel}\Omega_{\rm m0}$
$0.10-0.58$ BGS 0.01 0.17 0.0 0.18 IM2 0.02 $-$ 0.0 0.02 IM2$\otimes$BGS -0.15
0.4 0.01 0.26 $0.90-1.80$ H$\alpha$ 0.01 5.59 0.0 5.6 $0.35-3.05$ IM1 -0.01
$-$ 0.01 0.01 $0.60-3.05$ IM1$\otimes$H$\alpha$ -0.09 6.24 0.01 6.15
$0.10-3.05$ (IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) -0.01 3.46 0.0
3.45 IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 0.0 6.13 0.0 6.13

Table 6: As in Table 5, for the spectral index $n_{\rm s}$.

Redshift Survey $\delta^{\rm D}n_{\rm s}$ $\delta^{\rm L}n_{\rm s}$
$\delta^{\rm P}n_{\rm s}$ $\delta^{\rm rel}n_{\rm s}$ $0.10-0.58$ BGS 0.0
-0.18 0.0 -0.19 IM2 -0.01 $-$ 0.0 -0.01 IM2$\otimes$BGS 0.03 -0.32 -0.01 -0.30
$0.90-1.80$ H$\alpha$ -0.02 -7.57 -0.01 -7.60 $0.35-3.05$ IM1 -0.01 $-$ 0.01
0.01 $0.60-3.05$ IM1$\otimes$H$\alpha$ 0.10 -8.96 -0.01 -8.87 $0.10-3.05$
(IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 0.02 -4.76 0.0 -4.74
IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 0.02 -7.65 -0.01 -7.63

Table 7: As in Table 5, for the Hubble parameter $H_{0}$.

Redshift Survey $\delta^{\rm D}H_{0}$ $\delta^{\rm L}H_{0}$ $\delta^{\rm
P}H_{0}$ $\delta^{\rm rel}H_{0}$ $0.10-0.58$ BGS 0.0 0.19 0.0 0.20 IM2 0.02
$-$ 0.0 0.02 IM2$\otimes$BGS 0.05 0.39 0.01 0.44 $0.90-1.80$ H$\alpha$ 0.02
7.42 0.0 7.44 $0.35-3.05$ IM1 0.0 $-$ 0.01 0.01 $0.60-3.05$
IM1$\otimes$H$\alpha$ -0.10 8.26 0.01 8.17 $0.10-3.05$
(IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 0.0 4.35 0.01 4.36
IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 0.0 7.34 0.01 7.36

Table 8: As in Table 5, for the dark energy equation of state $w$.

Redshift Survey $\delta^{\rm D}w$ $\delta^{\rm L}w$ $\delta^{\rm P}w$
$\delta^{\rm rel}w$ $0.10-0.58$ BGS 0.03 -0.14 -0.01 -0.12 IM2 0.0 $-$ 0.0 0.0
IM2$\otimes$BGS -0.25 -0.43 -0.01 -0.69 $0.90-1.80$ H$\alpha$ -0.03 -8.59 -0.0
-8.63 $0.35-3.05$ IM1 -0.03 $-$ 0.01 -0.02 $0.60-3.05$ IM1$\otimes$H$\alpha$
0.05 -9.63 0.0 -9.58 $0.10-3.05$
(IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 0.07 -4.97 -0.02 -4.93
IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ 0.06 -7.52 -0.03 -7.48

Table 9: As in Table 5, for the primordial power spectrum amplitude $A_{\rm
s}$.

Redshift Survey $\delta^{\rm D}A_{\rm s}$ $\delta^{\rm L}A_{\rm s}$
$\delta^{\rm P}A_{\rm s}$ $\delta^{\rm rel}A_{\rm s}$ $0.10-0.58$ BGS 0.01
-0.21 -0.01 -0.21 IM2 -0.02 $-$ 0.0 -0.02 IM2$\otimes$BGS -0.18 -0.33 0.0
-0.52 $0.90-1.80$ H$\alpha$ -0.03 -7.71 0.0 -7.74 $0.35-3.05$ IM1 0.0 $-$
-0.01 -0.01 $0.60-3.05$ IM1$\otimes$H$\alpha$ 0.10 -8.39 -0.01 -8.30
$0.10-3.05$ (IM2$\otimes$BGS)$\oplus$(IM1$\otimes$H$\alpha$) 0.0 -4.23 -0.01
-4.24 IM2$\otimes$BGS$\otimes$IM1$\otimes$H$\alpha$ -0.01 -7.13 -0.01 -7.15

As expected, neglecting the Doppler and potential effects does not bias any
best-fit values significantly. The largest bias is $\sim 25\%$ on $w$ in the
low-$z$ multi-tracer IM2$\otimes$BGS, which is below $1\sigma$. For the
lensing effect, the same applies at low redshifts. Once again, it is only a
Euclid-like H$\alpha$ survey, and its combinations with the other surveys,
that leads to significant biases in all parameters when lensing is neglected.
Although the trend is the same as found previously, the relative biases are
generally smaller. This may be caused by the reduction in the parameter space
volume when $f_{\rm NL}$ is fixed to zero. In any case, such marginal
reduction should be interpreted qualitatively, given the approximation used to
compute the bias. The take-home message is that for high-$z$ spectroscopic
surveys, lensing magnification must be included for unbiased measurements of
the standard cosmological parameters.

##  4 Conclusion

In this paper, we extended our investigation in [1] of the constraining power
of combinations of next-generation large-scale structure surveys in the
optical and radio: DESI-like BGS and Euclid-like H$\alpha$ galaxy surveys,
together with SKAO-like 21cm intensity mapping surveys in lower- and higher-
frequency bands. Our choice was motivated by: high redshift resolution (to
detect the Doppler effect); good coverage of redshifts in the range $0<z<2$; a
negligible cross-shot noise between optical and 21cm intensity samples; and
the very different systematics affecting optical and 21cm radio surveys.

In [1], we included all relativistic observational effects on the power
spectrum and used a multi-tracer analysis to forecast the precision on $f_{\rm
NL}$ and to determine the detectability of the relativistic effects. Here we
focused on the potential theoretical systematic bias on measurements of
$f_{\rm NL}$ and standard cosmological parameters, which can arise if the
relativistic effects are neglected in the modeling. The observable angular
power spectra $C^{AB}_{\ell}(z_{i},z_{j})$ are used in the analysis, since
they naturally include relativistic light-cone effects, wide-angle effects,
and correlations between all redshift bins, and do not require an Alcock-
Paczynski correction.

We first performed a Fisher analysis to estimate the expected precision on the
cosmological parameters – using the correct theoretical model, which includes
lensing, Doppler and potential effects, in addition to the standard redshift-
space distortion effect. Table 2 shows that the multi-tracer significantly
improves on single-tracer precision, due to the combination of information and
the elimination of cosmic variance. The contour plots in Figure 1 visually
demonstrate the improvement in precision from combining low- and high-$z$
survey combinations, as well as showing the breaking of degeneracies between
several parameters. The same qualitative features apply to the precision on
$f_{\rm NL}$, which was computed in [1] and is shown in Table 4.

Then we investigated what happens when we use the incorrect theoretical model,
i.e. when we neglect one or more of the relativistic effects in the model.
This leads to a theoretical systematic that threatens accuracy – by biasing
the best-fit (or measured) values of the parameters, as given by (2.7), (2.14)
and (2.15). The question is: how large is this bias for $f_{\rm NL}$ and the
cosmological parameters? If the bias is $<1\sigma$, the relativistic effect
can be neglected if necessary; otherwise it must be included. Each parameter
in the model that is not fixed will be biased by disregarding a relativistic
effect. The more free parameters there are, the greater the parameter space
volume and hence the larger the potential bias.

The results on best-fit bias are summarised in Table 3–Table 9 and Figure 2.
When we separate the relativistic effects, we see that only the neglect of
lensing leads to a bias above $1\sigma$ on $f_{\rm NL}$ and cosmological
parameters, while the neglect of Doppler leads to at most a 25% bias. If we
are pressed to save computation time, we can therefore neglect the Doppler and
potential effects. This is confirmed in Figure 3.

It is clear that lensing effects cannot be neglected for the full multi-tracer
combination considered here – or for any multi-tracer combination involving
the H$\alpha$ survey, including the H$\alpha$ survey on its own. The special
role of the H$\alpha$ survey is due to: (a) its high redshift reach which
boosts the lensing effects, as shown in Figure 4; (b) the fact that the 21cm
intensity surveys are unaffected by lensing magnification, although they can
contribute to the lensing of galaxies in cross-correlations (see [1]). The BGS
galaxy survey can also detect the lensing effect, but only at low
significance, given its low redshift reach.

We confirmed that the same qualitative statements apply in the case of the
bias on the cosmological parameters in a Gaussian universe, where $f_{\rm NL}$
is fixed at zero. The results are summarised in Table 5 to Table 9.

One might ask how lensing drives the bias on $f_{\rm NL}$, given that its
signal does not require ultra-large scales in order to be significant. The
point is that the lensing magnification contribution $(5s-2)\,\kappa$ is a
weighted average along the line of sight of the matter density contrast – and
therefore it can partially mimic a change in the Gaussian clustering bias,
which in turn can bias the amplitude of the $f_{\rm NL}$ contribution [6, 7].

Figure 4: Signal-to-noise $S(z_{i},z_{j})$ of the lensing magnification
contribution in the Euclid-like H$\alpha$ survey over its redshift range
$0.9<z<1.8$.

Our results are broadly consistent with previous work on galaxy surveys, in
particular [6, 13, 25, 5, 7, 29, 15], but we consider a different combination
of surveys and we use the full multi-tracer combination of four surveys.

In [8], the bias on cosmological parameters is negligible for spectroscopic
surveys but significant for photometric surveys. However, [8] uses the 2-point
correlation function, without cross-bin correlations, for spectroscopic
surveys. A similar result was found in [26] for spectroscopic surveys using
the angular power spectrum in a hybrid approach which, like [8], does not
consider correlations from large redshift differences. By contrast, we include
all cross-bin correlations amongst many thin bins. In Figure 4, we show the
lensing magnification contribution from each individual auto- and cross-bin
correlation of a Euclid-like H$\alpha$ survey. The signal-to-noise ratio $S$
is given by equation (4.2) of [1]. It is clear that widely separated cross-bin
correlations have the highest signal-to-noise. This accounts for our different
conclusion – and also explains why we agree with the result of [8] on
photometric surveys, for which they do include cross-bin correlations via a
tomographic analysis.

The key point is that, in order to avoid serious bias on the best-fit values
of $f_{\rm NL}$ and cosmological parameters, the effect of lensing
magnification on the galaxy power spectrum must be included in upcoming
surveys which cover high redshifts. The inclusion of lensing in the
theoretical modeling highlights the importance of good-precision estimates of
the lensing magnification bias parameter (see also [30, 31, 24, 32, 33, 25, 5,
7, 8, 15, 34]). The lensing magnification and associated parameter $s$ are
given by [31, 34]

$\Delta^{\rm Lens}=(5s-2)\,\kappa\quad\mbox{where}\quad
s={\partial\log\bar{n}_{\rm g}\over\partial m_{\rm c}}={2\over
5}\,{\phi\over\bar{n}_{\rm g}}\,.$ (4.1)

Here $\bar{n}_{\rm g}$ is the comoving number density at the source, which is
given by an apparent magnitude integral of the luminosity function $\phi$,
with limiting apparent magnitude $m_{\rm c}$. Measurements of the luminosity
function, therefore, provide an estimate of $s$, and the errors on this
estimate can be modelled by simulations.

The precision on the lensing and Doppler contributions would be washed away if
$s$ and $b_{e}$ are poorly measured. We have partially allowed for
uncertainties in $s$ and $b_{e}$ by marginalising over the lensing and Doppler
parameters $\varepsilon_{\rm L}$ and $\varepsilon_{\rm D}$. Based on the
analysis in [32], we can estimate that errors on $s$ and $b_{e}$ need to be
$\lesssim 10\%$ in order to preserve detectability of the lensing and Doppler
effects.

Finally, we note that our simplified analysis, based on Fisher forecasts,
means that our estimates of the impact of lightcone effects should be regarded
as optimistic. We have fully included uncertainties from cosmological
parameters and from the modelling of Gaussian clustering biases
$b_{A}(z_{i})$. These are important, but they have little impact on the multi-
tracer, as shown in Table 1. Observational systematics have not been
incorporated into our analysis. Systematics on ultra-large scales include
stellar contamination and dust extinction for galaxy surveys (see e.g. [21]),
and foreground contamination for intensity surveys (see e.g. [35, 22, 23]). We
have made some allowance for these systematics by excluding the largest scales
via the cut $\ell\geq\ell_{\rm min}=5$. In Figure 7 of [1] we showed that the
full multi-tracer constraints on $f_{\rm NL}$, $\varepsilon_{\rm L}$ and
$\varepsilon_{\rm D}$ are robust to an increase of $\ell_{\rm min}$, up to
$\sim 10-20$.

## Acknowledgments

We thank Ruth Durrer for a useful discussion. JV and RM acknowledge support
from the South African Radio Astronomy Observatory and the National Research
Foundation (Grant No. 75415). RM also acknowledges support from the UK Science
& Technology Facilities Council (STFC) (Grant No. ST/N000550/1). JF
acknowledges support from the UK STFC (Grant No. ST/P000592/1). This work made
use of the South African Centre for High-Performance Computing, under the
project Cosmology with Radio Telescopes, ASTRO-0945.

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