# The Effective Field Theory of Large-Scale Structure and Multi-tracer II:
redshift space and realistic tracers

Thiago Mergulhão  Henrique Rubira  Rodrigo Voivodic

###### Abstract

We extend the multi-tracer (MT) formalism of the effective field theory of
large-scale structure to redshift space, comparing the results of MT to a
single-tracer analysis when extracting cosmological parameters from
simulations. We used a sub-halo abundance matching method to obtain more
realistic multi-tracer galaxy catalogs constructed from N-body simulations.
Considering different values for the sample shot noise and volume, we show
that the MT error bars on $A_{s}$, $\omega_{\rm cdm}$, and $h$ in a full-shape
analysis are approximately $50\%$ smaller relative to ST. We find that
cosmological and bias coefficients from MT are less degenerate, indicating
that the MT parameter basis is more orthogonal. We conclude that using MT
combined with perturbation theory is a robust and competitive way to
accommodate the information present in the mildly non-linear scales.

##  1 Introduction

The large-scale structure of the Universe (LSS) is a key probe for cosmology.
Ongoing spectroscopic and photometric surveys, such as BOSS, eBOSS, and DES
[1, 2, 3] have enlarged our knowledge of structure formation and in the short
future DESI, Euclid, and LSST [4, 5, 6] data will also sum up to current
analysis. LSS data will be vital, for instance, to shed light on the tensions
in the values of the Hubble constant [7] and the amplitude of primordial
fluctuations.

The current state-of-the-art LSS analysis includes extracting data from the
(mildly) non-linear regime using the full-shape (FS) modeling of the galaxy
power spectrum and higher-order n-point functions. This approach builds upon
the effective field theory of LSS (EFTofLSS) [8, 9, 10, 11, 12, 13] and the
large-scale bias expansion of matter tracers [14, 15]. It has achieved
remarkable results when applied to BOSS data [16, 17, 18, 19, 20, 21, 22] and
when probing new physics [23, 24, 25, 26, 27, 28, 29, 30]. One of its
downsides is that it requires a few free parameters (counter-terms, bias, and
stochastic coefficients) to be fitted in data, although those parameters have
a clear physical motivation and guarantee the correct parametrization of
structure formation on the mildly non-linear regime. The quality of the EFT
modeling is therefore attached to how well those free parameters can be
determined by the data available.

In parallel, multi-tracer (MT) techniques are another way to boost the
information extracted from galaxy surveys and have been used to probe
primordial non-gaussianities and redshift-space distortions [31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43]. MT was also developed in the context of
the eBOSS, both in configuration and Fourier space, considering correlations
between luminous red galaxy (LRG) and emission line galaxy (ELG) [44, 45] and
also voids [46]. In those works, the principal motivation for using MT is that
it leads to cosmic variance cancellation.

As we have motivated in [47], the gains of doing multi-tracer go beyond cosmic
variance cancellation in linear scales. We have shown that a real-space FS
analysis of MT combined with EFTofLSS outperforms the usual single-tracer (ST)
approach, improving the constraints on cosmological parameters with error bars
approximately 60% smaller relative to ST. The MT improvement can be attributed
to a degeneracy break between cosmological and EFT parameters. Via the MT
splitting of a tracer into sub-samples, one can single out non-linear effects
in the gravitational evolution of each sub-species and add the cross power
spectrum as a new source of information. On the other hand, single-tracer
analysis smooths out non-linear dynamics for each tracer into a single
averaged specie. Therefore, despite MT having more free parameters, we
concluded that those parameters were better determined by the same data.

In this work, we extend the EFTofLSS results in redshift space to allow for
multiple tracers and test it by performing a set of Markov chain Monte Carlo
(MCMC) parameter extraction from simulations. We use the TheOne simulation
from the BACCO project [48, 49] together with the subhalo abundance matching
extended (SHAMe) method to populate halos with galaxies [50]. We also perform
a tracer split into two populations following the galaxy’s star formation rate
(SFR) provided by SHAMe, which is very similar to dividing galaxies by their
color [51]. This improvement makes the comparison to galaxy surveys more
reliable compared to our previous analysis [47], which focused mainly on halos
split by their masses. Moreover, we analyze the dependence of the MT method on
the shot noise of the sample and the volume of the box considered, providing
projections for MT performance for ongoing and future galaxy surveys. We find
a substantial improvement in the extraction of cosmological parameters (and
also in the bias parameters), ranging from $40\%$ to $80\%$ depending on the
scales and samples considered. Our results show that MT method strongly breaks
the degeneracy between cosmological and the EFT parameters, indicating that
there is considerable information gain in the tracer splitting.

This paper is structured as follows: we described the theoretical model of MT
in redshift space in Sec. 2. The simulation data, the covariance, and the MCMC
setup used are described in Sec. 3. We present the main results in Sec. 4 and
conclude in Sec. 5.

##  2 Theoretical model

In this section, we review the FS modeling of tracers in both real and
redshift space for a single-tracer. We then generalize it to accommodate
multiple tracers.

###  2.1 Single-tracer (ST)

#### Real space

The most general bias expansion for the overdensity of a tracer $A$,
$\delta^{A}$, in an operator basis $\mathcal{O}$ is given by [14, 15]

$\delta^{A}(\boldsymbol{x},\tau)=\sum_{\mathcal{O}}b_{\mathcal{O}}(\tau)\mathcal{O}(\boldsymbol{x},\tau)+\epsilon(\boldsymbol{x},\tau)+\sum_{\mathcal{O}}\epsilon_{\mathcal{O}}(\boldsymbol{x},\tau)\mathcal{O}(\boldsymbol{x},\tau)\,,$
(2.1)

where $\epsilon(\boldsymbol{x},\tau)$and
$\epsilon_{\mathcal{O}}(\boldsymbol{x},\tau)$ are stochastic fields and
$b_{\mathcal{O}}(\tau)$ are the tracer (time-dependent) bias coefficients. Up
to the third order in the fields, we find the following set of relevant
operators

$\mathcal{O}\in\left\\{\delta,\delta^{2},\delta^{3},\mathcal{G}_{2}[\Phi_{g}],\delta\mathcal{G}_{2}[\Phi_{g}],\mathcal{G}_{3}[\Phi_{g}],\Gamma_{3}[\Phi_{g},\Phi_{v}],\nabla^{2}\delta\right\\}\,,$
(2.2)

with $\delta$ being the matter overdensity, $\mathcal{G}_{2}$ an
$\mathcal{G}_{3}$ being the second and third-order Galileon operators and
$\Gamma_{3}$ a combination of Galileon operators for the gravitational
$\Phi_{g}$ and velocity potential $\Phi_{v}$. The last term in the set above,
$\nabla^{2}\delta$, accounts for the fact that tracer formation occurs over a
finite region [15, 52] and it will lead to terms proportional to
$k^{2}\delta$.

Using the set of operators in Eq. (2.1), the auto power-spectrum of tracer $A$
can be computed as [14]

$\displaystyle P^{AA}(z,k)$ $\displaystyle=$
$\displaystyle\left[b^{A}_{1}(z)\right]^{2}\left[P_{\rm lin}(k)+P_{\rm
1L}(k)\right]+b^{A}_{1}(z)b^{A}_{2}(z)\,\mathcal{I}_{\delta^{2}}(k)+2b^{A}_{1}(z)b^{A}_{\mathcal{G}_{2}}(z)\,\mathcal{I}_{\mathcal{G}_{2}}(k)$
$\displaystyle+$
$\displaystyle\left[2b^{A}_{1}(z)b^{A}_{\mathcal{G}_{2}}(z)+\frac{4}{5}b^{A}_{1}(z)b^{A}_{\Gamma_{3}}(z)\right]\mathcal{F}_{\mathcal{G}_{2}}(k)+\frac{1}{4}\left[b^{A}_{2}(z)\right]^{2}\,\mathcal{I}_{\delta^{2}\delta^{2}}(k)$
$\displaystyle+$
$\displaystyle\left[b^{A}_{\mathcal{G}_{2}}(z)\right]^{2}\,\mathcal{I}_{\mathcal{G}_{2}\mathcal{G}_{2}}(k)+b^{A}_{2}(z)b^{A}_{\mathcal{G}_{2}}(z)\,\mathcal{I}_{\delta^{2}\mathcal{G}_{2}}(k)+P_{\rm
ct}^{AA}(k)+P_{\varepsilon^{A}\varepsilon^{A}}(k)\,,$

where $P_{\rm lin}$ and $P_{\rm 1L}$ are respectively the linear matter
spectrum and its one-loop correction. For a complete expression of the
operators in Eq. (2.2) and the spectra in Eq. (2.1), see [53]. Hereafter, we
omit the time dependence. The $P_{\rm ct}$ term is a sum of two different
contributions: the dark matter sound speed counter-term $c_{s}^{2}$ and the
intrinsic halo-formation scale $R^{A}_{*}$

$P_{\rm
ct}^{AA}(k)=-2\left[\left(b^{A}_{1}\right)^{2}\frac{c_{s}^{2}}{k^{2}_{\mathrm{NL}}}+b^{A}_{1}\left(R^{A}_{*}\right)^{2}\right]k^{2}P_{\rm
lin}\equiv-2b^{A}_{\nabla^{2}\delta}b^{A}_{1}\frac{k^{2}}{k_{\rm
norm}^{2}}P_{\rm lin}\,.$ (2.4)

In the last step the two parameters $k_{\mathrm{NL}}$ and $1/R^{A}$,
respectively the non-linear scale for matter and the typical halo scale, were
rewritten as a single arbitrary scale, fixed to be $k_{\rm
norm}=0.1\,h\,\mathrm{Mpc}^{-1}$. We also redefined the bias parameter
combination as $b^{A}_{\nabla^{2}\delta}$.

Finally, the last term in Eq. (2.1) is a stochastic contribution which can be
written as an expansion in even powers of $k$ [15]:

$P_{\varepsilon^{A}\varepsilon^{A}}(k)=\frac{1}{\bar{n}_{A}}\left(1+c^{AA}_{\rm
st,0}+c^{AA}_{\rm st,2}\frac{k^{2}}{k_{\rm norm}^{2}}\right)\,$ (2.5)

with $\bar{n}_{A}$ being the number density of the tracer $A$. Note that the
stochastic parameters as defined above will capture deviations from perfect
Poisson noise.

#### Redshift space

We now move to calculate the structure of the auto-correlator for the tracer
$A$ in redshift space, later expanding it in multipoles. After changing
coordinates to redshift space, up to third-order in perturbations, the $AA$
spectrum is written as [54]

$\displaystyle P^{AA}(k,\mu)=$
$\displaystyle\left[Z^{A}_{1}(\k{)}\right]^{2}P_{\rm lin}(k)+2\int_{{\bf
q}}\left[Z^{A}_{2}({\bf q},\k{-}{\bf q})\right]^{2}P_{\rm lin}(|\k{-}{\bf
q}|)P_{\rm lin}(q)+$ (2.6) $\displaystyle 6Z_{1}^{A}(\k{)}P_{\rm
lin}(k)\int_{{\bf q}}Z_{3}^{A}({\bf q},-{\bf q},\k{)}P_{\rm
lin}(q)+P_{\text{ct}}^{AA}(k,\mu)+P_{\varepsilon^{A}\varepsilon^{A}}(k,\mu)\,,$

in which the $Z$ kernels are described in [54] and $\mu=\hat{z}\cdot\vec{k}/k$
is the angle w.r.t the line-of-sight direction $\hat{z}$. Following the
parametrization of [54], the counter-terms are111Note that there are different
parametrizations of the counter-terms and bias coefficients in redshift space
[21]. In principle, those differences may lead to different constraints in
cosmological parameters due to prior and volume effects [28, 55]. Those
differences are however small in the limit in which the data set is
constraining enough.

$\displaystyle P^{AA}_{\mathrm{ct}}(k,\mu)$ $\displaystyle=$ $\displaystyle
2P_{11}Z^{\rm A}_{1}\frac{k^{2}}{k_{\rm norm}^{2}}\left[c^{A}_{{\rm
ct},20}+c^{A}_{{\rm ct},22}\mu^{2}+c^{A}_{{\rm ct},24}\mu^{4}+c^{A}_{{\rm
ct},26}\mu^{6}\right.$ (2.7) $\displaystyle\left.+\,c^{A}_{{\rm
ct},44}\frac{k^{2}}{k_{\rm norm}^{2}}\mu^{4}+c^{A}_{{\rm
ct},46}\frac{k^{2}}{k^{2}_{\rm norm}}\mu^{6}\right].$

Similar to Eq. (2.4), we keep only a single normalization scale $k_{\rm
norm}$. Notice that the redshift-space term $P_{\rm ct}(k,\mu)$ contains
higher-order terms in $k^{2}$ when compared to its real-space version Eq.
(2.4) and [54]. Those terms serve as a proxy to capture higher-order
contributions of fingers-of-god (FoG) [56, 57, 21]. We comment later on the
importance of those terms for different tracers, highlighting the contribution
of the $\mu^{6}$ term in the context of MT.

The last term in Eq. (2.6) comes from the stochastic contribution
$\epsilon^{A}_{\rm RS}$, which also includes a velocity component

$\displaystyle\epsilon^{A}_{\rm
RS}=\delta^{A}_{\epsilon}+f\mu^{2}\theta^{A}_{\epsilon}\;,$ (2.8)

where $\theta$ is the velocity divergence and $f$ is the log derivative of the
growth factor $D$ with respect to the scale factor $f=d\log{D}/d\log{a}$. Its
power spectrum is given by222Regarding the $\theta^{A}_{\epsilon}$
contribution, due to momentum and mass conservation, we expect its first
$k^{0}$ contribution to be the same as the matter one [15]. The first (non-
degenerate) contribution then should scale as $k^{2}$. Moreover, one should
also consider stochastic contributions from the renormalized contact
operators, but it happens that up to $k^{2}$ contributions, these are
degenerate with the other $k^{2}$ contributions.

$P_{\varepsilon^{A}\varepsilon^{A}}(k,\mu)=\frac{1}{\bar{n}_{A}}\left[1+c^{AA}_{\mathrm{st},00}+c^{AA}_{\mathrm{st},20}\frac{k^{2}}{k_{\rm
norm}^{2}}+c^{AA}_{\mathrm{st},22}\frac{k^{2}}{k_{\rm
norm}^{2}}f\mu^{2}\right]\;.$ (2.9)

Therefore, a single-tracer analysis up to one-loop in perturbation theory in
redshift space requires, in addition to the cosmological parameters, the fit
of the following set of parameters:

$\left\\{b^{A}_{1},\,b^{A}_{2},\,b^{A}_{\mathcal{G}_{2}},\,b^{A}_{\Gamma_{3}},\,c^{A}_{{\rm
ct},20},\,c^{A}_{{\rm ct},22},\,c^{A}_{{\rm ct},24},\,c^{A}_{{\rm
ct},26},\,c^{A}_{{\rm ct},44},\,c^{A}_{{\rm
ct},46},\,c^{AA}_{\mathrm{st},00},\,c^{AA}_{\mathrm{st},20},\,c^{AA}_{\mathrm{st},22}\right\\}\;.$
(2.10)

We discuss the relevance of each term later in the text.

We can now project the tracer spectrum into $\ell$-multipoles

$\displaystyle P^{AA}(k,\mu)=\sum_{\ell\;{\rm even}}{\cal
L}_{\ell}(\mu)P^{AA}_{\ell}(k)\,,$ (2.11)

with

$\displaystyle P^{AA}_{\ell}(k)\equiv\frac{2\ell+1}{2}\int_{-1}^{1}d\mu\,{\cal
L}_{\ell}(\mu)P^{AA}(k,\mu)\,,$ (2.12)

to obtain the following map for the projected counter-terms onto the non-
projected ones:

$\displaystyle P^{AA}_{\mathrm{ct,\;\ell=0}}(k)$ $\displaystyle=$
$\displaystyle\frac{2}{105}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[35\left(f+3b^{A}_{1}\right)c^{A}_{\mathrm{ct},20}+7\left(3f+5b^{A}_{1}\right)c^{A}_{\mathrm{ct},22}\right.$
$\displaystyle+$
$\displaystyle\left.3\left(5f+7b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44}\right)+\frac{35}{21}\left(7f+b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)\right]\;,$ $\displaystyle
P^{AA}_{\mathrm{ct,\;\ell=2}}(k)$ $\displaystyle=$
$\displaystyle\frac{4}{21}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[7fc^{A}_{\mathrm{ct},20}+\left(6f+7b^{A}_{1}\right)c^{A}_{\mathrm{ct},22}\right.$
$\displaystyle+$
$\displaystyle\left.\left(5f+6b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44})\right)+\frac{20}{33}\left(28f+33b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)\right]\;,$ $\displaystyle
P^{AA}_{\mathrm{ct,\;\ell=4}}(k)$ $\displaystyle=$
$\displaystyle\frac{16}{385}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[11fc^{A}_{\mathrm{ct},22}+\left(15f+11b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44}\right)\right.$ $\displaystyle+$
$\displaystyle\left.\frac{15}{13}\left(14f+13b^{A}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)\right]\;.$

We extract each non-linear term of the non-linear spectrum using the publicly
available code CLASS-PT [53] (including the IR-resummation based on [58]).
Finally, we can already foresee the importance of the $\mu^{6}$ terms: that
there are in total six free terms $c^{A}_{{\rm ct},20}$, $c^{A}_{{\rm
ct},22}$, $c^{A}_{{\rm ct},24}$, $c^{A}_{{\rm ct},26}$, $c^{A}_{{\rm ct},44}$
and $c^{A}_{{\rm ct},46}$ to contemplate the two possible scalings
($k^{2}P_{\rm lin}$ and $k^{4}P_{\rm lin}$) for the three multipoles
considered here.

###  2.2 Multi-tracer (MT)

#### Real space

In this section, we generalize the ST expressions to multiple tracers both in
real and redshift space. When having two tracers $A$ and $B$, their auto-
spectra will be the same as those defined in the previous section.
Nevertheless, the cross-power spectrum will have a different form [47]:

$\displaystyle P^{AB}(k)$ $\displaystyle=$ $\displaystyle
b^{A}_{1}b^{B}_{1}\left[P_{\rm lin}(k)+P_{\rm
1L}(k)\right]+\frac{1}{2}\left(b^{A}_{1}b^{B}_{2}+b^{B}_{1}b^{A}_{2}\right)\mathcal{I}_{\delta^{2}}(k)$
(2.16) $\displaystyle+$
$\displaystyle\left(b^{A}_{1}b^{B}_{\mathcal{G}_{2}}+b^{B}_{1}b^{A}_{\mathcal{G}_{2}}\right)\mathcal{I}_{\mathcal{G}_{2}}(k)+\left[\left(b^{A}_{1}b^{B}_{\mathcal{G}_{2}}+b^{B}_{1}b^{A}_{\mathcal{G}_{2}}\right)+\frac{2}{5}\left(b^{A}_{1}b^{B}_{\Gamma_{3}}+b^{B}_{1}b^{A}_{\Gamma_{3}}\right)\right]\mathcal{F}_{\mathcal{G}_{2}}(k)$
$\displaystyle+$
$\displaystyle\frac{1}{4}b^{A}_{2}b^{B}_{2}\mathcal{I}_{\delta^{2}\delta^{2}}(k)+b^{B}_{\mathcal{G}_{2}}b^{A}_{\mathcal{G}_{2}}\mathcal{I}_{\mathcal{G}_{2}\mathcal{G}_{2}}(k)+\frac{1}{2}(b^{A}_{2}b^{B}_{\mathcal{G}_{2}}+b^{B}_{2}b^{A}_{\mathcal{G}_{2}})\mathcal{I}_{\delta^{2}\mathcal{G}_{2}}(k)$
$\displaystyle+$ $\displaystyle P_{\rm
ct}^{AB}(k)+P_{\varepsilon^{A}\varepsilon^{B}}(k)\,,$

where we have for $P_{\rm ct}^{AB}$ and for the cross-stochastic term
$P_{\varepsilon^{A}\varepsilon^{B}}$

$\displaystyle P_{\rm ct}^{AB}(k)$ $\displaystyle=$ $\displaystyle-k^{2}P_{\rm
lin}(k)\left[2\frac{c_{s}^{2}b^{A}_{1}b^{B}_{1}}{k_{\mathrm{NL}^{2}}}+b^{A}_{1}\left(R^{B}_{*}\right)^{2}+b^{B}_{1}\left(R^{A}_{*}\right)^{2}\right]$
(2.17) $\displaystyle=$
$\displaystyle-\left(b^{A}_{\nabla^{2}\delta}b^{B}_{1}+b^{B}_{\nabla^{2}\delta}b^{A}_{1}\right)k^{2}P_{\rm
lin}\;,$ $\displaystyle P_{\varepsilon^{A}\varepsilon^{B}}(k)$
$\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\bar{n}_{A}\bar{n}_{B}}}\left[c^{AB}_{\mathrm{st},0}+c^{AB}_{\mathrm{st},2}\frac{k^{2}}{k^{2}_{\mathrm{norm}}}\right]\,.$
(2.18)

The cross-stochastic term in MT is often neglected since the stochastic
contribution depends on the formation history of each tracer on very small
scales and is therefore uncorrelated [15]. However, exclusion effects and
satellite galaxies effects [59] may lead to non-Poissonian noise and increase
the cross-correlation between the stochastic components for tracers $A$ and
$B$ generating e.g. terms proportional to powers of $k^{2}$. In the first work
[47], we found a cross-stochastic term consistent with zero and that the
inclusion of that term does not deteriorate parameter extraction. In this
work, again, we investigate the impact of that cross-stochastic term, which
turns out to be very mild (see Sec. 4.1).

#### Redshift space

After symmetrizing Eq. (2.6), we get

$\displaystyle P^{AB}(k,\mu)$ $\displaystyle=$ $\displaystyle
Z^{A}_{1}(k)Z^{B}_{1}(k)P_{\rm lin}(k)+2\int_{{\bf q}}Z^{A}_{2}({\bf
q},\k{-}{\bf q})Z^{B}_{2}({\bf q},\k{-}{\bf q})P_{\rm lin}(z,|\k{-}{\bf
q}|)P_{\rm lin}(q)$ (2.19) $\displaystyle+$ $\displaystyle
3Z_{1}^{A}(\k{)}P_{\rm lin}(k)\int_{{\bf q}}Z_{3}^{B}({\bf q},-{\bf
q},\k{)}P_{\rm lin}(q)+3Z_{1}^{B}(\k{)}P_{\rm lin}(k)\int_{{\bf
q}}Z_{3}^{A}({\bf q},-{\bf q},\k{)}P_{\rm lin}(q)$ $\displaystyle+$
$\displaystyle
P_{\text{ct}}^{AB}(k,\mu)+P_{\varepsilon^{A}\varepsilon^{B}}(k,\mu)\,,$

where we have for the counter-terms

$\displaystyle P_{\rm ct}^{AB}(k,\mu)$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[Z^{A}_{1}\left(c^{B}_{{\rm ct},20}+c^{B}_{{\rm
ct},22}\mu^{2}+c^{B}_{{\rm ct},24}\mu^{4}+c^{B}_{{\rm
ct},44}\frac{k^{2}}{k_{\rm norm}^{2}}\mu^{4}+\right.\right.$ $\displaystyle+$
$\displaystyle\left.\left.c^{B}_{{\rm ct},26}\mu^{6}+c^{B}_{{\rm
ct},46}\frac{k^{2}}{k_{\rm norm}^{2}}\mu^{6}\right)+A\leftrightarrow
B\right]\,,$

and for the stochastic term

$\displaystyle P_{\varepsilon^{A}\varepsilon^{B}}(k,\mu)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\bar{n}_{A}\bar{n}}_{B}}\left[c^{AB}_{\rm
st,00}+c^{AB}_{\rm st,20}\frac{k^{2}}{k_{\rm norm}^{2}}+c^{AB}_{\rm
st,22}\frac{k^{2}}{k_{\rm norm}^{2}}f\mu^{2}\right]\,.$ (2.21)

We study the impact of including this cross-stochastic term in Sec. 4.1.

The counter-terms for the cross-power spectrum, after using the projection
(2.12), are

$\displaystyle P^{AB}_{\mathrm{ct,\;\ell=0}}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{105}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[35\left(f+3b^{B}_{1}\right)c^{A}_{\mathrm{ct},20}+7\left(3f+5b^{B}_{1}\right)c^{A}_{\mathrm{ct},22}\right.$
$\displaystyle+$
$\displaystyle\left.3\left(5f+7b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44}\right)+\frac{35}{21}\left(7f+b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)\right.$ $\displaystyle+$
$\displaystyle\left.A\leftrightarrow B\right]\;,$ $\displaystyle
P^{AB}_{\mathrm{ct,\;\ell=2}}(k)$ $\displaystyle=$
$\displaystyle\frac{2}{21}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[7fc^{A}_{\mathrm{ct},20}+\left(6f+7b^{B}_{1}\right)c^{A}_{\mathrm{ct},22}\right.$
$\displaystyle+$
$\displaystyle\left.\left(5f+6b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44})\right)+\frac{20}{33}\left(28f+33b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)\right.$ $\displaystyle+$
$\displaystyle\left.A\leftrightarrow B\right]\;,$ $\displaystyle
P^{AB}_{\mathrm{ct,\;\ell=4}}(k)$ $\displaystyle=$
$\displaystyle\frac{8}{385}\frac{k^{2}}{k_{\rm norm}^{2}}P_{\rm
lin}(k)\left[11fc^{A}_{\mathrm{ct},22}+\left(15f+11b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},24}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},44}\right)\right.$ $\displaystyle+$
$\displaystyle\left.\frac{15}{13}\left(14f+13b^{B}_{1}\right)\left(c^{A}_{\mathrm{ct},26}+\frac{k^{2}}{k_{\rm
norm}^{2}}c^{A}_{\mathrm{ct},46}\right)+A\leftrightarrow B\right]\;.$

The full set of parameters in redshift space (in addition to the cosmological
parameter) for MT is then:

$\displaystyle\left\\{b^{A}_{1},\,b^{A}_{2},\,b^{A}_{\mathcal{G}_{2}},\,b^{A}_{\Gamma_{3}},\,c^{A}_{{\rm
ct},20},\,c^{A}_{{\rm ct},22},\,c^{A}_{{\rm ct},24},\,c^{A}_{{\rm
ct},26},\,c^{A}_{{\rm ct},44},\,c^{A}_{{\rm
ct},46},\,c^{AA}_{\mathrm{st},00},\,c^{AA}_{\mathrm{st},02},\,c^{AA}_{\mathrm{st},22},\,\right.$
$\displaystyle
b^{B}_{1},\,b^{B}_{2},\,b^{B}_{\mathcal{G}_{2}},\,b^{B}_{\Gamma_{3}},\,c^{B}_{{\rm
ct},20},\,c^{B}_{{\rm ct},22},\,c^{B}_{{\rm ct},24},\,c^{B}_{{\rm
ct},26},\,c^{B}_{{\rm ct},44},\,c^{B}_{{\rm
ct},46},\,c^{BB}_{\mathrm{st},00},\,c^{BB}_{\mathrm{st},02},\,c^{BB}_{\mathrm{st},22},$
$\displaystyle\left.c^{AB}_{\mathrm{st},00},\,c^{AB}_{\mathrm{st},02},\,c^{AB}_{\mathrm{st},22}\right\\}\;.$
(2.25)

The inclusion of that many new free coefficients slows down the MCMC
computations. We make use of a Taylor expansion over the cosmological
parameters for the computation of the linear power spectrum, similar to
Appendix A of [47]. We comment more on that in Sec. 3.3. As we discuss later
in the text, the high number of free parameters for MT does not deteriorate
the constraints, since those parameters are more orthogonal and better
determined.

###  2.3 Combining tracer bias coefficients

Since the bias and stochastic parameters encapsulate information from
different fields in the cases of MT and ST, it does not make sense to compare
them directly. When comparing ST and MT, it is useful to define the effective
tracer [47]. This effective tracer is a way to provide a direct (and fair)
comparison between the parameters extracted in ST and MT. We define the
effective tracer overdensity as

$\delta^{\rm eff}=\frac{1}{\bar{n}}\sum_{i}\bar{n}^{i}\delta^{i}\;,$ (2.26)

where $\bar{n}=\sum_{i}\bar{n}^{i}$. Therefore, the effective parameters when
considering two tracers are a weighted mean of each tracer coefficient

$\displaystyle b^{\mathrm{eff}}_{\left[\boldsymbol{\mathcal{O}}\right]}$
$\displaystyle=$
$\displaystyle\frac{1}{\bar{n}}\left(\bar{n}^{A}b^{\mathrm{A}}_{\left[\boldsymbol{\mathcal{O}}\right]}+\bar{n}^{B}b^{\mathrm{B}}_{\left[\boldsymbol{\mathcal{O}}\right]}\right)\,,$
(2.27) $\displaystyle c^{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{\bar{n}}\left[\bar{n}^{A}c^{AA}+\bar{n}^{B}c^{BB}+2\sqrt{\bar{n}^{A}\bar{n}^{B}}c^{AB}\right]\,.$
(2.28)

Unless stated otherwise, we refer to the free coefficients of the effective
tracer when showing the MT bias and stochastic parameters in Sec. 4.

Figure 1: On the left, the power spectrum multipoles ($\ell=0,2,4$) measured
in the simulation for $\bar{n}=\bar{n}_{\rm low}$. We show the auto spectra of
tracers $A$ and $B$, their cross-power spectrum $AB$, and the single-tracer
case $A+B$. Notice the suppression in the spectrum for the quadrupole of $B$,
indicating that the effect of FoG for the tracer $B$ is stronger than for $A$.
On the right, the galaxy number $N_{g}$ as a function of the stellar formation
rate (SFR) and the stellar mass, $M_{\rm stell}$. The red lines indicate the
level curves $0.3$, $0.5$ and $0.9$ with respect to the peak. The dark solid
and dashed lines line shows the cut for the sample with $\bar{n}_{\rm
high}=0.003\,[h/{\rm Mpc}]^{3}$ and $\bar{n}_{\rm low}=0.0003\,[h/{\rm
Mpc}]^{3}$, respectively.

##  3 Data and MCMC setup

In this section, we start by describing the simulation data. Later, we discuss
the covariance matrix in redshift space for multi-tracer, the priors and the
MCMC setup.

###  3.1 Simulation

We use the dark matter simulations TheOne from the BACCO project [48, 49], in
a $1.440\,\mathrm{Gpc}/h$ box with $4320^{3}$ particles at $z=0$. The fiducial
cosmological parameters are $\Omega_{\mathrm{cdm}}=0.259$,
$\Omega_{\mathrm{b}}=0.048$, $h=0.68$, $n_{s}=0.96$ and $\sigma_{8}=0.9$. The
simulations were performed with the paired & fixed method of [60], which
strongly suppresses the cosmic variance. That allow us to reescale the volume
of the sample by adjusting the covariance [61]. Since the simulation volume is
$V_{s}\approx 3\,[\mathrm{Gpc}/h]^{3}\approx V_{\rm BOSS}/2$, we rescale the
covariance to fit the BOSS volume $V_{\rm BOSS}\sim 6\,[\mathrm{Gpc}/h]^{3}$
(see e.g. Table 6 of [15]) and consider $V_{\rm BOSS}$ as the fiducial volume
of our work. We discuss the covariance in Sec. 3.2 and the effect of the
sample volume in Sec. 4.2.

We populate the (sub)halos from TheOne with galaxies using the subhalo
abundance matching extended (SHAMe) empirical method of [50]333For the SHAMe
parameters, we used the values for number density $\bar{n}=0.001\,[h/{\rm
Mpc}]^{3}$ in Table 1 of [50].. After that, in order to provide two different
values for the total number density of the galaxies we perform two different
cuts in the galaxy catalogue according to their stellar mass $M_{\rm stell}$.
We keep only the most massive objects to obtain: $\bar{n}_{\rm
high}=0.003\,[h/{\rm Mpc}]^{3}$ and $\bar{n}_{\rm low}=0.0003\,[h/{\rm
Mpc}]^{3}$, with the latter being close to the number density of both BOSS and
DESI LRG samples (see again Table 6 of [15]). We discuss the impact of the
total sample shot noise in Sec. 4.2. The SHAMe method also attributes a star
formation ratio (SFR) parameter to each galaxy assuming the SFR depends on the
dark matter accretion rate of the host halo, akin to [62]. Thus, after
performing the first cut in $M_{\rm stell}$ to fix $\bar{n}$, we perform a
second cut in the SFR to split the sample into two tracers $A$ and $B$ such
that $\bar{n}_{A}=\bar{n}_{B}=\bar{n}/2$. This cut resembles a color cut since
blue (red) galaxies present a larger (lower) SFR [51]. Notice that this is a
more realistic scenario to test MT gains when compared to our first work [47],
which has mostly focused on halos split by their masses. We summarize the cuts
in stellar mass and SFR in Table 1 and on the right panel of Fig. 1, where we
display the number of galaxies $N_{g}$ as a function of the SFR and the
stellar mass $M_{\rm stell}$. The dark solid line shows the cut for the sample
with $\bar{n}_{\rm high}=0.003\,[h/{\rm Mpc}]^{3}$. The dark dashed line
refers to the cut to obtain the sample with $\bar{n}_{\rm low}=0.0003\,[h/{\rm
Mpc}]^{3}$. Note that the sample with $\bar{n}_{\rm low}$ corresponds to a cut
$M_{\rm stell}\gtrsim 11\times 10^{10}\,M_{\odot}/h$ and that the sample with
$\bar{n}_{\rm high}$ corresponds to a cut $M_{\rm stell}\gtrsim 10.3\times
10^{10}\,M_{\odot}/h$.

Galaxy set |  SFR [$M_{\odot}/$yr] | $\bar{n}\,[(\textrm{Mpc}/h)^{-3}]$
---|---|---
$A$ | $\gtrsim 10^{-4}$ | $0.0015$
$B$ | $\lesssim 10^{-4}$ | $0.0015$
$A+B$ |  | $0.0030$

Galaxy set |  SFR [$M_{\odot}/$yr] | $\bar{n}\,[(\textrm{Mpc}/h)^{-3}]$
---|---|---
$A$ | $\gtrsim 10^{-1}$ | $0.00015$
$B$ | $\lesssim 10^{-1}$ | $0.00015$
$A+B$ |  | $0.00030$

Table 1: Galaxy number density for tracers $A$ and $B$ in the MT scenario and
for the total tracer $A+B$ density used for ST. We first perform a cut in the
stellar mass $M_{\rm stell}$ attributed by SHAMe to get $\bar{n}_{\rm high}$
and $\bar{n}_{\rm low}$. Later, we perform a cut in SFR to split the tracers
into two populations $A$ and $B$. The left tabel refers to $\bar{n}_{\rm
low}=0.0003\,[h/{\rm Mpc}]^{3}$ and the right tabel to $\bar{n}_{\rm
high}=0.003\,[h/{\rm Mpc}]^{3}$.

We display on the left panel of Fig. 1 the monopole, quadrupole, and
hexadecapole for each tracer, $A$ and $B$, their cross-spectra ($AB$), and the
single-tracer scenario ($A+B$). Note that splitting the ST case into two
tracers makes evident that the amplitude of the FoG effect for tracer $B$ is
stronger than for tracer $A$. We see that as suppression in the spectrum,
especially for higher multipoles. We can relate that to the fact that the
tracer split we perform is very similar to the color split of a galaxy sample
into blue and red galaxies. It is widely known that red samples reside in more
virialized overdensities and therefore present larger peculiar velocities when
compared to the blue ones [63, 64, 1, 65]. Since part of the multi-tracer
samples may experience stronger FoG suppression, it is crucial to study the
impact of including higher-order $\mu^{2}$ counter-terms to account for that.
We discuss this impact in Sec. 4.1.

###  3.2 Covariance

We calculate in this section the covariance matrix for multi-tracer in
redshift space. This calculation is based on [66] and in [35], which
calculates the covariance for the angular spectrum.444We acknowledge and are
very thankful to Enea di Dio for providing his derivation for the covariance
matrix for multi-tracer in redshift space. Defining the power spectrum of two
(discrete) fields $\delta_{A}$ and $\delta_{B}$ with its cross shot noise
included

$\langle\delta_{A}(\boldsymbol{k})\delta_{B}^{\ast}(\boldsymbol{k}^{\prime})\rangle=(2\pi)^{3}\delta_{D}(\boldsymbol{k}-\boldsymbol{k}^{\prime})P^{AB}(\boldsymbol{k})\,,$
(3.1)

we can calculate the covariance by trivially generalizing the real space
covariance of [67, 68] to multi-tracer [47]

${\rm
Cov}\left[P^{AB}(\boldsymbol{k}),P^{CD}(\boldsymbol{k}^{\prime})\right]=\frac{(2\pi)^{3}}{V_{s}}\left[\delta_{\rm
D}(\boldsymbol{k}-\boldsymbol{k}^{\prime})P^{AD}(\boldsymbol{k})P^{BC}(\boldsymbol{k})+\delta_{\rm
D}(\boldsymbol{k}+\boldsymbol{k}^{\prime})P^{AC}(\boldsymbol{k})P^{BD}(\boldsymbol{k})\right]\,,$
(3.2)

where $V_{s}$ is the sample volume. Notice that we kept only the Gaussian
contribution ($PP$), neglecting the trispectrum part [68, 69]. Since we are
interested in scales $k<0.2\,h/$Mpc (and more concretely focusing on $k\sim
0.14\,h/$Mpc), the non-Gaussian contribution to the covariance is very small
[68, 69, 70]. Using the projection (2.12), we can now write down the
covariance for the projected multipoles as

$\displaystyle{\rm
Cov}\left[P^{AB}_{\ell}(\boldsymbol{k}),P^{CD}_{\ell^{\prime}}(\boldsymbol{k}^{\prime})\right]$
$\displaystyle=$
$\displaystyle\frac{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}{4V_{s}}\frac{\delta_{D}\left(k-k^{\prime}\right)}{2\pi
k^{2}}\int
d\Omega_{\hat{\mathbf{k}}}d\Omega_{\hat{\mathbf{k}}^{\prime}}\mathcal{L}_{\ell}\left(\mu_{k}\right)\mathcal{L}_{\ell^{\prime}}\left(\mu_{k^{\prime}}\right)$
(3.3)
$\displaystyle\left[P^{AC}\left(\mathbf{k}\right)P^{BD}\left(-\mathbf{k}\right)\delta_{D}\left(\hat{\mathbf{k}}-\hat{\mathbf{k}}^{\prime}\right)+P^{AD}\left(\mathbf{k}\right)P^{BC}\left(-\mathbf{k}\right)\delta_{D}\left(\hat{\mathbf{k}}+\hat{\mathbf{k}}^{\prime}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}{4V_{s}}\frac{\delta_{D}\left(k-k^{\prime}\right)}{k^{2}}\sum_{\ell_{1}\ell_{2}}\int
d\mu\mathcal{L}_{\ell}\left(\mu\right)\mathcal{L}_{\ell_{1}}\left(\mu\right)\mathcal{L}_{\ell_{2}}\left(-\mu\right)$
$\displaystyle\left[P_{\ell_{1}}^{AC}\left(k\right)P^{BD}_{\ell_{2}}\left(k\right)\mathcal{L}_{\ell^{\prime}}\left(\mu\right)+P^{AD}_{\ell_{1}}\left(k\right)P^{BC}_{\ell_{2}}\left(k\right)\mathcal{L}_{\ell^{\prime}}\left(-\mu\right)\right]\,,$

where $\Omega_{\hat{\mathbf{k}}}$ is the integral over the solid angle formed
w.r.t the $\hat{\mathbf{k}}$ direction. Again, we only kept the $PP$
contribution. We can simplify the Legendre polynomial products by writing

$\displaystyle\mathcal{L}_{\ell_{1}}\left(\mu\right)\mathcal{L}_{\ell_{2}}\left(\mu\right)$
$\displaystyle=$
$\displaystyle\sum_{\ell_{3}}\left(2\ell_{3}+1\right)\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\mathcal{L}_{\ell_{3}}\left(\mu\right)\,,$ (3.6)

where the $2\times 3$ matrices are the Wigner 3-j symbols. Therefore, we have
that

$\displaystyle\int
d\mu\mathcal{L}_{\ell}\left(\mu\right)\mathcal{L}_{\ell^{\prime}}\left(\mu\right)\mathcal{L}_{\ell_{1}}\left(\mu\right)\mathcal{L}_{\ell_{2}}\left(-\mu\right)$
$\displaystyle=$ $\displaystyle
2\left(-1\right)^{\ell_{2}}\sum_{\ell_{3}}\left(2\ell_{3}+1\right)\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\,,$ (3.11) $\displaystyle\int
d\mu\mathcal{L}_{\ell}\left(\mu\right)\mathcal{L}_{\ell^{\prime}}\left(-\mu\right)\mathcal{L}_{\ell_{1}}\left(\mu\right)\mathcal{L}_{\ell_{2}}\left(-\mu\right)$
$\displaystyle=$ $\displaystyle
2\left(-1\right)^{\ell^{\prime}+\ell_{2}}\sum_{\ell_{3}}\left(2\ell_{3}+1\right)\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\,,$ (3.16)

which we can use to write the final covariance for MT in redshift space

$\displaystyle{\rm
Cov}\left[P^{AB}_{\ell}(\boldsymbol{k}),P^{CD}_{\ell^{\prime}}(\boldsymbol{k}^{\prime})\right]$
$\displaystyle=$
$\displaystyle\frac{\left(2\ell+1\right)\left(2\ell^{\prime}+1\right)}{2V_{s}}\frac{\delta_{D}\left(k-k^{\prime}\right)}{k^{2}}$
(3.22)
$\displaystyle\sum_{\ell_{1}\ell_{2}\ell_{3}}\left(-1\right)^{\ell_{2}}\left(2\ell_{3}+1\right)\left(\begin{array}[]{ccc}\ell_{1}&\ell_{2}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}\left(\begin{array}[]{ccc}\ell&\ell^{\prime}&\ell_{3}\\\
0&0&0\end{array}\right)^{2}$
$\displaystyle\qquad\left[P_{\ell_{1}}^{AC}\left(k\right)P_{\ell_{2}}^{BD}\left(k\right)+\left(-1\right)^{\ell^{\prime}}P_{\ell_{1}}^{AD}\left(k\right)P_{\ell_{2}}^{BC}\left(k\right)\right]\,.$

Notice that, since the simulations described in Sec. 3.1 were performed with
the paired & fixed method [60], we can (artifitially) reescale the volume
$V_{s}$ of the sample considered in the covariance [61].

###  3.3 Priors and MCMC setup

Table 2 describes the priors chosen for the parameters considered in our MCMC
analysis. The boundaries of the uniform (flat) priors are based on previous
work and preliminary analysis of our pipeline with the data. We checked that
all parameter posteriors are within the priors considered (see also [47] for a
very broad discussion on the different choices of priors in the context of
MT). On top of the free terms from Eq. (2.2), we explore the posteriors for
three cosmological parameters: the density of cold dark matter $\omega_{\rm
cdm}=h^{2}\Omega_{\rm cdm}$, the Hubble parameter $h$ and the (rescaled)
amplitude of fluctuations $A_{s}\times 10^{9}$.

Parameter |  Flat Prior
---|---
$\omega_{\rm cdm}$ | $[0.09,0.14]$
$h$ | $[0.57,0.77]$
$A_{s}\times 10^{9}$ | $[1.5,4]$
$b_{1}$ | $[0.6,2]$
$b_{2}$ | $[-10,10]$
$b_{\mathcal{G}_{2}}$ | $[-10,10]$
$b_{\Gamma_{3}}$ | $[-15,15]$

Parameter |  Flat Prior
---|---
$c_{\rm ct,20}$ | $[-10,10]$
$c_{\rm ct,22}$ | $[-10,10]$
$c_{\rm ct,24}$ | $[-10,10]$
$c_{\rm ct,26}$ | $[-50,50]$
$c_{\rm ct,44}$ | $[-10,10]$
$c_{\rm ct,46}$ | $[-50,50]$
$c_{\rm st,00}$ | $[-2,2]$
$c_{\rm st,02}$ | $[-20,20]$
$c_{\rm st,22}$ | $[-50,50]$

Table 2: Priors on the different parameters considered in our MCMC analysis.
For a broader discussion of priors in the context of MT, see [47].

We sampled the posterior using the MCMC method and two different Python
packages: emcee [71] and pocoMC [72, 73]. Sampling the MT likelihood can be
challenging, since it has approximately $30$ parameters and some degeneracies.
We, therefore, decided to take a conservative approach and cross-checked that
these two samplers lead to consistent results. For the emcee, we used $20$
walkers per parameter and used the Gelman-Rubin criteria [74] to determine its
convergence (with four parallel chains and scale reduction factor
$\epsilon<0.03$). For pocoMC, we used its standard configurations and built-in
convergence criteria, with $5000$ particles. Both emcee and pocoMC led to
similar posteriors, with some minor differences in some of the nuisance
parameters. We used the emcee chains to make the figures in this work.

Finally, each MCMC iteration requires running a numerical Boltzmann solver to
get the linear matter power spectrum, and additional loop integrals from
perturbation theory. For that, we used CLASS-PT [53], a modified version of
CLASS [75]. Considering the large number of free parameters considered, each
iteration of CLASS-PT would be numerically expensive. Therefore, we Taylor
expanded all cosmology-dependent terms in our analysis following Appendix A of
[47]. The Taylor expansion boosts the evaluation of each MCMC iteration by a
factor of $\approx 100$ relative to running CLASS-PT.

##  4 Results

In this section we explain the main results of the paper. We start by
discussing the relevance of the higher-in-$\mu$ coefficients in the counter-
terms to parametrize the FoG for MT and the importance of the cross-stochastic
coefficients. Later, we consider the behaviour of MT for different shot noises
and box volumes. In the last part, we explain and quantify the benefits of MT
analysis compared to ST. We show results for $z=0$, different box sizes and
sample shot noise.

###  4.1 Higher-in-$\mu$ and cross-stochastic terms

Figure 2: On the left, we show the effect on the parameter posteriors
($1\sigma$) when including higher-in-$\mu$ terms in Eq. (2.7). On the right,
the effect of including the cross-stochastic terms of Eq. (2.21) for MT. In
both panels we consider $k_{\rm max}=0.14h/$Mpc and the largest value for the
tracer number density $\bar{n}_{\rm high}$.

We begin by investigating which terms are relevant when considering MT. On the
left panel of Fig. 2, we study the relevance of higher-order-in-$\mu$
coefficients for the counter-term in Eqs. (2.7) and (2.2) when gradually
adding the higher-in-$\mu$ parameters. We fix the maximum Fourier mode that
was used in the analysis to $k_{\rm max}=0.14h/$Mpc and the total tracer
number density $\bar{n}=\bar{n}_{\rm high}$. Moreover, all the other terms in
Eq. (2.10) for ST and (2.2) for MT are included. Notice that for ST (red
lines), including $\mu^{4}$ is vital, shifting the posterior towards the right
direction. It agrees with the analysis of other works [53, 76, 21] that found
those terms to be relevant as a proxy for FoG effects. For MT (blue lines), we
notice that the inclusion of a term proportional to $\mu^{6}$ is also
important. Neglecting this term for MT leads to a small shift ($1.5\sigma$) in
the amplitude of fluctuations $A_{s}$. We checked that this bias persists for
different values of $k_{\rm max}$. The need for those extra counter-terms is
to some extend expected, since the split into different tracers can uncover
non-linear behaviour that were smoothed out from the power spectrum of the
$A+B$ sample. We can see that, for instance, different intensities of FoG
suppression for tracers $A$ and $B$ (see left panel of Fig. 1). Including
high-order-in-$\mu$ terms is therefore vital to describe the tracer $B$
population, which presents stronger FoG suppression. We highlight again that
the importance of the $\mu^{6}$ terms is to contemplate the two possible
scalings ($k^{2}P_{\rm lin}$ and $k^{4}P_{\rm lin}$) for the three multipoles
considered (total of six degrees of freedom per tracer) with six free
coefficients per tracer (see Eq. (2.7)). Notice, however, that the inclusion
of the $\mu^{6}$ term does not deteriorate either ST or MT posteriors, which
motivates us to keep this term in further analysis. We comment on the
degeneracy between those terms later in the paper.

Another relevant set of contributions to be considered for MT is the
stochastic term for the cross-spectra of MT. We study on the right panel of
Fig. 2 the effect of including the stochastic terms, following Eq. (2.21). As
explained before, this term is often neglected since the tracers $A$ and $B$
do not correlate on small scales. However, exclusion effects may lead to a
non-vanishing cross-stochastic contribution [59]. We see that the inclusion of
this term does not lead to any substantial difference when constraining the
cosmological parameters, confirming the results in real space [47]. The
posteriors when including this cross-stochastic terms (blue solid) are very
similar to those when neglecting it (blue dotted), both being very smaller
than the ST scenario. For completeness, we include this cross-stochastic
contribution for MT in further analysis. Notice that we fixed $k_{\rm
max}=0.14h/$Mpc and $\bar{n}_{\rm high}=0.003\,[h/{\rm Mpc}]^{3}$ for Fig. 2
but we checked that the result is very similar for the sample with large shot
noise and $\bar{n}_{\rm low}=0.0003\,[h/{\rm Mpc}]^{3}$.

###  4.2 Survey volume and sample shot noise

Figure 3: Impact of shot noise on the $1\sigma$ regions of the posterior for
$k_{\rm max}=0.14h/$Mpc (left) and $k_{\rm max}=0.2h/$Mpc (right). We consider
two different values for the tracer number density: $\bar{n}=\bar{n}_{\rm
low}=0.0003\,[h/{\rm Mpc}]^{3}$ (dashed) that has a larger shot noise and
$\bar{n}=\bar{n}_{\rm high}=0.003\,[h/{\rm Mpc}]^{3}$ (solid) that has a lower
shot noise.

A potential disadvantage of doing MT is that each sub-sample will have a
larger shot noise. In order to check if the shot noise is indeed a limitation,
we now show in Fig. 3 the impact of the shot noise for multi-tracer and
single-tracer on the cosmological parameter posteriors. We use two different
values for the total tracer number density, as described in Sec. 3.1:
$\bar{n}_{\rm high}=0.003\,[h/{\rm Mpc}]^{3}$ (solid) and $\bar{n}_{\rm
low}=0.0003\,[h/{\rm Mpc}]^{3}$ (dashed). For tracers $A$ and $B$, we have
$\bar{n}_{A}=\bar{n}_{B}=\bar{n}/2$. The left panel shows the result for
$k_{\rm max}=0.14h/$Mpc. We notice that the parameter posteriors for MT for
different $\bar{n}$ are slightly shifted from each other, but agree on
$1\sigma$ with the fiducial values. For ST, the Hubble parameter $h$ presents
a small bias ($\sim 1\sigma$) for the sample with larger shot noise. On the
right panel, we set $k_{\rm max}=0.2h/$Mpc. In this case, the sample with
larger shot noise ($\bar{n}_{\rm low}=0.0003\,[h/{\rm Mpc}]^{3}$) presents a
$\sim 1\sigma$ bias in $A_{s}$ both for MT and ST. It is important to
highlight that, at $z=0$, higher-order terms from perturbation theory might
start to be relevant on those scales. Overall, we can conclude that MT results
are consistently better relative to ST for both $\bar{n}_{\rm high}$ and
$\bar{n}_{\rm low}$, the latter being close to the shot noise considered for
the red galaxies for eBOSS and DESI. The persistence of MT improvements for
both number densities indicates that their origin is not solely due to cosmic
variance cancellation, since that cancellation is highly dependent on
$\bar{n}$ and expected to be more relevant for tracers with large densities
[37, 31, 33].

Finally, we discuss the perspectives of MT analysis when increasing the survey
volume. As explained in Sec. 3, the simulation was performed using the paired
& fixed method of [60], which substantially suppresses the cosmic variance. It
allow us to safely rescale the sample volume, $V_{s}\approx
3\,[\mathrm{Gpc}/h]^{3}$, to higher values by simply adjusting the covariance
(3.22) [61]. On the right panel of Fig. 4 we show the constraints from MT and
ST for different volumes, which are in units of the volume of the BOSS LRG
galaxy sample. For comparison, notice that $V_{s}\approx V_{\rm BOSS}/2\approx
V_{\rm DESI}/13$. The right panel of Fig. 4 indicates that, as expected,
increasing the simulation volume led to a substantial reduction in the error
bars, without introducing any bias and significant shift in the posterior. In
summary, MT method can also provide better constraint for larger sample
volumes, such as those from DESI.

###  4.3 MT compared to ST

Figure 4: On the left, the cosmological parameter posteriors extracted (at
$z=0$) when considering the tracers $A$ and $B$ alone, compared to the MT and
ST scenarios. On the right, the cosmological parameters extracted for ST and
MT when varying the volume considered for the covariance (3.22). We fix for
those figures $k_{\rm max}=0.14\,h\mathrm{Mpc}^{-1}$ and $\bar{n}=\bar{n}_{\rm
high}$.

We now move to estimate and understand the improvements of MT compared to ST.
We start by comparing on the left panel of Fig. 4: ST, MT and the constraints
extracted when considering the tracers $A$ and $B$ alone. We first notice that
MT substantially outperforms the other constraints, while ST (which includes
both tracers $A$ and $B$ together) presents comparable constraints as the
tracers $A$ and $B$ alone. One would expect that combining the tracers $A$ and
$B$ into a single species (ST) would improve the constraints, since the number
of objects is increased by a factor of two. However, this is not the case
since the shot noise on the scales considered ($k_{\rm max}=0.14h/$Mpc) is too
small compared to the signal (for both $\bar{n}_{\rm low}$ and $\bar{n}_{\rm
high}$). It means that the error bars are dominated by the number of modes
available in that volume, which is the same for ST, MT and tracers $A$ and $B$
alone. We can therefore conclude that the total sample ($A+B$) presents
similar amount of information as the individual sub-samples alone. On the
other hand, the fact that MT analysis of these same samples (tracers $A$ and
$B$) led to tighter constraints on the cosmological parameter indicates that
there is an information gain. A key source of information that is not present
in the ST case is the cross power spectrum between the tracers. In [47], we
have shown that the cross power spectrum helps to break degeneracies between
bias parameters (e.g., by constraining the product
$b_{\mathcal{O}}^{A}b_{\mathcal{O}}^{B}$). Throughout this section, we aim to
provide more arguments to show that MT can be a way to extract more powerful
constraints in redshift space.

Figure 5: Evolution of the error in different parameters as a function of
$k_{\rm max}$ for MT and ST. The shaded regions indicate the error for
$A_{s}$, $\omega_{\rm cdm}$, $h$ and $b_{1}$. The lines indicate the relative
size of the error bars and the values of the cosmological parameters are
normalized by the fiducial values from the simulation.

To further compare the MT improvements relative to ST, we show in Fig. 5 their
results as a function of $k_{\rm max}$, fixing $\bar{n}=\bar{n}_{\rm high}$.
For $k_{\rm max}\geq 0.16\,h$/Mpc, we see that the error bars of MT are
substantially better: $\sigma_{\rm MT}/\sigma_{\rm ST}\sim 2/3$ for
$\omega_{\rm cdm}$, $A_{s}$ and $h$. For $k_{\rm max}\leq 0.14\,h$/Mpc, the MT
improvement is even better, with $\sigma_{\rm MT}/\sigma_{\rm ST}\sim 6/10$
for $\omega_{\rm cdm}$ and $A_{s}$ and $\sigma_{\rm MT}/\sigma_{\rm ST}<1/2$
for $h$. At low $k$, the signal is less affected by the increasing shot noise
from MT. When increasing $k$, there is a non-trivial $k$ scaling due to how
both MT and ST incorporate information and are able to determine the extra
free parameters from the EFTofLSS and the bias expansion. After have included
the $\mu^{4}$ terms for ST and $\mu^{6}$ terms for MT, we do not see any
indication of strong bias in the results for the range of $k_{\rm max}$
considered.

To help to understand why MT outperforms ST, we display in Fig. 6 the MCMC
posteriors for all parameters, comparing MT and ST for $k_{\rm max}=0.14h/$Mpc
and $\bar{n}=\bar{n}_{\rm high}$. As described in Sec. 2.3, the MT counter-
terms, stochastic and bias parameters displayed refer to the effective tracer
values. Notice that MT outperforms ST for almost all parameters. In addition
to the relative gain in the cosmological parameters, we notice a substantial
improvement in the bias terms $b_{1},b_{2},b_{\mathcal{G}_{2}}$ and
$b_{\Gamma_{3}}$, ranging from a factor 1.5 to 3 better if compared to ST. The
stochastic parameters are also better determined for MT. It brings us to the
first important conclusion: splitting one population into two tracers has led
to a better estimate of the non-linear part of the matter spectrum for each
tracer despite having more free parameters.

Moreover, Fig. 6 points to a (almost exact) degeneracy between $c_{\rm ct}$’s
for ST indicating that a few of those terms are not really necessary in the ST
analysis. We include those terms for completeness but, as commented above, we
explicitly checked that the results are very similar when dropping those terms
from ST. For multi-tracer, on the other hand, they are required in order to
have unbiased results for the cosmological parameters (see Fig. 2), despite
still being partially degenerate with other terms, as we discuss below. We can
also spot a degeneracy between $c_{\rm ct}^{20}$ both with $h$ and
$\omega_{\rm cdm}$ for ST, which can partially explain the $1\sigma$ deviation
of $h$ seen in Fig. 5 at low $k_{\rm max}$. That may be induced by volume
effect since it goes away for larger volumes (see Fig. 4).

Figure 6: Comparison between MT and ST in redshift space using the full set of
parameters in Eqs. (2.7) and (2.2) for $k_{\rm max}=0.14h/$Mpc, $\bar{n}_{\rm
high}$ and $z=0$.

To conclude, we show in Fig. 7 the correlation matrix between the free
parameter for ST (left) and MT (right). The matrix is more diagonal in the
case of MT, especially in the cross-correlation between $c_{\rm ct}$’s and
$\\{A_{s},\,\omega_{\rm cdm},h\\}$, explaining part of the gains of MT
compared to ST. Notice also that $b_{1}$ is less correlated with other
parameters for MT. In addition, we highlight a strongly-correlated block (both
for MT and ST) for the parameters $\\{c_{\rm ct}^{22},\,c_{\rm
ct}^{24},\,c_{\rm ct}^{44},\,c_{\rm ct}^{26},\,c_{\rm ct}^{46}\\}$. The
stochastic coefficients behave similarly for MT and ST. As we already
mentioned, part of this gain can be attributed to the extra information coming
from the cross spectrum, which constraints the product
$b_{\mathcal{O}}^{A}b_{\mathcal{O}}^{B}$. We also compare the correlations for
$\bar{n}_{\rm high}$ (top) and $\bar{n}_{\rm low}$ (bottom). Notice that for
the case with smaller shot noise ($\bar{n}_{\rm high}$) we find more
correlations between the parameters if compared to the high shot noise
scenario ($\bar{n}_{\rm low}$). For both values of $\bar{n}$ the correlation
matrix for the effective tracer of MT is more diagonal. All that leads us to
another important conclusion: the MT basis of parameters is more orthogonal
than the ST basis, explaining part of the gains from MT. By breaking a
population into distinct tracers, given the same data, the bias parameters and
counter-term can encapsulate more information about the specific history and
dynamics of each tracer. The combination of both populations into a single
population blurs the data into an averaged tracer (see left panel of Fig. 1)
and, consequently, less information is absorbed into each free coefficient.
For instance, see the $b_{1}$ in the left panel of Fig. 4: the two tracers
present a different linear bias and the combination of both populations leads
to an average tracer that erases part of the information gained when
considering both populations separately.555Note that both MT and ST present a
linear bias that has an intermediate value of $b_{1}$ compared to considering
$A$ and $B$ alone. However, in the MT case, the $b_{1}$ shown is from the
effective tracer defined in Sec. 2.3 and not of the average tracer as ST. In
practice, it means that while a single coefficient is fitted for ST, for MT we
fit two coefficients using the auto and cross-spectra together, and later we
take the average. We also highlight that MT’s gains were observed despite the
difference in the linear bias values for tracers $A$ and $B$ being relatively
minor ($b_{1}^{A}\sim 0.8$ and $b_{1}^{B}\sim 1.5$). It indicates that the
gain goes beyond the $b_{1}$ difference and linear scales. MT is capable of
making a clear estimate of the free coefficients for each specific sub-
population, leading to a substantial improvement in the cosmological
constraints.

Figure 7: Correlation matrix between cosmological parameters, bias, stochastic
coefficients and counter-terms for ST (left) and MT (right) for $k_{\rm
max}=0.14h/$Mpc. The top and bottom panels display values for the two shot
noises considered in this work, $\bar{n}_{\rm high}$ (top) and $\bar{n}_{\rm
low}$ (bottom). Although in the ST case there is a high correlation between
$\\{A_{s},\,\omega_{\rm cdm},\,h\\}$ and the other parameters, in the MT model
part of these correlations are broken. It indicates that the MT parameter
basis is more orthogonal.

##  5 Conclusions

In this work, we extended the results of [47] to redshift space, showing that
the multi-tracer is still a relevant tool not only to beat the cosmic variance
on linear scales but to break degeneracies between free parameters on the
mildly non-linear regime. We also improved the previous analysis by
considering more realistic galaxies generated by the state-of-the-art SHAMe
method and a SFR tracer split.

In addition, we found that the FoG effects can be starker for different
populations when doing the MT tracer splitting. We concluded that the higher-
in-$\mu$ parameters in the EFT counter-terms are necessary to absorb those
effects and extract unbiased cosmological parameters in an MT analysis. In
addition, the relevance of those higher-in-$\mu$ terms indicate that it may be
interesting to consider in the future the clustering wedges statistics from
[77, 16] for MT. Furthermore, we saw that the cross-stochastic terms did not
deteriorate the MT improvement, in agreement with [47].

We summarize in Fig. 8 the main results of this work, comparing MT error bars
relative to ST for all bias and cosmological parameters for
$\bar{n}=\bar{n}_{\rm high}$ and different values of $k_{\rm max}$. We see
that the error bars for MT are, on average, $50\%$ smaller for $A_{s}$, $h$,
and $\omega_{\rm cdm}$. For the bias parameters, the error bars for MT are
between 1.5 to 3 times better, with this improvement being smaller for $b_{1}$
and larger for $b_{\mathcal{G}_{2}}$ and $b_{\Gamma_{3}}$. Those improvements
are consistent with the conclusions of the first paper [47], with the
difference that we are now able to break the degeneracy between $A_{s}$ and
$b_{1}$ in redshift space.

Figure 8: The multi-tracer (MT) improvements relative to the single-tracer
(ST) scenario in a full-shape power-spectrum analysis. We display the size of
the error bars of MT compared to ST for different values of $k_{\rm max}$. Our
results show a consistent improvement of MT over ST for both cosmological
(upper panel) and bias parameters (bottom panel). For the cosmological
parameters, there is a consistent improvement $\sigma_{\rm ST}/\sigma_{\rm
MT}\sim 1.5$. The improvements are even more striking for the bias parameters,
especially for $b_{\Gamma_{3}}$ and $b_{\mathcal{G}_{2}}$, with MT error bars
two or three times smaller than ST.

A common criticism to the EFTofLSS framework relies on a large number of free
parameters present in the theory. This leads to a fundamental question: when
is it reasonable and justified to add new free parameters to an analysis? The
success of EFTofLSS when providing competitive cosmological constraints [17,
18, 19, 20, 21, 22] shows that this criticism of EFT does not hold. The extra
parameters added are physical and are exactly what guarantees the robustness
of the theory by parametrizing the UV regime in a self-controlled manner.
Those parameters are (relatively) well measured in data, which is one of the
reasons why FS provides competitive constraints. In general, a $n$-tracer
analysis will have about $n$-times more coefficients within the EFTofLSS
framework. Again, we found that for the case considered in this work, those
extra parameters do not jeopardize the constraints, but improve them. Despite
sounding counter-intuitive to have a more constraining theory with more free
parameters, the presence of the cross power spectrum between the tracers
reduces the correlation between the EFT and the cosmological coefficients,
which ultimately induce tighter constraints on both set of parameters. Those
parameters are better determined for MT (Fig. 6) and present fewer
degeneracies (Fig. 7). In contrast, part of the information encoded in
intermediate scales for each sub-tracer would be smoothed out when doing a
single-tracer analysis (see the left panel of Fig. 1).

To conclude, MT provides an exciting perspective to maximize the information
from galaxy surveys. There are several ways to split a sample of galaxies into
sub-samples, and the MT gains might depend on how it is performed. In this
work, we focused on a SFR split. It would be interesting in future work to
explore other ways to split the tracers, such as considering the local
environment density [78]. Furthermore, this work is a fundamental step before
applying MT to data, using simulations to test for the first time the EFTofLSS
pipeline for MT in redshift space.

## Acknowledgments

We are very thankful to Enea di Dio for providing his derivation for the
covariance matrix for multi-tracer in redshift space. We also acknowledge
Florian Beutler, John Peacock, and Pedro Carrilho for valuable comments, and
Raul Abramo for his participation in the early stages of this work. HR
acknowledges Mathias Garny and Fabian Schmidt for useful discussions and
comments on the draft. HR is supported by the Deutsche Forschungsgemeinschaft
under Germany’s Excellence Strategy EXC 2094 ‘ORIGINS’. (No. 390783311). TM is
supported by the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement 853291).

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