$h$ |  | $0.6766\pm 0.0070$ | $0.43$ |  | $0.6759\pm 0.0070$ | $0.33$ |
$\omega_{cdm}$ |  | $0.1197^{+0.0017}_{-0.0012}$ | $0.20$ |  | $0.1195\pm 0.0014$ | $0.35$ |
$\ln(10^{10}A_{s})$ |  | $3.027^{+0.027}_{-0.034}$ | $0.30$ |  | $3.040\pm 0.031$ | $0.12$ |
Table 6: Beyond baseline Analyses - $V_{25}$. 1-dimensional posterior means and 0.68 c.i. for Full-Modelling (FM) and ShapeFit (SF) using the MinF bias setup. We also show the deviation from the value of the simulations $\Delta\bm{\Omega}/\sigma$. The em dash “—” indicates inconclusive results, as the posterior distribution saturates the prior for that parameter.  |  | FM |  | SF |
---|---|---|---|---|---
|  | 68% c.i. | $\,\Delta\Omega/\sigma_{\Omega}$ |  | 68% c.i. | $\,\Delta\Omega/\sigma_{\Omega}$ |
Baseline, $V_{5}$: |  |  |  |  |  |  |
$h$ |  | $0.6744^{+0.0050}_{-0.0049}$ | $0.17$ |  | $0.6739^{+0.0047}_{-0.0051}$ | $0.06$ |
$\omega_{cdm}$ |  | $0.1191^{+0.0032}_{-0.0028}$ | $0.31$ |  | $0.1191^{+0.0023}_{-0.0027}$ | $0.38$ |
$\ln(10^{10}A_{s})$ |  | $3.040^{+0.037}_{-0.048}$ | $0.08$ |  | $3.048\pm 0.040$ | $0.30$ |
Hexadecapole, $V_{5}$: |  |  |  |  |  |  |
$h$ |  | $0.6741\pm 0.0048$ | $0.11$ |  | $0.6738\pm 0.0048$ | $0.04$ |
$\omega_{cdm}$ |  | $0.1190^{+0.0030}_{-0.0026}$ | $0.36$ |  | $0.1188\pm 0.0022$ | $0.52$ |
$\ln(10^{10}A_{s})$ |  | $3.045^{+0.030}_{-0.046}$ | $0.24$ |  | $3.058^{+0.043}_{-0.036}$ | $0.55$ |
$\omega_{b}$ (flat prior), $V_{5}$: |  |  |  |  |  |  |
$\omega_{b}$ |  | $0.0222^{+0.0024}_{-0.0046}$ | $0.06$ |  | — | — |
$h$ |  | $0.672^{+0.022}_{-0.037}$ | $0.04$ |  | — | — |
$\omega_{cdm}$ |  | $0.1188^{+0.0059}_{-0.0094}$ | $0.16$ |  | — | — |
$\ln(10^{10}A_{s})$ |  | $3.038^{+0.034}_{-0.048}$ | $0.04$ |  | $3.018^{+0.081}_{-0.11}$ | $0.19$ |
Varying $n_{s}$, $V_{5}$: |  |  |  |  |  |  |
$n_{s}$ |  | $0.947\pm 0.030$ | $0.59$ |  | $0.970^{+0.047}_{-0.054}$ | $0.17$ |
$h$ |  | $0.6760\pm 0.0057$ | $0.41$ |  | $0.6751^{+0.0075}_{-0.0084}$ | $0.19$ |
$\omega_{cdm}$ |  | $0.1222^{+0.0056}_{-0.0065}$ | $0.36$ |  | $0.121^{+0.012}_{-0.014}$ | $0.16$ |
$\ln(10^{10}A_{s})$ |  | $3.006\pm 0.073$ | $0.82$ |  | $3.052^{+0.090}_{-0.12}$ | $0.14$ |
$\nu\Lambda$CDM, $V_{5}$: |  |  |  |  |  |  |
$M_{\nu}$ |  | $0.175^{+0.073}_{-0.15}$ | $0.42$ |  |  |  |
$h$ |  | $0.6750^{+0.0053}_{-0.0047}$ | $0.28$ |  |  |  |
$\omega_{cdm}$ |  | $0.1221^{+0.0034}_{-0.0050}$ | $0.50$ |  |  |  |
$\ln(10^{10}A_{s})$ |  | $3.056^{+0.038}_{-0.052}$ | $0.44$ |  |  |  |
$k\Lambda$CDM ($\ell=0,2,4$), $V_{5}$: |  |  |  |  |  |  |
$\Omega_{k}$ |  | $-0.014\pm 0.025$ | $0.55$ |  | $-0.002\pm 0.024$ | $0.07$ |
$h$ |  | $0.679\pm 0.010$ | $0.55$ |  | $0.6744\pm 0.0082$ | $0.09$ |
$\omega_{cdm}$ |  | $0.1193^{+0.0029}_{-0.0025}$ | $0.25$ |  | $0.1190\pm 0.0024$ | $0.44$ |
$\ln(10^{10}A_{s})$ |  | $3.001\pm 0.085$ | $0.17$ |  | $3.049\pm 0.086$ | $0.17$ |
$w$CDM ($\ell=0,2,4$), $V_{5}$: |  |  |  |  |  |  |
$w$ |  | $-1.036\pm 0.061$ | $0.60$ |  | $-1.014\pm 0.059$ | $0.14$ |
$h$ |  | $0.682\pm 0.014$ | $0.61$ |  | $0.677^{+0.012}_{-0.014}$ | $0.23$ |
$\omega_{cdm}$ |  | $0.1194^{+0.0030}_{-0.0026}$ | $0.21$ |  | $0.1190^{+0.0021}_{-0.0026}$ | $0.44$ |
$\ln(10^{10}A_{s})$ |  | $3.017^{+0.056}_{-0.070}$ | $0.31$ |  | $3.043\pm 0.063$ | $0.10$ |
Table 8: Beyond baseline Analyses - $V_{5}$. 1-dimensional posterior means and
0.68 c.i. for Full-Modelling (FM) and ShapeFit (SF) using the MinF bias setup.
Same notation as in table 5.

##  8 Conclusions

The cosmological LSS contains a wealth of valuable information encoded in the
distribution of matter, serving as a powerful observational tool to constrain
the cosmological parameters governing our Universe. DESI is currently
embarking on the creation of the largest map of the Universe to date, offering
us the opportunity to test our cosmological theories with unprecedented
precision. However, this endeavor presents a challenge on the theoretical
front, demanding more accurate and comprehensive models to extract the maximal
cosmological information from the galaxy catalogues. In anticipation of the
DESI Year-1 data release, as well as forthcoming releases, we investigate the
systematic errors introduced by the full-shape analyses of the galaxy power
spectrum using the Folps code and pipelines for both the Full-Modelling (or
direct-fit) and the compressed ShapeFit methods.

Our analysis employs mocks created from the AbacusSummit set of simulations,
comprising three tracers: LRGs, ELGs, and QSOs, distributed at redshifts
$z=0.8$, $z=1.1$, and $z=1.4$, respectively. These simulations consist of 25
realizations, each with a volume of $8\,\,h^{-3}\text{Gpc}^{3}$. We conduct
our analyses using volumes $V=8,\,40$ and $200\,\,h^{-3}\text{Gpc}^{3}$.
Despite potential doubts about the reliability of the simulations at the
largest volume ($200\,h^{-3}\text{Gpc}^{3}$), we opt to use it due to
significant agreements between theory and simulation in our baseline analysis,
where all estimated parameters fall well within the $1\sigma$ limits. Thus, it
proves valuable to test theoretical models in such a demanding scenario.

We explore the optimal range of wave-numbers to include in the analyses,
finding optimal results for the interval
$k\in(0.01,0.18)\,h\,\text{Mpc}^{-1}$, with similar outcomes when
$k_{\text{max}}=0.2\,h\,\text{Mpc}^{-1}$. We find that systematics in the
standard $\Lambda$CDM model are sub-dominant compared to the statistical
errors expected from DESI Year-1 data. Additionally, we bring attention to
different priors commonly used in the literature, cautioning against Gaussian
priors that can significantly impact the posterior distribution of the
parameter space without necessarily having a strong physical motivation. We
also examine the inclusion of the hexadecapole alongside the monopole and
quadrupole, finding slight improvements in constraints for some models beyond
the vanilla $\Lambda$CDM, particularly for $w$CDM and $k\Lambda$CDM.

Furthermore, we compare the effects of the restrictive BBN prior on the baryon
abundance, finding that the direct-fit approach can constrain it while
compressed methods cannot. This is likely due to the ability of Full-Modelling
to accurately capture the relative amplitude of the power spectrum wiggles, a
capability lacking in ShapeFit. We also investigate the impact of freeing
$n_{s}$, finding tighter and more accurate parameter estimations for Full-
Modelling compared to ShapeFit, attributed to a strong degeneracy between the
$m$ and $n$ parameters of ShapeFit, hindering their independent variation and
resulting in poor constraints on $\omega_{m}$ and $n_{s}$. On the other hand,
we find that ShapeFit constrains better the dark energy models, which seems to
be more evident for less constrictive data, in our case for smaller volumes in
the simulations.

We further introduce the neutrino mass as an additional parameter. We find
that the well-known degeneracy between $M_{\nu}$ and $h$ from CMB data seems
to disappear. However, and expected degeneracy with the overall power spectrum
amplitude worsens the constraints of $\omega_{cdm}$ and $A_{s}$. Notably, we
observe that the constraints on $M_{\nu}$ show limited sensitivity to the
tidal bias parameter $b_{s^{2}}$ and the third-order bias parameter $b_{\rm
3nl}$ demonstrating robustness in the constraints on $M_{\nu}$ under the
minimal and maximal freedom setups.

Our findings reveal that both minimal and maximal freedom setups yield
consistent results for LRGs and ELGs, suggesting that the bias for these HODs
is very close to being local in Lagrangian space. However, for QSOs, there are
indications that the minimal freedom setup may not be sufficient, and maximal
freedom may be required.

Finally, we explore the impact of the assumed template cosmology on ShapeFit
by analyzing the LRGs within the conservative volume of
$8\,\,h^{-3}\text{Gpc}^{3}$. We find that the influence of the template
cosmology on ShapeFit is relatively minor, especially for DESI Year-1.
However, for larger volumes, the impact of the template cosmology on ShapeFit
may become more significant, requiring further investigation. The differences
we observe are primarily driven by the effective shape parameter introduced by
ShapeFit, with BAO and RSD being minimally affected by the choice of template
cosmology. For changes in the template of less than 3%, the errors with
respect to the true cosmology remain below $\sim 0.2\sigma$. However, with a
10% change, the errors can increase, reaching up to $\sim 0.6\sigma$ in the
worst-case scenario. However, such a large change in the template would raise
warning flags through the scaling parameters deviating from unity in some
redshift bins.

In this work, we have shown that the novel effective field theory-based
package to efficiently evaluate the redshift space power spectrum in the
presence of massive neutrinos, Folps$\nu$, has the required robustness and
accuracy to be used in both full modelling and compressed variables (ShapeFit)
cosmological analyses of DESI galaxy and quasars catalogs.

##  9 Data availability

The data used in this analysis will be made public along the Data Release 1
(details in https://data.desi.lbl.gov/doc/releases/)

## Acknowledgments

HN and AA are supported by Ciencia de Frontera grant No. 319359 and PAPIIT
IG102123. HGM acknowledges support through the program Ramón y Cajal
(RYC-2021-034104) of the Spanish Ministry of Science and Innovation. HN, MV,
SR, and SF are supported by PAPIIT IN108321 and IN116024, and Proyecto PIFF.
SF and FR are supported by PAPIIT IN115424. MV is supported by Investigación
in Ciencia Básica CONAHCYT grant No. A1-S-13051.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of High-Energy Physics, under Contract No.
DE–AC02–05CH11231, and by the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility under the same contract.
Additional support for DESI was provided by the U.S. National Science
Foundation (NSF), Division of Astronomical Sciences under Contract No.
AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research
Laboratory; the Science and Technology Facilities Council of the United
Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation;
the French Alternative Energies and Atomic Energy Commission (CEA); the
National Council of Humanities, Science and Technology of Mexico (CONAHCYT);
the Ministry of Science and Innovation of Spain (MICINN), and by the DESI
Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the U. S. National Science Foundation, the U. S. Department of Energy, or
any of the listed funding agencies.

The authors are honored to be permitted to conduct scientific research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation.

##  Appendix A Analytical marginalization

The formalism presented here enables a significant reduction in computational
time, not on the side of model evaluation, but rather in the process of
exploring the posterior distributions. We can further reduce computational
time by analytically marginalizing over some nuisance parameters of the model.
Typically, our focus lies in analyzing the probability distribution of the
cosmological parameters and a few select nuisances, such as the linear bias
$b_{1}$. Nevertheless, numerous other nuisance parameters are often of little
interest to us. Hence, by marginalizing over these nuisance parameters, we
effectively reduce the dimensionality of the parameter space, which
facilitates the exploration of posterior distributions when employing
efficient sampling algorithms.

We start by classifying the nuisance parameters of the model into two types,
the bias $\vec{b}=\\{b_{1},b_{2},b_{s^{2}},b_{3\rm nl}\\}$, and the stochastic
and counterterm parameters
$\vec{\alpha}=\\{\alpha_{0},\alpha_{2},\alpha_{4},\alpha^{\rm
shot}_{0},\alpha^{\rm shot}_{2}\\}$. The latter has the property that they are
linear-order parameters at the level of the power spectrum model $P(k)$. This
feature allows to split the model prediction as

$P(k)=\sum_{i}\alpha_{i}\,P_{\alpha_{i}}(k)+P_{b}(k),$ (A.1)

where $P_{\alpha_{i}}$ represents the stochastic and counterterm
contributions, while $P_{b}$ refers to the remaining contributions to the
power spectrum. Note that the split in equation (A.1) is an exact expression,
rather than an approximation. As the next step, we marginalize over the
linear-order nuisance parameters $\vec{\alpha}$ of the model, causing the
likelihood function $L$ to become

$L(P_{D}|P_{b})=\int
d\alpha_{i}\,L(P_{D}|\\{P_{b},\,P_{\alpha_{i}}\\})\mathcal{P}(\alpha_{i}),$
(A.2)

where $P_{D}$ represents the data vector and $\mathcal{P}(\alpha_{i})$ denotes
the priors for the linear-order nuisances parameters, which for simplicity are
assumed to be Gaussians, following

$\mathcal{P}(\alpha_{i})=\frac{1}{\sqrt{2\pi\sigma_{i}^{2}}}\,\exp\left[-\frac{1}{2}\left(\frac{\alpha_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}\right],$
(A.3)

centered in $\mu_{i}$ with standard deviation given by $\sigma_{i}$. To
introduce uninformative priors we should take standard deviations with
sufficiently large values, as presented in figure 7.

The unmarginalized likelihood function $L(P_{D}|\\{P_{b},\,P_{\alpha_{i}}\\})$
is given by

$L(P_{D}|\\{P_{b},\,P_{\alpha_{i}}\\})=\exp\left[{-\frac{1}{2}\left(\sum_{i}\alpha_{i}\,P_{\alpha_{i}}+P_{b}-P_{D}\right)^{\text{T}}\textit{Cov}^{-1}\left(\sum_{j}\alpha_{j}\,P_{\alpha_{j}}+P_{b}-P_{D}\right)}\right],$
(A.4)

where Cov denotes the covariance matrix defined in equation (2.1). Putting all
the pieces together into equation (2.1), we find

$L(P_{D}|P_{b})=\int
d\alpha_{i}\,\exp\left(-\frac{1}{2}\sum_{i,j}\alpha_{i}\alpha_{j}\mathcal{A}_{ij}+\sum_{i}\alpha_{i}\mathcal{B}_{i}+\mathcal{C}_{0}\right),$
(A.5)

with

$\displaystyle\mathcal{A}_{ij}=P_{\alpha_{i}}^{T}\,\text{Cov}^{-1}\,P_{\alpha_{j}}+\frac{\delta_{ij}}{\sigma_{i}\sigma_{j}},$
(A.6)
$\displaystyle\mathcal{B}_{i}=-P_{\alpha_{i}}^{T}\,\text{Cov}^{-1}\,\Delta
P+\frac{\mu_{i}}{\sigma_{i}^{2}},$ (A.7)
$\displaystyle\mathcal{C}_{0}=-\frac{1}{2}\left(\Delta
P^{T}\,\text{Cov}^{-1}\,\Delta P+\frac{\mu^{2}}{\sigma^{2}}\right),$ (A.8)

where $\Delta P\equiv P_{b}-P_{D}$ is the residual between the contributions
of non-linear order parameters from the model and the data vector. Finally,
using the multivariate Gaussian integral, we found that the marginalized log-
likelihood $\mathcal{L}$ takes

$\mathcal{L}=\mathcal{C}_{0}+\frac{1}{2}\mathcal{B}_{i}\cdot\,\mathcal{A}^{-1}_{ij}\cdot\,\mathcal{B}_{i}-\frac{1}{2}\ln\left[\det(\mathcal{A}_{ij})\right].$
(A.9)

We have transformed the original likelihood function given by equation (A.4),
into a more complicated form, the equation (A.9). Although it may appear at
first glance to be a process with no advantages, as the new likelihood is more
computationally expensive compared to the original one, in practice, it is
quite different because we have eliminated the parameters
$\vec{\alpha}=\\{\alpha_{0},\alpha_{2},\alpha_{4},\alpha^{\text{\rm
shot}}_{0},\alpha^{\text{\rm shot}}_{2}\\}$ from the exploration of
posteriors, and when we use MCMC methods, the chains tend to converge much
more quickly in the case where marginalization is employed. This process
reduces drastically the number of steps required for the convergence of
chains. Moreover, we have to recall that no additional assumptions were made
to perform the marginalization presented above. Hence, the information of the
remaining parameters, i.e., the ones we are interested in, is not degraded by
this analytical marginalization process [132].

Figure 19 shows that our analytical marginalization implementation works
excellent, showing identical posterior results as without using it. To perform
this plot, we have assumed uniformative priors which are obtained in the limit
of infinite Gaussian widths $\sigma_{i}$ in eqs.(A.6)–(A.8).212121We have also
tested the consistency between using or not analytical marginalization for the
compressed Standard and ShapeFit analyses. However, due to space constraints,
we decided to present only the results obtained from the direct Full-Modelling
analysis. In table 9, we provide the statistics with the mean and a 68%
confidence intervals, for both cosmological and bias parameters parameters.
The table reveals the consistency between results obtained with and without
analytical marginalization. Additionally, we also present the means obtained
in each scenario (with and without marginalization) for the marginalized
linear-order nuisance parameters, demonstrating consistent findings across
both approaches.

Figure 19: Comparison of the constraints for the Full-Modelling analysis with and without analytical marginalization over the linear-order nuisances parameters $\vec{\alpha}=\\{\alpha_{0},\alpha_{2},\alpha^{\rm shot}_{0},\alpha^{\rm shot}_{2}\\}$. Here we set $\alpha_{4}=0$ because we are only fitting the monopole and quadrupole. These results correspond to the baseline analysis with $V_{25}=200\,\,h^{-3}\text{Gpc}^{3}$ and minimal freedom case.  Parameter | Without marginalizing | Marginalizing
---|---|---
$h$ | $0.6742\pm 0.0035$ | $0.6741\pm 0.0036$
$\omega_{cdm}$ | $0.1197^{+0.0017}_{-0.0014}$ | $0.1196^{+0.0018}_{-0.0013}$
$\omega_{b}$ | $0.0223\pm 0.00037$ | $0.0223\pm 0.00037$
$\ln(10^{10}A_{s})$ | $3.0336^{+0.013}_{-0.023}$ | $3.0325^{+0.013}_{-0.023}$
$b_{1}$ | $2.0030^{+0.032}_{-0.017}$ | $2.0047^{+0.031}_{-0.017}$
$b_{2}$ | $0.1014^{+0.80}_{-0.56}$ | $0.1190^{+0.69}_{-0.60}$
$\alpha_{0}$ | $7.6187$ | $7.1394$
$\alpha_{2}$ | $-47.1230$ | $-46.2287$
$\alpha^{\rm shot}_{0}$ | $0.0192$ | $0.0071$
$\alpha^{\rm shot}_{2}$ | $-9.0170$ | $-9.2193$
Table 9: Comparison of the means and 0.68 c.i. obtained for the cosmological
and bias parameters when marginalizing or not over the counterterm and
stochastic parameters. We also display the means on the counterterms and
stochastic parameters obtained in each case, showing consistent results
between them. The counterterm $\alpha_{4}$ was set to zero because, for this
test, we only fitted the monopole and quadrupole.

##  Appendix B ShapeFit implementation in FOLPS

ShapeFit, $V_{1}=8\,h^{-3}\text{Gpc}^{3}$:
$\vec{\Omega}^{\text{ref}}=\vec{\Omega}^{\text{true}}$

ShapeFit, $V_{1}=8\,h^{-3}\text{Gpc}^{3}$:
$\vec{\Omega}^{\text{ref}}=\vec{\Omega}^{\text{true}}+10\%$ in
$h,\,\omega_{cdm},\,A_{s}$

Figure 20:  1D marginalized posterior distributions for the ShapeFit
compressed parameters when using the approximate loop contributions given by
eq. (B.1) (solid lines) and when computing the full loop contributions from
equations (3.9) and (3.10) at each step during the likelihood evaluation
(dash-dotted lines). Here we use the LRG catalogues of the simulations
AbacusSummit, and the volume $V_{1}=8\,h^{-3}\text{Gpc}^{3}$. In the top
panel, we display the results when the reference cosmology used to create the
fixed template coincides with the cosmology of the simulations, and on the
bottom, we show the results when using a reference cosmology biased with
respect to the true values of simulations. Specifically, for the latter case,
the reference cosmology is increased by 10% in the cosmological parameters
$h$, $\omega_{cdm}$, and $A_{s}$.

In practice, ShapeFit employs a fixed template to determine the linear power
spectrum $P^{\text{ref}}_{L}(k)$ and the non-linear corrections (i.e., the
$P_{22}$-type and $P_{13}$-type loop integrals) at the reference cosmology. As
we explore compressed parameters, the reference linear power spectrum is then
transformed into $P^{\prime}_{L}(k)$ as a function of the ShapeFit parameters
$m$ and $n$, according to equation (5.12). Consequently, the non-linear
contributions need to be expressed in terms of these ShapeFit parameters. This
means that the $P_{22}$-type and $P_{13}$-type loop integrals should be
evaluated as functions of $m$ and $n$. To make this latter process faster, the
standard implementation of ShapeFit approximates the non-linear contributions
obtained from the fixed template via [23]

$\displaystyle
I_{mn}(k)=\left(\frac{P^{\prime}_{L}(k)}{P^{\text{ref}}_{L}(k)}\right)^{2}I^{\text{ref}}_{mn}(k),$
(B.1)

with $I_{mn}$ the pure one-loop pieces of the functions entering eq. (3.8).
Within this implementation, the one-loop contributions are computed only once
at the reference cosmology, denoted as $I^{\text{ref}}_{mn}$, instead of
computing them at each trial cosmology. That is, for a $P_{22}$-type one-loop
contribution, we approximate

$I^{\textit{$P_{22}$-type}}_{mn}(k)=\left(\frac{P^{\prime}_{L}(k)}{P^{\text{ref}}_{L}(k)}\right)^{2}\int
d^{3}p\,\mathcal{K}_{mn}(\bm{k},\bm{p})P_{L}^{\text{ref}}(|\bm{k}-\bm{p}|)P_{L}^{\text{ref}}(p)$
(B.2)

and similar for a $P_{13}$-type one. Thereafter, the power spectrum multipoles
are computed in the same way as the standard approach, following eq. (5.11).

The use of this approximation is an additional ingredient that may be useful
for a faster estimation of parameters. However, it is not necessary and one
can compute directly the loop integrals using the trial power spectrum. To
test the validity of eq. (B.1), we compare the compressed parameters obtained
with and without using the approximation. In figure 20 we exclusively display
the results for the LRG using the baseline settings, where the ShapeFit
parameter $n$ is set to zero. We decide to show the particular case of
$V_{1}=8\,h^{-3}\text{Gpc}^{3}$ because of its allowance for larger values on
the slope parameter $m$, where the transformed ShapeFit power spectrum
$P^{\prime}_{L}(k)$ exhibits a larger departure from the reference power
spectrum $P^{\text{ref}}_{L}(k)$, and thus the approximation is more likely to
fail. The top panel of figure 20 illustrates the special case where the
reference cosmology aligns with the cosmology of the simulations, while the
bottom panel depicts the case where they differ. The results are in excellent
agreement, indicating that the approximation works very well. It should be
noted that even for the biased cosmology case (bottom panel of figure 20) the
approximation is still accurate, showing that eq. (B.1) is reliable regardless
of the reference cosmology used for the template.

Finally, although we only present results for the LRG with
$V_{1}=8\,h^{-3}\text{Gpc}^{3}$, we also examined the approximation for the
other tracers and using the full volume $V_{25}=200\,h^{-3}\text{Gpc}^{3}$,
obtaining very good agreement across all cases. Consequently, for the level of
precision demanded by any realistic survey, the effects of using or not the
approximation of eq. (B.1) are negligible. Hereafter, in this work, we adopt
this approximation to compute the loop contributions.

##  Appendix C Testing the settings on FFTLog and IR-resummations

In this appendix, we present a couple of tests changing some settings of the
Modelling. The tests presented here are conducted using the LRGs tracer with
the volume $V_{25}=200\,\,h^{-3}\text{Gpc}^{3}$, that produce the smaller
statistical errors. Therefore, any differences in the model are more likely to
arise in this volume. Thus, these tests provides us a good inside of the
systematic effects introduced by the model when some of its settings are
changed. For these tests, we followed the Full-Modelling prescription
presented in the baseline analysis for the Min.F. case, as detailed in table
2.

We start by studying the impact caused by the number of sampling points
$N_{\text{FFTLog}}$ used when employing the FFTLog formalism to decompose the
linear real space power spectrum into a series of power laws, see eq. (4.1) of
[39]. In figure 21, we present the 1-dimensional marginalized posterior
distributions of the cosmological parameters using both the default value of
$N_{\text{FFTLog}}=128$ and a higher value of $N_{\text{FFTLog}}=256$, showing
that both cases yield very similar results. Therefore, the default value of
$N_{\text{FFTLog}}=128$ not only achieves good accuracy but also exhibits
excellent computational speed, as shown in figures 8 and 9 of [39].
Additionally, in the figure we present the results obtained using a code
called fkpt [99],222222https://github.com/alejandroaviles/fkpt which is based
on perturbation theory for $\Lambda$CDM and modified gravity theories using
fk-kernels [68, 39]. For this comparison, the effects of modify gravity are
not taken into account. The code is very similar to Folps, with the difference
that loop integrals are not integrated using FFTLog methods, but instead it
employs a brute force approach. We observe that fkpt and Folps codes are in
very good agreement, especially for $A_{s}$ and $\omega_{cdm}$, while there
are still some small differences in the parameter $h$. However, these
differences are inside $1\sigma$ fluctuations and they are not significant for
the chosen volume.

Figure 21:  Comparison of the 1-dimensional marginalized posterior
distributions for the cosmological parameters using two different values for
the number of sampling points in the FFTLog formalism, the default
$N_{\text{FFTLog}}=128$ (dotted blue line) and an increased
$N_{\text{FFTLog}}=256$ (solid black line). Additionally, we compare Folps
against another perturbation theory code known as fkpt (dash-dot green line),
showing that both codes are in accord.  Figure 22:  1-dimensional marginalized
posterior distributions for the cosmological parameters under changes in the
IR-resummation settings introduced in equations (3.21) - (3.23). We explore
the effects of varying the separation scale $k_{\text{IR}}$ and also compare
the approximation of keeping the BAO peak scale fixed at
$\ell_{\text{BAO}}\simeq 104\,h^{-1}\text{Mpc}$ and varying it as a function
of the cosmological parameters through $\ell_{\text{BAO}}=r_{\text{d}}$ during
the likelihood evaluation.

The second test we perform in this appendix investigates the impact caused by
the cutoff scale $k_{\text{IR}}$ introduced in eqs. (3.21)–(3.23) when
performing the IR-resummations. We compare the 1-dimensional marginalized
posterior distributions for the cosmological parameters when using the default
value of $k_{\text{IR}}=0.4\,h\,\text{Mpc}^{-1}$ (dotted blue line) with the
results obtained for different values of $k_{\text{IR}}$, as presented in
figure 22. From the figure, we observe that the effects on the cosmological
parameters caused by varying $k_{\text{IR}}$ are minimal, even when taking
$k_{\text{IR}}=0.05\,\,h\,\text{Mpc}^{-1}$ where we observe some differences,
mainly in $A_{s}$. However, the results are still in accord within $1\sigma$
fluctuations. We have to keep in mind that this test was done using $V_{25}$.
For smaller volumes, the results would likely coincide even more.

For the default value $k_{\text{IR}}=0.4\,\,h\,\text{Mpc}^{-1}$, we also
compare the effect of keeping the BAO scale $\ell_{\text{BAO}}\simeq
104\,h^{-1}\text{Mpc}$ fixed (dotted blue line) or varying it (dashed red
line) during the likelihood evaluation as a function of the cosmological
parameter through $\ell_{\text{BAO}}=r_{\text{d}}$, with the sound horizon
scale at the drag epoch $r_{\text{d}}$ given by the approximation [133]

$r_{\text{d}}=\frac{55.154\exp\left[-72.3\,\left(\omega_{\nu}+0.0006\right)^{2}\right]}{\omega_{cb}^{0.25351}\,\omega_{b}^{0.12807}}\,\text{Mpc}.$
(C.1)

From figure 22, we can observe whether the results of keeping the BAO scale
fixed or not are indistinguishable. Therefore, the approximation of keeping
the BAO scale fixed is quite good, at least for the baseline analysis, where
the baryon abundance is included via a narrow BBN prior, and the total
neutrino mass is fixed to its fiducial value.

##  Appendix D Author Affiliations

1Instituto de Ciencias Físicas, Universidad Autónoma de México, Cuernavaca,
Morelos, 62210, (México)

2Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México

3Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona (UB), c.
Martí i Franquès, 1, 08028 Barcelona, Spain.

4Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Martí
i Franquès 1, E08028 Barcelona, Spain

5Institut d’Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain

6Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA

7Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA

8University of Michigan, Ann Arbor, MI 48109, USA

9Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK

10Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK

11Departamento de Física, Instituto Nacional de Investigaciones Nucleares,
Carreterra México-Toluca S/N, La Marquesa, Ocoyoacac, Edo. de México C.P.
52750, México

12Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA

13Institute for Computational Cosmology, Department of Physics, Durham
University, South Road, Durham DH1 3LE, UK

14Department of Physics and Astronomy, The University of Utah, 115 South 1400
East, Salt Lake City, UT 84112, USA

15IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France

16Institute of Cosmology and Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK

17Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia

18Observatorio Astronómico, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio H, CP 111711 Bogotá, Colombia

19Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain

20Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA

21Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA

22The Ohio State University, Columbus, 43210 OH, USA

23Department of Astronomy, University of Florida, 211 Bryant Space Science
Center, Gainesville, FL 32611, USA

24Max-Planck-Institut für Extraterrestische Physik, Postfach 1312,
Giessenbachstrasse, 85748 Garching, Germany

25School of Mathematics and Physics, University of Queensland, 4072, Australia

26Department of Physics, The University of Texas at Dallas, Richardson, TX
75080, USA

27NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

28Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain

29Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

30University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA
94720, USA

31Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain

32Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH,
U.K

33Department of Physics & Astronomy, University of Wyoming, 1000 E.
University, Dept. 3905, Laramie, WY 82071, USA

34Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México

35Instituto Avanzado de Cosmología A. C., San Marcos 11 - Atenas 202.
Magdalena Contreras, 10720. Ciudad de México, México

36Department of Physics and Astronomy, University of Waterloo, 200 University
Ave W, Waterloo, ON N2L 3G1, Canada

37Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada

38Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada

39Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA

40Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA

41Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

42Department of Physics and Astronomy, Sejong University, Seoul, 143-747,
Korea

43CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain

44Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA

45SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA

46Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France

47National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China

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