# Robustness of baryon acoustic oscillation constraints for early-Universe
modifications to $\Lambda$CDM

José Luis Bernal  Department of Physics and Astronomy, Johns Hopkins
University, 3400 North Charles Street, Baltimore, Maryland 21218, USA  Tristan
L. Smith  Department of Physics and Astronomy, Swarthmore College, 500 College
Ave., Swarthmore, PA 19081, USA  Kimberly K. Boddy  Theory Group, Department
of Physics, The University of Texas at Austin, Austin, TX 78712, USA  Marc
Kamionkowski  Department of Physics and Astronomy, Johns Hopkins University,
3400 North Charles Street, Baltimore, Maryland 21218, USA

###### Abstract

Baryon acoustic oscillations (BAO) provide a robust standard ruler, and can be
used to constrain the expansion history of the Universe at low redshift.
Standard BAO analyses return a model-independent measurement of the expansion
rate and the comoving angular diameter distance as function of redshift,
normalized by the sound horizon at radiation drag. However, this methodology
relies on anisotropic distance distortions of a fixed, pre-computed template
(obtained in a given fiducial cosmology) in order to fit the observations.
Therefore, it may be possible that extensions to the consensus $\Lambda$CDM
add contributions to the BAO feature that cannot be captured by the template
fitting. We perform mock BAO fits to power spectra computed assuming
cosmological models which modify the growth of perturbations prior to
recombination in order to test the robustness of the standard BAO analysis. We
find no significant bias in the BAO analysis for the models under study
($\Lambda$CDM with a free effective number of relativistic species, early dark
energy, and a model with interactions between neutrinos and a fraction of the
dark matter), even for cases which do not provide a good fit to Planck
measurements of the cosmic microwave background power spectra. This result
supports the use of the standard BAO analysis and its measurements to perform
cosmological parameter inference and to constrain exotic models. In addition,
we provide a methodology to reproduce our study for different models and
surveys, as well as discuss different options to handle eventual biases in the
BAO measurements.

##  I Introduction

Baryon acoustic oscillations (BAO) appear due to the primordial sound waves
propagating in the tightly coupled photon-baryon plasma in the early Universe
until recombination (e.g., Peebles_perturbations ; Sunyaev_fluctuations ).
After recombination, when the interaction rate between photons and baryons
becomes inefficient due to Hubble expansion, the acoustic waves stop
propagating and the BAO are imprinted in the baryon distribution. Therefore,
BAO are present in the temperature anisotropies of the cosmic microwave
background (CMB), which allows a very precise inference of cosmological
parameters (see e.g., Planck18_pars ). In addition, their features also appear
in the matter (and subsequently the galaxy) distribution at low redshift,
although with lower significance. First detected in the galaxy power spectrum
around fifteen years ago firstBAO_SDSS ; firstBAO_2dF , BAO have been robustly
measured in galaxy, quasar, and Lyman-$\alpha$ density distributions reaching
percent-level precision (see e.g, Alam_bossdr12 ; GilMarin_QSOeBOSS ;
eBOSS_Lyalpha_auto ; eBOSS_Lyalpha_cross ).

The BAO features are characterized by a physical scale: the sound horizon at
radiation drag, $r_{\rm d}$. With this ‘standard ruler’, it is possible to
robustly map the expansion history of the Universe up to the redshift of
measurement Eisenstein_02ASP ; Blake_standardruler ; Seo_BAODE . Given that
BAO measurements depend on both $r_{\rm d}$ and the expansion history, they
are sensitive to early and late time physics in clean, distinct ways.

The BAO and type Ia supernovae (SNeIa) Betoule14_jla ; Scolnic_pantheon  have
become the main complement to CMB observations to constrain deviations from
the $\Lambda$CDM consensus model at low redshift. Moreover, since BAO
measurements form a standard ruler which connects the direct and inverse
distance ladders Cuesta:2014asa , they have been key in addressing the Hubble
constant ($H_{0}$) tension RiessH0_19 ; H0_holicow_19 ; Planck18_pars ;
Addison_BAOBBN ; Schoneberg_BAOBBN ; Verde_H0summary . As pointed out in Ref.
BernalH0  (and confirmed by independent analyses Poulin_H0 ; Aylor_sounds ),
the $H_{0}$ tension can be reframed as a mismatch between the anchors of the
direct and the inverse cosmic distance ladders ($H_{0}$ and $r_{\rm d}$,
respectively), with existing data constraining deviations of the evolution of
the expansion history at low redshift. This has led authors of Ref.
Know_H0hunter  to theorize that any extension to $\Lambda$CDM should modify
pre-recombination physics rather than the low-redshift Universe if it is to
alleviate the $H_{0}$ tension. Some examples include Refs. Poulin_EDE ;
Smith_EDE ; Kreisch_selfnu ; Ghosh_DNIH0 ; Lin_acousticDM ; Agrawal_R&R ;
Berghaus_H0friction ; Zumalacarregui_MGH0 .

Standard BAO analyses rely on templates of the summary statistic under study
computed assuming a fiducial cosmology (normally $\Lambda$CDM). They consist
of determining the anisotropic rescaling of the distance coordinates that the
template needs in order to fit observations, marginalizing over a plethora of
nuisance parameters to account for other deviations with respect to the
template. This approach allows for a tomographic, model-independent
determination of the expansion history of the Universe that can be used to
constrain different cosmological models.

This procedure has been proven to be extremely robust and flexible for models
predicting different expansion rates at late times (see e.g., Xu_anisoBAO ;
Aubourg15 ; VargasMagana_BAOtheosyst ; Carter_BAOtest ), and it successfully
models changes in $r_{\rm d}$ due to early-time modifications of the
cosmological model. However, there may be other contributions to the BAO
feature that are not captured by these rescaling and nuisance parameters, such
as phase shifts (which can be scale dependent) or a different scale dependence
of the amplitude of the oscillations. These additional contributions may bias
standard BAO analyses (in the case they are not properly modeled in the
template), but they can also be targeted to constrain beyond-$\Lambda$CDM
physics with BAO (as shown in Refs. Baumann_NeffBAO ; Baumann_NeffBAO_fisher ;
Baumann_NeffBAO_measure  for the case of extra relativistic species $N_{\rm
eff}$, and in Ref. Beutler_PrimBAO  for the case of oscillations in the
primordial power spectrum). Given the key role of BAO measurements in
constraining extensions to $\Lambda$CDM, it is essential to ascertain whether
cosmological-parameter inference from BAO analyses is free of unaccounted
systematic errors, especially when the requirements on the robustness of the
analyses will tighten for next-generation BAO measurements.

In this work we explore the impact that modifications of the BAO feature,
arising from changes in the growth of the matter density perturbations, may
have in a BAO analysis. Concretely, we consider the possibility that standard
template-fitting BAO analyses are not flexible enough to characterize these
modifications. If this were the case, the reported BAO measurements obtained
using a $\Lambda$CDM template might be biased if the true cosmology
corresponds to these models; moreover, a bias could render BAO measurements
unsuitable for constraining extended models.

This study is therefore timely and needed in order to ensure the robustness of
using BAO measurements to constrain models beyond $\Lambda$CDM. We focus on
$\Lambda$CDM, a model with a different number of relativistic species
($\Lambda$CDM$+N_{\rm eff}$), and two models that claim to ease or resolve the
Hubble tension: early dark energy (EDE) Poulin_EDE ; Smith_EDE  (although see
also Refs. Hill_EDE ; Ivanov_EDEEFT ; D'Amico_EFTEDE ; SmithEDELSS ;
PoulinEDEWL ) and dark neutrino interactions (DNI) Ghosh_DNIH0 ; Ghosh_DNIGW .
We perform BAO analyses using mock power spectrum measurements computed under
these models, for both ‘good’ and ‘bad’ fits to CMB observations. We use the
term ‘bad fits’ to refer to sets of cosmological parameters for which one of
the parameters has been chosen to have a fixed value at least $\gtrsim
3\sigma$ away from its best fit to Planck, while the rest of the parameters
are selected to maximize the posterior fulfilling such requirement. We also
include an extreme case for DNI, the model adding the largest difference in
the BAO feature with respect to the $\Lambda$CDM prediction. We find that the
standard BAO analysis returns results without significant biases, even when
considering the ‘bad fits’ mock power spectra: the biases in the BAO rescaling
parameters are $<0.2\sigma$ except for the extreme case of DNI, for which we
find a bias $\sim 1\sigma$). These results confirm the robustness of BAO
analyses and their appropriate use for cosmological parameter inference. An
alternative, general approach to estimate the bias on parameter inference can
be found in e.g., Ref. Bernal_Fisher .

We describe in detail our methodology for future application to other models
to check whether the standard BAO analysis is unbiased. In case a bias were to
be found, we discuss different courses of action to address the situation.
Nonetheless, we also provide approximate arguments for extrapolating our
results to other cosmological models.

Although this work focuses on the BAO feature, there is also cosmological
information contained in the broadband of the observed clustering. Redshift-
space distortions at linear scales are mostly sensitive to the product
$f\sigma_{8}$, where $f$ is the linear growth rate and $\sigma_{8}$ is the
root mean square of the matter fluctuations within $8\,{\rm Mpc}/h$ (where
$h=H_{0}/\left(100\,{\rm km\,s^{-1}Mpc^{-1}}\right)$). The degeneracy between
$f$ and $\sigma_{8}$ can be broken by e.g., using higher-order statistics,
such as the three-point correlation function Slepian_3PCFmodel  or the
bispectrum GilMarin_RSDBispectrumDR12 ; Gualdi_bispectrummultipoles , by
exploiting the clustering at smaller scales using perturbation theory (see
e.g., Desjacques_bias ) or by using phase correlations Wolstenhulme_phases ;
Byun_phase . Additional physics related to large modifications of the scale
dependence of the clustering can be strongly constrained, such as suppression
at small scales (see e.g., Drlica-Wagner_LSSTDM ; Bullock_smallscalesreview ;
Gluscevic_DMintWP ; Nadler_DMMW ; Abazajian_sterilenureview  and references
therein) or primordial non-Gaussianities (see e.g., Dalal_png07 ;
Matarrese_png08 ; Raccanelli:fNL ; Bernal_EMU ). Parameters driving features
with weaker scale dependence can also be constrained, but normally external
data or priors are required for these constraints to be competitive.

This paper is structured as follows: we discuss the origin and nature of the
cosmological information encoded in the BAO measurements in Section II; review
the standard BAO analysis of the power spectrum in Section III; introduce the
cosmological models considered, discuss the variations in the predicted BAO
feature, and estimate the bias in the BAO fit in Section IV; and discuss the
results and conclude in Sections V and VI, respectively. Afterwards, we
discuss the methodologies used to extract and isolate the BAO feature in
Appendix A; detail how we compute the mock measured power spectrum in Appendix
B; and illustrate the connection between the CMB power spectra and the BAO
feature in the matter power spectrum at late times with a toy model in
Appendix C.

##  II BAO Cosmology

In this section, we describe the cosmological information contained in the BAO
feature of the low-redshift clustering of tracers of matter density
perturbations. The analysis of large-scale structure clustering allows for
precise determination of cosmological parameters. By separating the clustering
at different scales into a smooth, featureless broadband and the BAO feature,
it is possible to extract robust cosmological information encoded in the BAO
scale.

###  II.1 Alcock-Paczynski effect and isotropic dilation

Observations measure the positions of different tracers of matter in terms of
redshifts and angular positions on the sky, which must then be transformed to
obtain three-dimensional clustering summary statistics (e.g., the correlation
function or power spectrum) as a function of spatial distances or the
corresponding Fourier mode wave numbers. Given an angular separation $\theta$
and a small redshift separation $\delta z$, the spatial comoving distance in
the transverse direction and along the line of sight are

$r_{\perp}=D_{M}(z)\theta,\qquad\qquad r_{\parallel}=\frac{c\delta z}{H(z)},$
(1)

respectively, where $D_{M}$ is the comoving angular diameter distance, $H$ is
the Hubble expansion rate, and $c$ is the speed of light. Eq. (1) can be
adapted to proper distances substituting $D_{M}$ by the proper angular
diameter distance $D_{A}=D_{M}/(1+z)$. There are three main effects that alter
these components of the observed distances: redshift-space distortions
Hamilton_RSDreview , the Alcock-Paczynski effect Alcock_paczynski  and the
isotropic dilation. Redshift-space distortions are a physical modification to
$r_{\parallel}$, due to the peculiar velocities of galaxies changing the
redshift of observed sources along the line of sight (hence changing their
position in redshift space with respect to the real space). These distortions
introduce anisotropies in the observed clustering, which is isotropic a
priori.

Assuming a background expansion history (obtained from a fiducial cosmology)
that differs from that of the true expansion rate of the Universe causes an
artificial distance distortion. The fiducial cosmology is used to compute
$D_{M}$ and $H$ in Eq. (1); therefore, the recovered $r_{\perp}$ and
$r_{\parallel}$ differ from the true distances. This distortion affects
$r_{\perp}$ and $r_{\parallel}$ in different ways, so it is possible to
decompose it into an isotropic and an anistropic component: the isotropic
dilation and the Alcock-Paczynski effect.

The Alcock-Paczynski effect and the isotropic dilation can be modeled by
rescaling factors, obtained when comparing the observed distances, which
assume the fiducial cosmology, and true distances: $r_{\perp,\parallel}^{\rm
true}=r^{\rm obs}_{\perp,\parallel}q_{\perp,\parallel}$ (or $k^{\rm
true}_{\perp,\parallel}=k^{\rm obs}_{\perp,\parallel}/q_{\perp,\parallel}$ in
Fourier space). Using Eq. (1), the rescaling parameters are

$q_{\perp}=\frac{D_{M}(z)}{\left(D_{M}(z)\right)^{\rm fid1}},\qquad
q_{\parallel}=\frac{\left(H(z)\right)^{\rm fid1}}{H(z)},$ (2)

where superscript ‘fid1’ denotes the assumed fiducial cosmology used in Eq.
(1). Using $q_{\perp}$ and $q_{\parallel}$, the isotropic dilation corresponds
to $\left(q_{\perp}^{2}q_{\parallel}\right)^{1/3}$, and the Alcock-Paczinski
effect is given by the ratio of $q_{\perp}$ and $q_{\parallel}$.

###  II.2 BAO scale

The Alcock-Paczysnki effect and the isotropic dilation are always
present111The only case in which the rescaling of distances in Eq. (2) is not
explicitly present in the analysis is if the clustering statistics used are
functions of redshifts and angles (see e.g., Ref. Bertacca_xi3dGR ). In this
case, the rescaling is implicitly embedded in the modeling of the clustering
statistics. in the measurement of the clustering statistics: it is inherent to
any measurement that depends on distance scales. Nonetheless, the BAO feature
is clearly distinguishable against the broadband of the summary statistic; it
manifests as oscillations in Fourier space or a peak in configuration space,
and large-scale clustering measurements have well-determined its location.

To extract the BAO scale from the observed target summary statistic, standard
BAO analyses employ a pre-computed template of the target summary statistic
generated assuming a given cosmology. This cosmology does not need to be the
same as that used in Eqs. (1) and (2). As we discuss in further detail in
Section III, using the template allows for the extraction of $r_{\rm d}$,
which is the only characteristic scale of matter clustering at low redshifts
(at larger scales than those corresponding to matter-radiation equality). The
BAO scale of the template might not match the true BAO scale; therefore, a
correction on $r_{\rm d}$ must be included when rescaling distances in order
to fit the observed BAO feature with the template. The correction is
isotropic, and the rescaling of distances becomes $r_{\perp,\parallel}^{\rm
th}=r^{\rm obs}_{\perp,\parallel}\alpha_{\perp,\parallel}$ (or $k^{\rm
th}_{\perp,\parallel}=k^{\rm
obs}_{\perp,\parallel}/\alpha_{\perp,\parallel}$), where

$\alpha_{\perp}=q_{\perp}\frac{\left(r_{\rm d}\right)^{\rm fid2}}{r_{\rm
d}},\qquad\alpha_{\parallel}=q_{\parallel}\frac{\left(r_{\rm d}\right)^{\rm
fid2}}{r_{\rm d}}$ (3)

provide a mapping between the observed distances (or wave numbers) and those
which enter our theoretical modeling, denoted by ‘th’.222There are alternative
parametrizations of these rescalings (or those in Eq. (2)), obtained through
combinations of $\alpha_{\perp}$ and $\alpha_{\parallel}$. Some examples focus
on the isotropic and anisotropic distortions $(\alpha,\epsilon)$ or on the
monopole and the $\mu^{2}$ moment of the two-point statistics
$(\alpha_{0},\alpha_{2})$ (see e.g., Refs. Xu_anisoBAO  and GilMarin_BAODR12 ,
respectively). The superscript ‘fid2’ in Eq. (3) refers to the fiducial
cosmology used to generate the template for the BAO analysis. It is important
to notice that the rescaling of $r_{\rm d}$ in Eq. (3) is not related to the
Alcock-Paczynski effect or the isotropic dilation. Hence, the rescaling
between observed distances and those entering our theoretical model introduced
in Eq. (3) is the combination of two non-physical effects: the redshift-
distance transformation and the ratio between the fiducial (for the fixed
template) and true $r_{\rm d}$ values.

Hereinafter we assume that the fiducial cosmology used to convert redshifts
into distances in Eq. (1) is the same as the one used to compute the template
of the clustering statistic, as is commonly the case in BAO studies. Under
this assumption, the rescaling parameters become

$\begin{split}\alpha_{\perp}=\frac{D_{M}(z)/r_{\rm d}}{\left(D_{M}(z)/r_{\rm
d}\right)^{\rm fid}},\qquad\alpha_{\parallel}=\frac{\left(H(z)r_{\rm
d}\right)^{\rm fid}}{H(z)r_{\rm d}},\end{split}$ (4)

where ‘fid’ corresponds to the fiducial cosmology that has been used to both
translate redshifts into distances and compute the fixed template.

###  II.3 Cosmological information in BAO

In order to avoid biasing the information obtained from the BAO feature, the
shape and amplitude of the broadband are marginalized over with the
introduction of nuisance parameters. After marginalization, the only remaining
cosmological information in the clustering statistics is related to the BAO
location and anisotropy, which is mostly encoded in the rescaling parameters.
This is an entirely geometric fit to the observations; hence, it has the
potential to be performed without being limited to any cosmological model
without loss of generality. Specifically, the rescaling parameters are the fit
parameters, and the resulting constraints are traditionally used in global
analyses to infer cosmological parameters of any cosmological model.

As evident from Eq. (2), the only cosmological information the Alcock-
Paczynski effect and the isotropic dilation are sensitive to is the late-time
expansion rate. By utilizing a fixed template in the analysis, BAO
measurements are also sensitive to pre-recombination physics through $r_{\rm
d}$ (Eq. (4)) in an agnostic and independent way, incorporating information
about both the expansion rate and the growth of matter perturbations. While
the isotropic dilation is completely degenerate with $r_{\rm d}$, the Alcock-
Paczynski effect (modeled by the ratio of $\alpha_{\perp}$ and
$\alpha_{\parallel}$ when a fixed template is used) is independent of the BAO
scale. Since
$\alpha_{\perp}/\alpha_{\parallel}=D_{M}H/\left(D_{M}H\right)^{\rm fid}$ does
not depend on $H_{0}$, the Alcock-Paczynski effect constrains the unnormalized
expansion history of the Universe, independently of the BAO scale.

Accessing early-time information about $r_{\rm d}$ enables the construction of
the inverse distance ladder Cuesta:2014asa  and allows for an independent
constraint on the product $r_{\rm d}H_{0}$ StandardQuantities , which is what
makes the BAO measurements key for the $H_{0}$ tension BernalH0 . While having
sensitivity to $r_{\rm d}$ is beneficial, the fixed template may reduce the
flexibility of the analysis and, therefore, the applicability of the
constraints on $\alpha_{\perp,\parallel}$ to cosmological models that deviate
from the fiducial cosmology in the early Universe. We explore this possibility
in Section IV.3.

Finally, there is also cosmological information contained not only in the
location of the BAO feature, but also in its shape and amplitude (see e.g.,
Baumann_NeffBAO ; Beutler_PrimBAO ). However, given that the significance of
the BAO feature with respect to the broadband in the density distribution at
low redshift is significantly smaller than in the CMB power spectra, the
constraining power of this information is limited with respect to the CMB
constraints. Nonetheless, for some cases, external priors on the rescaling
parameters can be used to practically fix their values and focus only on the
shape of the BAO. This approach has provided promising results to constrain
$N_{\rm eff}$ Baumann_NeffBAO_fisher ; Baumann_NeffBAO_measure  and
oscillations in the primordial power spectrum Beutler_PrimBAO , and it offers
an alternative (but not independent) approach to look for physics beyond
$\Lambda$CDM, complementary to direct CMB constraints.

##  III BAO analysis

In this section, we introduce the methodology that we use to extract the BAO
scale in this work. We follow the standard analysis used to obtain BAO
measurements from galaxy surveys, using the galaxy power spectrum in Fourier
space. A similar analysis can be performed in configuration space using the
correlation function (see e.g., Anderson_BOSSDR9  for the computation of the
templates).

###  III.1 Power spectrum template and methodology

The power spectrum $P(\bm{k})$ and the correlation function $\xi(\bm{s})$ are
equivalent estimators for two-point clustering statistics in Fourier and
configuration space, respectively, where $\bm{s}$ is the redshift space
distance and $\bm{k}$ is the associated wave number. The Legendre multipoles
of the power spectrum are given by

$P_{\ell}(k)=\frac{2\ell+1}{2}\int_{-1}^{1}{\rm d}\mu
P(k,\mu)\mathcal{L}_{\ell}(\mu),$ (5)

where $k\equiv\lvert\bm{k}\lvert$ is the module of the wave number vector and
$\mu$ is the cosine of the angle between the wave number vector and the line
of sight. The Legendre multipoles of the correlation function, $\xi_{\ell}$,
are defined in an analogous way, and related with $P_{\ell}$ by the Fourier
transform via

$\xi_{\ell}(s)=i^{\ell}\int\frac{k^{3}{\rm d}\log
k}{2\pi^{2}}P_{\ell}(k)j_{\ell}(ks),$ (6)

where $s\equiv\lvert\bm{s}\lvert$ is the module of the redshift space distance
and $j_{\ell}$ is the $\ell$-th order spherical Bessel function. Note that Eq.
(6) equally holds for real space distances and wave numbers. In this section,
we do not explicitly include the dependence on redshift, present in
practically all quantities, for the sake of brevity and readability; we do,
however, show the dependence on $k$ and $\mu$, for clarity.

The standard BAO analysis is based on fitting a template (pre-computed under a
fiducial cosmology) to the observations. This template is built in such a way
that the BAO feature is identifiable and isolated. In order to isolate the BAO
feature, the linear matter power spectrum $P_{\rm m}$ is decomposed into a
smooth component $P_{\rm m,sm}$ (i.e., the broadband, with no contribution
from the BAO) and an oscillatory contribution $O_{\rm lin}$. In this way, the
total matter power spectrum is given by $P_{\rm m}(k)=P_{\rm m,sm}(k)O_{\rm
lin}(k)$. There are different ways to extract $P_{\rm m,sm}$ from $P_{\rm m}$,
and we describe the methodology used here in Appendix A.333Our methodology is
a generalization of the approach introduced in Ref. Kirkby_BAOtemplate ,
accounting for the fact that $r_{\rm d}$ can have (very) different values
depending on the cosmology assumed.

The galaxy bias $b_{\rm g}$ and a factor encoding the effect of redshift-space
distortions $F_{\rm RSD}$ can be applied to $P_{\rm m,sm}$ in order to obtain
the anisotropic, smoothed galaxy power spectrum in redshift space

$P_{\rm sm}(k,\mu)=BF_{\rm RSD}^{2}(k,\mu)P_{\rm m,sm}(k),$ (7)

where $B$ is a constant absorbing $b_{\rm g}$ and potential variations on the
amplitude of $P_{\rm m,sm}$, and

$F_{\rm
RSD}(k,\mu)=\left(1+\beta\mu^{2}R\right)\frac{1}{1+0.5\left(k\mu\sigma_{\rm
FoG}\right)^{2}},$ (8)

where $\beta=f/b_{g}$, and the fingers of God small-scale suppression is
driven by the parameter $\sigma_{\rm FoG}$, whose value is related to the halo
velocity dispersion.444The damping due to the fingers of God can also be
modeled with a Gaussian function, providing similar results without losing
flexibility in the fit to the observations.

The actual amplitude of the BAO feature is reduced with respect to the linear
prediction due to non-linear collapse. In addition, non-linear clustering also
introduces a sub-percent shift in the BAO scale. However, these effects can be
partially reverted using density field reconstruction
Eisenstein_reconstruction ; Padmanabhan_recons . The effect of wrong
assumptions regarding the fiducial cosmology and galaxy bias on density field
reconstruction was studied in Ref. Sherwin_wrongrecon , where a negligible
shift in the BAO peak location was found. However, it was also found that
these small changes in the shape of the BAO feature and in the amplitude of
the quadrupole of the power spectrum might introduce small biases in the BAO
fit for future surveys. The factor $R$ in Eq. (8) models the partial removal
of redshift-space distortions produced by the density field reconstruction and
takes the following values: $R=1$ before reconstruction and
$R=1-\exp\left[-\left(k\Sigma_{\rm recon}\right)^{2}/2\right]$ after
reconstruction.555The specific functional form of $R$ after reconstruction
depends on the reconstruction formalism used Seo_BAOrecon . The one used in
this work corresponds to the ‘Rec-Iso’ convention in Ref. Seo_BAOrecon , which
accounts for the removal of redshift-space distortions during the process of
reconstruction. Nonetheless, the choice of reconstruction formalism does not
affect our results, since we use the same approach for the analysis and for
the computation of the mock power spectra that we analyze. On the other hand,
the non-linear damping of the BAO is modeled with an exponential suppression
applied to $O_{\rm lin}$ Eisenstein_BAOrobust ; Crocce_BAOdamp ;
Seo_nonlinearBAO ; Seo_nonlinearBAO10 . The damping affects the transverse and
line-of-sight directions differently; hence, we introduce two separate scales
$\Sigma_{\perp}$ and $\Sigma_{\parallel}$, respectively.

The final anisotropic galaxy power spectrum, accounting for the effect of non-
linearities on the BAO features and eventual density field reconstruction, can
be expressed as

$\begin{split}P(k,\mu)&=P_{\rm sm}(k,\mu)\,\times\\\
\times&\left[1+\left(O_{\rm
lin}(k)-1\right)e^{-\frac{k^{2}}{2}\left\\{\mu^{2}\Sigma_{\parallel}^{2}+\left(1-\mu^{2}\right)\Sigma_{\perp}^{2}\right\\}}\right]+\\\
+&P_{\rm shot}\,,\end{split}$ (9)

where $P_{\rm shot}=n_{\rm g}^{-1}$ (where $n_{\rm g}$ is the mean comoving
number density of galaxies) is a scale-independent contribution arising from
the fact that we use discrete tracers of the matter density field, such as
galaxies. The template for the BAO analysis is generated with Eq. (9).

As shown in Eq. (9), it is clearer to express the anisotropic power spectrum
as function of $k$ and $\mu$, instead of $k_{\perp}$ and $k_{\parallel}$. The
rescaling of distances appearing in Eq. (4) can be transformed to $k$ and
$\mu$ as Ballinger_scaling96

$\begin{split}&k^{\rm true}=\frac{k^{\rm
obs}}{\alpha_{\perp}}\left[1+\left(\mu^{\rm obs}\right)^{2}\left(F_{\rm
AP}^{-2}-1\right)\right]^{1/2},\\\ &\mu^{\rm true}=\frac{\mu^{\rm obs}}{F_{\rm
AP}}\left[1+\left(\mu^{\rm obs}\right)^{2}\left(F_{\rm
AP}^{-2}-1\right)\right]^{-1/2},\end{split}$ (10)

where $F_{\rm AP}\\!\equiv\\!\alpha_{\parallel}/\alpha_{\perp}$.

Given the large scales probed, the line of sight changes with each pointing
and cannot be considered parallel to any Cartesian axis. Hence, it is not
possible to obtain a well-defined $\mu$ for the observations, which makes a
direct measurement $P(k,\mu)$ impossible. However, one can directly measure
the Legendre multipoles of the anisotropic power spectrum using, e.g., the
Yamamoto estimator YamamotoEstimator .666There are other compression options,
such as the so-called angular wedges Kazin_wedges . Then, the observed power
spectrum multipoles are modeled as

$\begin{split}&P_{\ell}(k^{\rm
obs})=\frac{2\ell+1}{2\alpha_{\perp}^{2}\alpha_{\parallel}}\times\\\
&\times\int_{-1}^{1}{\rm d}\mu^{\rm obs}P(k^{\rm true},\mu^{\rm
true})\mathcal{L}_{\ell}(\mu^{\rm obs})+A_{\ell}(k),\end{split}$ (11)

where $\mathcal{L}_{\ell}$ is the Legendre polynomial of degree $\ell$,
$P(k^{\rm true},\mu^{\rm true})$ is computed using Eqs. (9) and (10), and a
$\left(r_{\rm d}^{\rm fid}/r_{\rm d}\right)^{3}$ term has been absorbed into
the constant factor $B$ of $P_{\rm sm}$ in Eq. (7). Different polynomials
$A_{\ell}(k)$ are added to each one of the power spectrum multipoles. These
polynomials are added not only to marginalize over uncertainties related with
non-linear clustering, but in particular to account for the possibility that
the broadband of the template $P_{\rm m,sm}$ does not match the actual one.
These polynomials have the form

$A_{\ell}(k)=a_{\ell,1}k^{-3}+a_{\ell,2}k^{-2}+a_{\ell,3}k^{-1}+a_{\ell,4}+a_{\ell,5}k^{n},$
(12)

where $n=1$ and $n=2$ before and after density field reconstruction,
respectively Beutler_BAODR12 .

Note that the standard BAO analysis can be performed with slightly different
models of the signal that incorporate, for example, different nonlinear
evolution and procedures to isolate the BAO feature. These slight differences
are discussed in Ref. Hinton_BARRY  and do not affect our results or
conclusions.

In summary, BAO-only analyses include the following parameters:

$\left\\{\alpha_{\perp},\alpha_{\parallel},B,\beta,\bm{a}_{\ell},\sigma_{\rm
FoG},\Sigma_{\perp},\Sigma_{\parallel}\right\\},$ (13)

where $\bm{a}_{\ell}$ are the coefficients of $A_{\ell}$ in Eq. (12). All but
the two first parameters $\alpha_{\perp}$ and $\alpha_{\parallel}$ are
nuisance parameters. There are thus 17 parameters in the analysis of the
monopole and quadrupole of the power spectrum (22 parameters if the
hexadecapole is also included). In this work we consider the monopole and
quadrupole of the post-reconstruction galaxy power spectrum. The values of the
parameters listed in Eq. (13) that we use to compute the template are:
$\alpha_{\perp}=\alpha_{\parallel}=1$, $B=b_{\rm g}^{2}$, $\beta=f/b_{\rm g}$,
$\bm{a}_{\ell}=0$, $\sigma_{\rm FoG}=10$ Mpc$/h$, $\Sigma_{\perp}=2$ Mpc$/h$
and $\Sigma_{\parallel}=4$ Mpc$/h$. The value of $b_{\rm g}$ depends on the
survey considered, $f$ depends on the redshift and cosmology assumed, and we
choose the values for $\Sigma_{\perp}$ and $\Sigma_{\parallel}$ following Ref.
Eisenstein_BAOrobust . The specific values chosen for the fiducial parameter
do not change our results.

###  III.2 Covariance and likelihood

Galaxy surveys normally rely on an estimation of the variation of the power
spectrum using a suite of mock catalogs (see e.g., Kitaura_Patchy ; White_qpm
) computed on the fiducial cosmology to obtain the covariance of the
multipoles of the power spectrum. Using galaxy mocks makes it easier to model
selection effects, observational mask, survey geometry, and other
observational systematics that need to be taken into account to avoid a bias
in the measurement Ross_BOSSsyst . However, motivated by the ever growing
number of mocks required to lower the noise of the covariance below the
statistical errors, there are proposals to obtain precise covariance matrices
directly from the data (see e.g., Oconnell_covmat_nomocks ). Since in this
work we aim to treat BAO analyses in general and do not need exquisite
precision on the covariance, we use an analytic approximation. Neglecting mode
coupling due to the non-linear collapse and the observational mask, we
approximate the covariance per $k$ and $\mu$ bin as

$\sigma^{2}(k,\mu)=\frac{P^{2}(k,\mu)}{N_{\rm modes}(k,\mu)},$ (14)

where $N_{\rm modes}$ is the number of modes per bin in $k$ and $\mu$ in the
observed volume $V_{\rm obs}$, given by

$N_{\rm modes}(k,\mu)=\frac{k^{2}\Delta k\Delta\mu}{8\pi^{2}}V_{\rm obs},$
(15)

where $\Delta k$ and $\Delta\mu$ are the widths of the $k$ and $\mu$ bins,
respectively. Note that $P_{\rm shot}$ is implicitly included in Eq. (14),
since it is part of the modeling of $P(k,\mu)$ (see Eq. (9)). In the case of
shot-noise subtracted power spectrum measurements, $P_{\rm shot}$ should be
removed from Eq. (9) and added explicitly to $P(k,\mu)$ in Eq. (14).

Once we have defined the covariance per $k$ and $\mu$ bin, we can compute the
covariance of the power spectrum multipoles. This covariance can be decomposed
in sub-covariance matrices for each multipole, plus the sub-covariance
matrices between different multipoles. For multipoles $\ell$ and
$\ell^{\prime}$, the sub-covariance matrix under the Gaussian assumption is
given by (see Ref. Grieb_covmat  for a detailed derivation)

$\begin{split}\mathcal{C}_{\ell\ell^{\prime}}(k_{i},k_{j})&=\delta_{ij}\frac{(2\ell+1)(2\ell^{\prime}+1)}{2}\times\\\
&\times\int_{-1}^{1}{\rm
d}\mu\sigma(k_{i},\mu)^{2}\mathcal{L}_{\ell}(\mu)\mathcal{L}_{\ell^{\prime}}(\mu)\,,\end{split}$
(16)

where $\delta_{ij}$ is the Kronecker delta. Finally, it is necessary to
incorporate observational effects (in both the clustering summary statistic
and its covariance), which are mainly due to the geometry of the survey
itself, in the theoretical modeling of the observed multipoles of the power
spectrum. The geometry of the survey (the footprint and the fact that some
regions may be observed more times than others) affects the selection of
galaxies and is modeled using a mask in configuration space (which becomes a
convolution in Fourier space). While modeling and properly implementing the
mask (see Refs. Wilson_mask ; Beutler_BOSSDR12_BAO ; Beutler_BOSSDR12_RSD  for
detailed discussions) are key for obtaining reliable conclusions from power
spectrum measurements, they do not have an effect on the focus of our study:
the flexibility of the analysis and its validity regarding cosmologies beyond
$\Lambda$CDM at the level of perturbations. Moreover, the effect of the mask
on the BAO analysis is small and more important for redshift-space-distortion
measurements. Therefore, we do not account for observational effects in this
work.

Taking all this into account, we can compare the theoretical prediction of the
measured power spectrum with respect to a fiducial cosmology with actual
observations. Hence, the corresponding likelihood is

$\log L\propto\left(\bm{P}^{\rm th}-\bm{P}^{\rm
data}\right)\mathcal{C}^{-1}\left(\bm{P}^{\rm th}-\bm{P}^{\rm
data}\right)^{T},$ (17)

where $\bm{P}$ denotes the concatenation of all the multipoles of the power
spectrum considered in the analysis, $\bm{P}^{\rm th}$ and $\bm{P}^{\rm data}$
are the theoretical prediction and the measured power spectrum, respectively,
the superscript $T$ refers to the transpose operator, and all quantities are
evaluated at the observed values of $k$ (following Eq. (11)).

##  IV Estimating the bias on BAO measurements

In this section we explore the impact that extensions to $\Lambda$CDM have on
the BAO feature. After introducing the models considered, we study the changes
in $O_{\rm lin}$ with respect to the fiducial cosmology, individually varying
different parameters. Afterwards, we perform mock BAO analyses as detailed in
Section III to evaluate the flexibility and validity of BAO results for the
extended cosmologies.

###  IV.1 Cosmological models

As a point of reference, we consider $\Lambda$CDM with one massive neutrino
species ($m_{\nu}=0.06$ eV) and its standard parameters: the physical baryon
and cold dark matter density parameters today, $\Omega_{\rm b}h^{2}$ and
$\Omega_{\rm cdm}h^{2}$, respectively, the reduced Hubble constant $h$, the
spectral index $n_{\rm s}$, and the amplitude of the primordial power spectrum
of scalar modes $A_{\rm s}$. We take the fiducial cosmology in this section to
be the same used for our analysis in Section IV.3: $\Lambda$CDM with parameter
values listed in Table 11. In addition to $\Lambda$CDM, we focus on two models
that have been suggested as potential solutions of the Hubble tension: EDE and
DNI. We also consider a $\Lambda$CDM model with a different number of
relativistic species than the prediction of the standard model $N_{\rm
eff}=3.046$, $\Lambda$CDM$+N_{\rm eff}$, where $N_{\rm eff}$ is an additional
parameter of this model.

The EDE model proposes the existence of a scalar field $\phi$ that is
initially frozen (due to Hubble friction) at some field value and becomes
dynamical after the Hubble parameter drops below some critical value. If the
EDE becomes dynamical before recombination, it leads to a decrease in size of
the photon-baryon sound horizon, providing a resolution to the Hubble tension
Poulin_EDE ; Smith_EDE . Therefore, an EDE resolution to the Hubble tension
leads to a decrease in $r_{\rm d}$, while keeping CMB power spectra angular
scales and peak heights fixed via small shifts in the standard $\Lambda$CDM
parameters. In particular we explore the oscillating EDE model, described in
Ref. Smith_EDE . The scalar field evolves along an axion-like potential $V$ of
the form

$V(\phi)=\Upsilon^{4}\left[1-\cos\left(\phi/\lambda\right)\right]^{n_{\rm
axion}},$ (18)

where $\Upsilon$ is the normalization of the potential and $\lambda$ is the
decay constant of the scalar field. The additional parameters beyond those of
$\Lambda$CDM considered for this model are the critical redshift $z_{\rm
EDE}^{c}$ when the EDE becomes dynamical, the fraction $f_{\rm EDE}$ of EDE in
the total energy density of the Universe at the critical redshift, the initial
scalar field displacement $\Theta_{\rm EDE}\equiv\phi_{i}/\lambda$, and the
power $n_{\rm axion}$ of the oscillating factor of the potential for the
scalar field. The shift in $r_{\rm d}$ is mainly controlled by $f_{\rm EDE}$,
with a weaker dependence on $z_{\rm EDE}^{c}$. The shape of the potential
(which is controlled by $n_{\rm axion}$) determines the rate at which the EDE
dilutes after $z_{\rm EDE}^{c}$, and $\Theta_{\rm EDE}$ mainly controls the
evolution of the EDE perturbations (see also Lin:2019qug ; Smith_EDE ).

The DNI model proposes an interaction between neutrinos and a small fraction
$f_{\rm DNI}$ of the dark matter $\chi$ with mass $m_{\chi}$ in order to solve
the $H_{0}$ tension Ghosh_DNIH0 . The scattering cross section
$\sigma_{\chi\nu}$, taken to be independent of the neutrino temperature, is
parametrized as $u_{\rm DNI}\propto\sigma_{\chi\nu}/m_{\chi}$. In this model,
rather than decreasing the size of the sound horizon, the new interaction
inhibits the free-streaming of neutrinos, which lowers the standard phase
shift $\varphi$ induced by neutrinos in the BAO. The peak structure of the CMB
power spectra can be roughly described by the position of the $p$-th peak

$\ell_{p,{\rm peak}}\simeq(p\pi-\varphi)\frac{D_{M}(z_{*})}{r_{*}},$ (19)

where the subscript $*$ denotes quantities at the time of recombination. In
this model, $r_{*}$ does not change with respect to $\Lambda$CDM,777Although
the redshift of last scattering $z_{*}$ and the redshift of the end of
radiation drag $z_{d}$ do not coincide, we expect minimal model dependence in
the difference between $r_{*}$ and $r_{\rm d}$. Therefore, we consider them as
encapsulating redundant information. so that $D_{M}(z_{*})$ needs to
compensate the reduction of the phase shift in order to reproduce the measured
location of the CMB peaks. Given that the physical matter, radiation, and
cosmological constant energy densities are very constrained by other CMB
features, the required decrease in $D_{M}(z_{*})$ is achieved by increasing
$H_{0}$. Since this model modifies the $\Lambda$CDM neutrino-induced phase
shift in the BAO and the standard BAO analysis does not account for such
effects, DNI might lead to a significantly biased BAO constraint. Note that,
as indicated in Ref. Ghosh_DNIH0 , this model assumes massless neutrinos (with
no implementation for massive neutrinos). Therefore, we will also consider
massless neutrinos when assuming DNI cosmologies.

We use the public Boltzmann code CLASS Lesgourgues:2011re ; Blas:2011rf
,888http://class-code.net/ as well as its modifications to include the EDE and
DNI999https://github.com/subhajitghosh-phy/CLASS_DNI models, to compute the
matter power spectra and other cosmological quantities needed.

###  IV.2 Cosmological dependence of the shape of $\boldsymbol{O_{\rm lin}}$

Figure 1: BAO feature $O_{\rm lin}$ in the matter power spectrum for varying
cosmological parameters (with $k$ rescaled by $r_{\rm d}/r_{\rm d}^{\rm
fid}$), compared with the fiducial prediction (red dashed lines), and the
corresponding ratio (i.e., modified over fiducial cosmology) in the lower
panels. The insets show $r_{\rm d}/r_{\rm d}^{\rm fid}$ for each case, and the
model under consideration is given in the upper left corner of each panel. We
keep all $\Lambda$CDM parameters to their fiducial values except for when each
of them is varied, as indicated in the corresponding color bars. The intervals
limited by white lines in the color bars correspond to the 68% confidence
level constraints of each parameter from Planck observations, and additional
data sets for EDE and DNI. For the panels corresponding to EDE, we use $f_{\rm
EDE}=0.2$, $\log_{10}z_{\rm EDE}^{c}=3.5$, $\Theta_{\rm EDE}=2.8$, and $n_{\rm
axion}=3$, unless otherwise indicated. For the DNI model, we use $f_{\rm
DNI}=0.02$ and $u_{\rm DNI}=5$, unless otherwise indicated. DNI constraints
are reported in terms of $f_{\rm DNI}u_{\rm DNI}$, hence they are adapted to
the fixed values of $u_{\rm DNI}$ and $f_{\rm DNI}$ assumed in this figure.
Note the change in scale of the $y-$axis for the lower sections of each panel.

We now consider how the BAO feature changes with respect to the fiducial under
these different cosmological models with varying parameters. In particular, we
are concerned with changes in the evolution of perturbations before
recombination, which affect $\alpha_{\perp}$ and $\alpha_{\parallel}$ through
$r_{\rm d}$. Once we account for modifications of $r_{\rm d}$, we can
investigate changes in the BAO feature beyond its characteristic scale.
Changes to the background expansion history at low redshift affect $q_{\perp}$
and $q_{\parallel}$; however, it has been shown that the standard BAO analysis
is robust to such changes Aubourg15 ; VargasMagana_BAOtheosyst ;
Carter_BAOtest . Therefore, we do not model the Alcock-Paczynski effect nor
the isotropic dilation in this section and focus on the intrinsic undistorted
shape of $O_{\rm lin}$.

In order to study the BAO feature, we compare $O_{\rm lin}(k)$ predictions for
the modified and fiducial cosmologies. If only the peak position shifts away
from the fiducial, while the overall pattern remains the same, the result is a
rescaling of $k$ in the argument of $O_{\rm lin}$ due to the variation of
$r_{\rm d}$ (see Eq. (4)). In this case, all $O_{\rm lin}(k/[r_{\rm d}/r_{\rm
d}^{\rm fid}])$ would be the same upon adjusting $r_{\rm d}$, where we have
incorporated the $r_{\rm d}$ dependence in the rescaling of $k$ explicitly.
If, however, the pattern of $O_{\rm lin}$ changes, then the standard BAO
analysis does not sufficiently capture differences between the fiducial and
the modified cosmologies.

We show this comparison in Fig. 1, where we individually vary all the relevant
parameters of the cosmologies discussed in Section IV.1, keeping the rest of
the parameters fixed. In order to study the variations thoroughly, we show
both the absolute values of $O_{\rm lin}(k/[r_{\rm d}/r_{\rm d}^{\rm fid}])$
in the upper panels, as well as the ratio $O_{\rm lin}(k/[r_{\rm d}/r_{\rm
d}^{\rm fid}])/\left(O_{\rm lin}(k/[r_{\rm d}/r_{\rm d}^{\rm
fid}])\right)^{\rm fid}$ to compare with the fiducial in the lower panels.
Note that at low redshift, the results shown in Fig. 1 are independent of
redshift, since all redshift dependence is contained in $P_{\rm m,sm}$, except
for the redshift evolution of the non-linear damping of the BAO, which we do
not model in this work.

The first two panels of Fig. 1 show $O_{\rm lin}$, varying $\Omega_{\rm
b}h^{2}$ and $\Omega_{\rm cdm}h^{2}$ under $\Lambda$CDM. We can see that
$r_{\rm d}$ changes considerably for the parameter ranges explored (since the
sound speed of the plasma and the matter content of the Universe change).
However, after rescaling $r_{\rm d}$, the only significant change in $O_{\rm
lin}$ is the amplitude of the wiggles, with no appreciable phase shift or
further change of the BAO feature. Although the amplitude of the BAO is
affected by non-linearities Eisenstein_BAOrobust , the non-linearities are
modeled with an exponential decay dependent on $\Sigma_{\perp}$ and
$\Sigma_{\parallel}$ (Eq. (9)), which we marginalize over. There might be some
scale dependence of the BAO amplitude that is not covered by this modeling,
but the fact that there is no change in the position of the BAO implies that
the constraints on $\alpha_{\perp}$ and $\alpha_{\parallel}$ should not be
biased.

For $O_{\rm lin}$ under $\Lambda$CDM$+N_{\rm eff}$, shown in the third panel
of Fig. 1, the amplitude is also slightly modified, but there is also a phase
shift introduced in the BAO wiggles. This phase shift can be seen in the
skewness of the oscillations for a range of $N_{\rm eff}$
Baumann_NeffBAO_fisher , as well as in the fact that the $k$-values where the
ratio of $O_{\rm lin}$ (after rescaling $r_{\rm d}$) crosses 1 change with
$N_{\rm eff}$. In addition, $N_{\rm eff}$ changes $r_{\rm d}$ significantly,
due to the modification of the expansion history of the Universe before
recombination.

The next four panels of Fig. 1 show $O_{\rm lin}$ for EDE cosmologies, varying
$f_{\rm EDE}$, $z_{\rm EDE}^{\rm c}$, $\Theta_{\rm EDE}$, and $n_{\rm axion}$.
The BAO feature is most sensitive to changes in $f_{\rm EDE}$. Although the
amplitude of $O_{\rm lin}$ does not change significantly, there is a small
phase shift introduced as $f_{\rm EDE}$ grows. A phase shift is also
noticeable when varying the other EDE parameters, especially for $n_{\rm
axion}$ (note that currently the whole range for $n_{\rm axion}$ covered by
the colorbar is within the 68% confidence level marginalized constraint
reported in Ref. Smith_EDE ). As noted in Ref. Poulin_EDE , varying $z_{\rm
EDE}^{c}$ mainly affects the height of the first acoustic peak of the CMB
power spectra as well as the ratio $r_{\rm d}/r_{\rm Silk}$, where $r_{\rm
Silk}$ is the Silk damping scale; thus, it has a noticeable effect on the
amplitude ratio between different peaks.

Finally, the last two panels of Fig. 1 show $O_{\rm lin}$ for DNI cosmologies,
varying $u_{\rm DNI}$ and $f_{\rm DNI}$. The value of $r_{\rm d}$ remains the
same, but the interaction between neutrinos and a fraction of the dark matter
modifies the fiducial phase shift in the BAO feature. This change is larger
for larger values of $u_{\rm DNI}$ or $f_{\rm DNI}$, evident from directly
comparing curves of $O_{\rm lin}$ in the upper sections of the panels (and in
e.g., Fig. 1 of Ref. Ghosh_DNIH0  for the CMB power spectra). For this model,
the ratio of $O_{\rm lin}$ with respect to the fiducial prediction may be
misleading. Although the phase shift in $O_{\rm lin}(k/[r_{\rm d}/r_{\rm
d}^{\rm fid}])$ is towards higher $k$ for larger values of $u_{\rm DNI}$ or
$f_{\rm DNI}$, the amplitude of the BAO is also larger, causing the
oscillations of the ratio of $O_{\rm lin}$ to shift towards lower $k$.

| $100\times\Omega_{\rm b}h^{2}$ | $\Omega_{\rm cdm}h^{2}$ | $h$ | $n_{\rm s}$ | $10^{9}\times A_{\rm s}$ | $N_{\rm eff}$ | $f_{\rm EDE}$ | $\log_{10}z_{\rm EDE}^{c}$ | $\Theta_{\rm EDE}$ | $n_{\rm axion}$ | $u_{\rm DNI}$ | $f_{\rm DNI}$ | $\Delta\chi^{2}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
Fiducial | 2.237 | 0.1200 | 0.6736 | 0.9649 | 2.100 | 3.046 | - | - | - | - | - | - | -
$\Lambda$CDM 1 | 2.229 | 0.1212 | 0.6680 | 0.9608 | 2.077 | 3.046 | - | - | - | - | - | - | 2.0
$\Lambda$CDM 2 | 2.261 | 0.1160 | 0.6900 | 0.9709 | 2.140 | 3.046 | - | - | - | - | - | - | 18.8
$\Lambda$CDM$+N_{\rm eff}$ 1 | 2.228 | 0.1154 | 0.6614 | 0.9587 | 2.072 | 2.796 | - | - | - | - | - | - | 1.3
$\Lambda$CDM$+N_{\rm eff}$ 2 | 2.287 | 0.1231 | 0.7061 | 0.9801 | 2.116 | 3.400 | - | - | - | - | - | - | 18.0
EDE 1 | 2.251 | 0.1320 | 0.7281 | 0.9860 | 2.191 | 3.046 | 0.132 | 3.531 | 2.72 | 2.60 | - | - | 7.8
EDE 2 | 2.261 | 0.1413 | 0.7552 | 0.9928 | 2.221 | 3.046 | 0.200 | 3.545 | 2.43 | 2.34 | - | - | 50.1
DNI 1 | 2.253 | 0.1180 | 0.7023 | 0.9492 | 2.019 | 3.046 | - | - | - | - | 18.6 | $10^{-3}$ | 6.4
DNI 2 | 2.230 | 0.1212 | 0.7200 | 0.9600 | 2.111 | 3.046 | - | - | - | - | 15.0 | 0.02 | 338

Table 1: Cosmological models considered in this work and their corresponding
cosmological parameter values. The ‘Fiducial’ model is used to generate the
template for the BAO fit. All other models are used as target cosmologies to
generate the mock power spectra. Models denoted with ‘1’ correspond to the
good fits to Planck, while models ‘2’ are poor fits, as described in the text.
Note that $\Omega_{\rm cdm}h^{2}$ represents the total dark matter density and
includes the portion of dark matter that interacts with neutrinos in the DNI
model. We find the minimum $\chi^{2}$ for each cosmology and the corresponding
best-fit nuisance parameters using
iMinuit.111111https://iminuit.readthedocs.io/en/stable/

###  IV.3 Bias in $\boldsymbol{\alpha_{\perp}}$ and
$\boldsymbol{\alpha_{\parallel}}$

Now that we have established that there are cosmological models that can
produce $O_{\rm lin}$ patterns that do not match the fiducial prediction under
a rescaling of $r_{\rm d}$ (or the BAO amplitude), we are ready to evaluate
the bias on the standard BAO fit that may arise under these cosmologies. In
order to do so, we perform mock BAO analyses using the formalism described in
Section III, with the aim of evaluating the potential bias introduced in the
determination of $\alpha_{\perp}$ and $\alpha_{\parallel}$. Such a bias could
affect the constraints on the expansion history of the Universe and hence the
inference of the cosmological parameters. Although it has been demonstrated
Aubourg15 ; VargasMagana_BAOtheosyst ; Carter_BAOtest  that BAO-inferred
distances are extremely robust to modifications on the expansion history of
the Universe at late times, these distances might not be robust against
changes in the acoustic oscillations due to the growth of perturbations before
recombination.

Whether or not differences in the BAO pattern with respect to the fiducial
prediction such as those highlighted in Fig. 1 introduce a potential bias on
the BAO measurements depends on the flexibility of the BAO analysis. If a bias
were found for a given cosmological model, current BAO measurements (which
assume a $\Lambda$CDM fiducial) would not be suitable for constraining that
model.

In order to perform the mock BAO analysis, we create a mock power spectrum for
each specific underlying cosmology under consideration, as if it were measured
from the observed galaxy distribution. Details on how we compute the mock
power spectra can be found in Appendix B. We perform the standard BAO analysis
on each mock spectrum using the same fiducial $\Lambda$CDM cosmology. We run
Monte Carlo Markov chains using the Monte Carlo sampler emcee emcee
121212https://github.com/dfm/emcee to perform the BAO fit with the likelihood
in Eq. (17).

Once we obtain the marginalized constraint in the
$\alpha_{\perp}$-$\alpha_{\parallel}$ plane for each mock analysis, we can
locate the point in parameter space at which the value of the marginalized
posterior peaks to compare with the true point. We compute
$\Delta\chi^{2}=-2\Delta\log{L}$ between these two points in the
$\alpha_{\perp}$-$\alpha_{\parallel}$ plane assuming a Gaussian marginalized
posterior and calculate the corresponding confidence level for a two-
dimensional $\chi^{2}$ distribution. This procedure is equivalent to computing
the cumulative probability of the $\chi^{2}$ distribution up to the value of
interest. Finally, we relate the calculated confidence level to a number of
standard deviations $\sigma$ for a Gaussian distribution. We estimate the bias
then to be this number of $\sigma$.

In Table 11, we specify the values of the cosmological parameters for both the
assumed fiducial cosmology needed for the BAO analysis and all of the target
cosmologies used to generate the mock power spectra. We use for the
cosmological parameters of our fiducial cosmology the mean values of the
$\Lambda$CDM analysis (including temperature, polarization, and lensing data)
to Planck Planck18_pars . In addition, we consider two different sets of
cosmological parameters for all models: the best fit to Planck data (or Planck
data combined with other data sets) and a poor fit. We refer to these two sets
with the labels ‘1’ and ‘2,’ respectively, both individually (e.g., in the
model name) and collectively. For the latter, we choose one of the parameters
to be at least $\gtrsim 3\sigma$ away from its nominal best-fit value and set
the remaining parameters to their new best-fit values to Planck, obtained from
keeping the chosen parameter fixed.

In addition to the the standard $\Lambda$CDM parameters and $N_{\rm eff}$, we
consider the extra parameters of EDE and DNI. For $\Lambda$CDM and
$\Lambda$CDM$+N_{\rm eff}$, we use best-fit cosmologies obtained from Planck’s
publicly-available Monte-Carlo Markov
chains,131313http://pla.esac.esa.int/pla/ and obtain poor fits from the same
chains fixing $H_{0}=69.00\,{\rm km\,s^{-1}Mpc^{-1}}$ for $\Lambda$CDM and
$N_{\rm eff}=3.4$ for $\Lambda$CDM$+N_{\rm eff}$. The EDE 1 and DNI 1
cosmologies are taken from the reported best fits for the considered data sets
in Refs. Smith_EDE  and Ghosh_DNIH0 , respectively. EDE 2 is obtained from the
Monte-Carlo Markov chain computed for Ref. Smith_EDE , fixing $f_{\rm EDE}$ to
0.2, while DNI 2 is the best fit to Planck data for fixed $u_{\rm DNI}=15$ and
$f_{\rm DNI}=0.02$, found using MontePython Audren13_mp . The DNI 2 case has
been chosen as a very extreme case to test the flexibility of the standard BAO
analysis: its $\Delta\chi^{2}$ for Planck power spectra with respect to the
$\Lambda$CDM mean values (i.e., our fiducial case) is $\sim 338$.

In order to mimic the power of current and next-generation galaxy surveys, we
use four different sets of survey specifications, inspired by BOSS
Alam_bossdr12  and DESI desi . These specifications are important, because
they determine the effective redshift at which the power spectrum is measured
and computed, as well as its covariance matrix. The specifications for these
surveys are listed in Table 2.

Survey | $z$ | $\Omega_{\rm sky}\,\left[{\rm deg}^{2}\right]$ | $V\,\left[\left({\rm Gpc}/h\right)^{3}\right]$ | $10^{4}n_{g}\,\left[\left({\rm Mpc}/h\right)^{-3}\right]$ | $b_{\rm g}$
---|---|---|---|---|---
BOSS LOWZ | 0.32 | 10000 | 1.27 | 2.85 | 1.85
BOSS CMASS | 0.57 | 10000 | 3.70 | 2.10 | 1.85
DESI 1 | 0.80 | 14000 | 6.80 | 12.3 | 1.86
DESI 2 | 1.00 | 14000 | 8.73 | 6.41 | 1.50

Table 2: Specifications for each of the galaxy surveys considered in the mock
BAO analyses. The columns give the effective redshift $z$ at which the power
spectrum is measured, the sky coverage $\Omega_{\rm sky}$ of the survey, the
physical volume $V$ covered by the survey, the mean galaxy density $n_{g}$,
and the galaxy bias $b_{\rm g}$.

Figure 2: The 68% and 95% confidence level marginalized constraints in the
$\alpha_{\perp}$-$\alpha_{\parallel}$ plane, shown with their maximum-
posterior and true values (represented as circles and stars, respectively),
for different mock galaxy power spectra computed in $\Lambda$CDM,
$\Lambda$CDM+$N_{\rm eff}$, EDE, and DNI, as indicated in the legend (rows),
and for BOSS LOWZ and CMASS, as well as DESI at $z=0.8$ and $z=1.0$ (columns;
note the different scales in the columns in the left and in the right). For
all analyses, we use a template computed under a fiducial that matches the
mean values of the $\Lambda$CDM parameters from Planck’s analysis. Blue
contours refer to the cosmologies numbered as ‘1’ (best fit from Planck, as
well as external data sets for EDE and DNI) and red contours to cosmologies
numbered as ‘2’. Cosmological parameters and survey specifications are listed
in Tables 11 and 2, respectively.

Marginalized constraints in the $\alpha_{\perp}$-$\alpha_{\parallel}$ plane
are shown in Fig. 2, comparing the values for which the posterior distribution
peaks and the true values of these parameters (marked by circles and stars,
respectively). Each panel shows the results for a given galaxy survey and
cosmological model. We show cosmologies 1 in blue and cosmologies 2 in red.
Note that, since each model has its own prediction for $D_{M}/r_{\rm d}$ and
$Hr_{\rm d}$ (and these quantities change with redshift), each panel shows
different true values for $\alpha_{\perp}$ and $\alpha_{\parallel}$. We show
results assuming finished surveys (BOSS Alam_bossdr12 ) in the two left
columns, while the two right columns correspond to future surveys (DESI desi
).

The true values of $\alpha_{\perp}$ and $\alpha_{\parallel}$ lie well within
the 68% confidence level contours in Fig. 2, indicating that there is no
significant bias in the inference of $\alpha_{\perp}$ and $\alpha_{\parallel}$
for any of the models in the finished surveys, even when the fiducial model
and the model under study have very different predictions for $r_{\rm d}$ or
$O_{\rm lin}$. Moreover, the largest bias found for our extreme DNI 2 example
is $\sim 1\sigma$, for DESI 1 survey (and $\sim 0.8\sigma$ for DESI 2 survey).
This bias is introduced from the change in the phase shift produced by the
interaction between neutrinos and dark matter, as demonstrated in Fig. 1, and
expected from the discussion in IV.1. However, note that for DNI 1, the bias
in the BAO measurements is always $\lesssim 0.2\sigma$.

Finally, in addition to biasing the best fit, inaccurate modeling can also
induce a misestimation of the parameter uncertainties (see e.g, Bellomo_Fisher
). However, we find that not only there is no significant bias in the best-fit
values of $\alpha_{\perp}$ and $\alpha_{\parallel}$, but their uncertainties
do not change. We can appreciate this when comparing the marginalized
constraints shown in Fig. 2 for cosmologies 1 and 2. In each panel, the
uncertainties for each of the cases are the same. This means that the BAO
standard analysis does not misestimate the covariance between $\alpha_{\perp}$
and $\alpha_{\parallel}$, even if the assumed fiducial cosmology is very far
from the cosmology corresponding to the observations.

##  V Discussion

The results shown in Section IV.3 are clear and powerful. The BAO standard
analysis is flexible enough to capture different growth-of-structure histories
prior to recombination for the models explored in this work. Moreover, our
results indicate that the inference of $\alpha_{\perp}$ and
$\alpha_{\parallel}$ using a template can be unbiased even for cosmologies
that differ significantly from the fiducial cosmology used to compute the
template. In most of the cases, this fiducial cosmology would come from the
best fit to the CMB power spectra.

These results, as well as the discussion included in Section II, provide a
strong motivation to keep using template-fitting methodologies to perform BAO-
only analysis: this approach is robust, flexible, and general enough so that
it allows agnostic studies of cosmology. Therefore, the results of standard
BAO analyses can be used to infer cosmological parameters without introducing
any bias in the subsequent constraints. Moreover, compressing the BAO
information in terms of $\alpha_{\perp}$ and $\alpha_{\parallel}$ reduces the
probability of introducing observational systematics and reduces computational
costs on cosmological parameter inference, which proves key as the amount of
observations grows.

In this work we have focused on the BAO fit to the galaxy power spectrum.
However, other tracers can be used to infer the matter power spectrum;
regardless of which tracer is used, BAO imprints are present in the
perturbations of all of them. Therefore, our study can be extrapolated to the
power spectrum of the fluctuations of the number of quasars, the
Lyman-$\alpha$ forest, line-intensity mapping, etc., as well as their cross-
correlations. The particularities of the analysis applied to each specific
tracer are related with observational systematics and not with the underlying
cosmology; hence, they do not affect the extrapolation of the conclusions of
this work.

Finally, although we have focused on four specific cosmological models, our
results can be extrapolated to other cosmological models. As explored in
Appendix C, the BAO feature present on the matter density distribution at the
time of recombination (indirectly probed by CMB observations to great
precision) survives unmodified until the present. This is true modulo non-
linear collapse of perturbations, which still preserves some of the features
encoded in the BAO, as shown in Ref. Baumann_NeffBAO .

Therefore, we conclude that using a template computed for a good fit to the
CMB observations should be suitable for a BAO analysis. This assessment might
not hold, however, if the evolution of dark matter and baryon overdensities is
influenced by beyond-$\Lambda$CDM physics post recombination, such as baryon-
dark matter interactions (e.g., see Refs. Dvorkin:2013cea ; Boddy:2018wzy  for
a description of the modified Boltzmann equations). Moreover, general
relativity is assumed in density-field reconstruction (see e.g.,
Eisenstein_reconstruction ), hence only BAO measurements obtained before
applying density-field reconstruction shall be used to constrain cosmological
models that modify gravity.

###  V.1 Constraining cosmological models with BAO

Cosmological parameter inference using $\alpha_{\perp}$-$\alpha_{\parallel}$
constraints would be completely accurate only if the BAO measurements are
unbiased for every power spectrum computed at every point of the parameter
space of the model. However, exploring these biases for the whole parameter
space is unfeasible. We overcome this limitation by studying concrete cases
far from the assumed fiducial cosmology. These cases represent the tails of
the posterior, where residual small biases in the BAO constraints would not
modify the bulk of the distribution nor introduce a significant bias in the
parameter inference.

On the other hand, it is expected that the bias in $\alpha_{\perp}$ and
$\alpha_{\parallel}$ decreases as we move in the parameter space from the
region corresponding to these cosmologies towards the fiducial cosmology.
Therefore, an unbiased analysis of the extreme cases that we consider further
supports the use of $\alpha_{\perp}$ and $\alpha_{\parallel}$ constraints for
joint parameter inference of cosmological models beyond $\Lambda$CDM with
other cosmological probes, such as the CMB power spectra. This means that our
results indicate that previously reported BAO measurements in the form of
$\alpha_{\perp}$ and $\alpha_{\parallel}$ constraints are valid for
constraining not only late-time deviations of the expansion history of the
Universe, but also early-Universe extensions to $\Lambda$CDM, without the need
to refit the galaxy power spectrum.

Figure 3: Hubble expansion rate over $(1+z)^{3/2}$(top) and comoving angular
diameter distance over $(1+z)^{3/2}$ (bottom) as a function of redshift,
weighted by the ratio between the actual sound horizon at radiation drag and
its fiducial value ($\Lambda$CDM best fit to Planck). We show predictions for
$\Lambda$CDM (blue) and DNI (green), as well as existing measurements from
BOSS Alam_bossdr12  and eBOSSGilMarin_QSOeBOSS ; eBOSS_Lyalpha_auto ;
eBOSS_Lyalpha_cross . Note that the error bars shown here do not include the
covariance between measurements.

Our findings ease the worry regarding the validity of reported BAO
measurements assuming a $\Lambda$CDM template for models beyond $\Lambda$CDM
with large deviations at the perturbation level. This concern has prompted to
some studies to omit BAO measurements from the set of cosmological data used
to constrain cosmological models (see e.g., Ref. Ghosh_DNIH0 ). We argue that
BAO measurements can and should be included, especially given their importance
in constraining proposed solutions to the $H_{0}$ problem BernalH0 ;
Aylor_sounds ; Know_H0hunter .

In particular, let us consider how the best-fit DNI model obtained in Ref.
Ghosh_DNIH0  without including BAO measurements fits the BAO observations. In
Fig. 3, we show the expansion history as predicted by the best-fit
$\Lambda$CDM cosmology to Planck 2018 and the best-fit DNI model to data from
Planck 2015 Planckparameterspaper , WiggleZ dark energy survey
Parkinson_wigglez , and SH0ES RiessH0_19  (the data set considered in Ref.
Ghosh_DNIH0 ). We also show existing measurements from BOSS Alam_bossdr12  and
eBOSS GilMarin_QSOeBOSS ; eBOSS_Lyalpha_auto ; eBOSS_Lyalpha_cross ,
demonstrating that $\Lambda$CDM provides a much better fit. The DNI prediction
implies a $\chi^{2}=5.9$ larger than the $\Lambda$CDM prediction for the
combined likelihood of BOSS Alam_bossdr12 , 2dFGS Beutler11  and SDSS DR7 MGS
Ross15  (the BAO data set considered in Planck analyses). We conclude that BAO
measurements disfavor the reported best-fit DNI model. We expect that BAO
measurements favor DNI models with lower $H_{0}$ than the reported best fit
(and corresponding lower $u_{\rm DNI}$ and $f_{\rm DNI}$), closer to the
$\Lambda$CDM constraint.

This expectation is common to any model that modifies pre-recombination
physics without altering the $\Lambda$CDM prediction of $r_{\rm d}$. This is
due to the strong, model-independent constraint set by BAO on $r_{\rm d}h$
StandardQuantities . This result hints that only introducing a phase shift in
the acoustic perturbations without modifying $r_{\rm d}$ is not enough, and
that a $r_{\rm d}$ needs to be lower to solve the $H_{0}$ tension BernalH0 ;
Poulin_H0 ; Aylor_sounds ; Know_H0hunter .

We note however that cosmological parameter inference could be systematically
affected by coherent small biases in the determination of $\alpha_{\perp}$ and
$\alpha_{\parallel}$ at several redshifts. We explore this possibility
adapting the estimation of the systematic bias in parameter inference due to
inaccurate approximations presented in Ref. Bernal_Fisher  to our needs. The
systematic shift in the cosmological parameters $\bm{\vartheta}$ when the
measured rescaling parameters $\bm{\alpha}^{\rm m}$ are biased with respect to
their true values $\bm{\alpha}^{\rm tr}$ can be estimated as

$\begin{split}\Delta\bm{\vartheta}=&\left(\sum_{S}F_{S}\right)^{-1}\times\\\
&\times\sum_{S,i,j}\left(\nabla_{\vartheta}\alpha_{S;i}\right)\left({\rm
Cov_{S}}\right)^{-1}_{ij}\left(\alpha^{\rm m}_{j}-\alpha^{\rm
tr}_{S;j}\right),\end{split}$ (20)

where $\Delta\bm{\vartheta}=\bm{\vartheta}^{\rm m}-\bm{\vartheta}^{\rm tr}$
(with $\bm{\vartheta}^{\rm m}$ denoting the parameters inferred from the
observations without accounting for the bias in the analysis), $F$ is the
Fisher matrix of the cosmological parameters, Cov is the covariance matrix of
the measurements of the rescaling parameters, $i$ and $j$ are the entries of
this covariance and the $\bm{\alpha}$ vectors, and $S$ refers to each of the
surveys or independent measurements included in the analysis.

Consider the DNI cosmologies from Table 11, with the measurements and biases
reported in Fig. 2. In this case, we have four independent measurements and
two relevant cosmological parameters: the total matter density parameter
$\Omega_{\rm M}$ today and $H_{0}$. We obtain each Fisher matrix transforming
the corresponding $\left({\rm Cov_{S}}\right)^{-1}$ to the $\Omega_{\rm M}$ \-
$H_{0}$ space. We find biases of $\sim-0.1\sigma$ ($\sim 0.1\sigma$) and
$\sim-1.4\sigma$ ($\sim-0.3\sigma$) for $\Omega_{\rm M}$ and $H_{0}$,
respectively, for the DNI 2 (1) cosmology. When only the BOSS surveys are
considered, these biases reduce to $\sim-0.01\sigma$ ($\sim-0.03\sigma$) and
$\sim-0.2\sigma$ ($\sim-0.01\sigma$). All the biases are compared with the
corresponding uncertainties from the cosmological parameter inference.

Here, we consider the bias obtained when using only the BAO measurements from
the surveys used in this work; the effective bias affecting $\bm{\vartheta}$
when combining additional cosmological probes can be estimated in a similar
way following Ref. Bernal_Fisher . Nonetheless, when different probes are
combined, small systematic errors may lead to significant biases in the joint
parameter inference. The study performed here to discuss potential biases on
the cosmological parameters can be adapted to estimate the associated
potential systematic error budget sourced by employing standard BAO
measurements to constrain models beyond $\Lambda$CDM. This error budget may
vary from case to case, depending on the cosmological model under study and
the actual observations used, and might be important for the analysis of
future BAO measurements.

###  V.2 Amending biases on standard BAO analyses

If there is a situation in which a significant bias on $\alpha_{\perp}$ or
$\alpha_{\parallel}$ is indeed found when applying the analysis presented in
this work, the power spectrum must be reanalyzed to measure the BAO scale.
There are a few different options about how to proceed.

One option is to find a template assuming a different fiducial cosmology that
minimizes the bias seen in the analysis and refit the BAO with this template.
In this case, $\alpha_{\perp}$ and $\alpha_{\parallel}$ would be given by Eq.
(3). This new fiducial cosmology will be most likely beyond $\Lambda$CDM.
However, the impact of the feature not captured by a rescaling may depend on
cosmological parameters. In this case, a fixed template without modeling such
a feature would still fail to remove the bias in $\alpha_{\perp}$ and
$\alpha_{\parallel}$.

Hence, a second option consists of adding freedom to the analysis. In case the
extra contribution to $O_{\rm lin}$ can be robustly modeled, it is possible to
include it in the BAO analysis with one or more new nuisance parameters that
control its impact. Using this approach, however, would weaken the constraints
on $\alpha_{\perp}$ and $\alpha_{\parallel}$, given the addition of extra
nuisance parameters.

Finally, if a reliable model describing the extra feature cannot be found, the
analysis can be performed changing the template at each step of the MCMC. This
procedure is related to alternative methodologies to extract cosmological
information from the BAO feature that do not rely on a pre-computed template
(see e.g., Refs. Damico_EFTBAO ; Ivanov_EFTBOSS ). These methods do not aim to
obtain a model-independent measurement of the expansion history of the
Universe, nor can they extract agnostic independent information of early-time
physics through $r_{\rm d}$, since the template readjusts during the fitting
procedure. In these cases, a prior is needed to break the degeneracy between
at least $\Omega_{\rm cdm}h^{2}$ and $\Omega_{\rm b}h^{2}$. This prior may
either come from CMB observations and be directly applied to $r_{\rm d}$, or
be obtained from primordial deuterium and helium abundances assuming standard
Big Bang nucleosynthesis and be applied to $\Omega_{\rm b}h^{2}$ and $N_{\rm
eff}$.

###  V.3 BAO cosmology beyond $\boldsymbol{\alpha_{\perp}}$ and
$\boldsymbol{\alpha_{\parallel}}$

Even if there is no bias in the BAO measurements, there can still be
cosmological signatures that are not captured by the modeling of the rescaling
of the template as described in Section III. This is the case for
$\Lambda$CDM+$N_{\rm eff}$, for instance: varying $N_{\rm eff}$ changes the
phase shift of the BAO in a way that cannot be reproduced by a rescaling with
$\alpha_{\perp}$ or $\alpha_{\parallel}$. Additional freedom is needed in
order to consistently include this effect in the BAO analysis. This was
explored in Ref. Baumann_NeffBAO_fisher , where an extra parameter was
included as a multiplicative factor of a $k$-dependent function that modeled
the amount of phase shift.

Once this additional freedom is included in the analysis, instead of
considering the extra parameter as a nuisance and marginalizing over it, it
can also be treated as the parameter of interest. In this case, one would
marginalize over the rest of the parameters to constrain the amplitude of the
extra feature introduced in $O_{\rm lin}$. This is the case for the phase
shift induced by $N_{\rm eff}$ in Ref. Baumann_NeffBAO_measure , where a phase
shift was detected at 95% confidence level after applying CMB priors on
$\alpha$. Since most of these effects are expected to be isotropic (affecting
only $O_{\rm lin}$ and not the Alcock-Paczynski effect or the redshift-space
distortions), it may be preferable to perform an isotropic BAO analysis in
order to reduce degeneracies between parameters.

##  VI Conclusions

BAO provide a robust probe of the expansion history of the Universe at low
redshift, as well as a calibrator of the sound horizon at radiation drag.
Recently, the importance of BAO measurements has been highlighted again with a
crucial role in the resolution of the $H_{0}$ tension, which can be reframed
as a mismatch between the two anchors of the cosmic distance ladder: $H_{0}$
and $r_{\rm d}$ BernalH0 . These are precisely the two quantities which set
the normalization of the expansion history of the Universe as constrained by
BAO measurements.

Standard BAO analyses are robust and model independent, since they rely on a
template fitting using rescaling parameters, which encode the cosmological
information of interest. In addition, differences in the shape between the
measured power spectrum and the pre-computed template (sourced by the choice
of a wrong fiducial cosmology or uncertainties in small-scale clustering) are
marginalized over using a plethora of nuisance parameters. As extensions of
$\Lambda$CDM become more complex in the search of new physics or in the
attempt to resolve cosmological tensions, this approach might not be flexible
enough to cover these cosmologies, rendering BAO measurement unsuitable for
studying such models.

In this work, we have assessed the flexibility of the standard BAO analysis
under different cosmological models: $\Lambda$CDM, $\Lambda$CDM+$N_{\rm eff}$,
EDE, and DNI. We have performed mock analyses of galaxy power spectra, using
the same $\Lambda$CDM template for all of them. We find that, for the models
explored in this work, there is no bias in the BAO analyses for existing data.
Moreover, the maximum bias found for future surveys is $\sim 1\sigma$ for an
extreme DNI model which provides a very bad fit to Planck data. Therefore, our
findings reinforce the model independence and robustness of the standard BAO
analysis, and they support using reported BAO measurements in terms of
$\alpha_{\perp}$ and $\alpha_{\parallel}$ (obtained using a $\Lambda$CDM
template) to constrain these models. This will be further supported by the
implementation of blinding analyses in future galaxy surveys Brieden_blinding
.

The models considered in this work have been chosen as a representative sample
of models that aim to solve the $H_{0}$ tension through modifications prior to
recombination. Nonetheless, our results can be extrapolated to other models
that do not impact the clustering of dark matter and baryons with new physics
after recombination. Moreover, futuristic surveys may reach a level of
precision for which the low significance of the biases found in this work is
enough to affect the measurement. If there is a model with a modified
perturbation history that changes the BAO feature beyond rescaling factors, we
advocate for the methodology described in this work to ascertain whether
reported constraints in the $\alpha_{\perp}$-$\alpha_{\parallel}$ plane are
valid for cosmological parameter inference. In case a bias in the BAO fit is
found, we have suggested possible ways forward to add flexibility to the
modeling of the power spectrum and perform an unbiased BAO measurement.

Next-generation galaxy surveys like DESI desi , Euclid euclid  and SKA redbook
will improve upon existing BAO measurements through higher precision
observations and reaching higher redshifts. Additionally, future line-
intensity mapping experiments promise to extend BAO measurements up to $z\sim
9$ with competitive precision Bernal_IM ; Bernal_IM_letter . Checking the
validity of BAO analyses for exotic models is of the utmost importance in
order to exploit the promising outlook for measurements in the near future,
especially given the importance of BAO as a cosmological probe.

###### Acknowledgements.

We thank Licia Verde, Héctor Gil-Marín and Vivian Poulin for useful
discussions, and Graeme Addison and Alvise Raccanelli for comments on the last
stages of the manuscript. JLB is supported by the Allan C. and Dorothy H.
Davis Fellowship. TLS is supported by the Research Corporation and NASA ATP
grant number 80NSSC18K0728.

##  Appendix A Extraction of the smooth power spectrum $\boldsymbol{P_{\rm
m,sm}}$

The idea behind the extraction of the smooth matter power spectrum $P_{\rm
m,sm}$ from the total matter power spectrum $P_{\rm m}$ is conceptually
simple: it is based on removing the wiggles produced by the BAO. However,
rather than performing a brute-force smoothing of $P_{\rm m}$ or interpolating
using only the zero points between the wiggles, one can choose more stable and
efficient options. There are several procedures to extract $P_{\rm m,sm}$,
mainly divided in two broad groups. One option relies on the computation of
the matter power spectrum without including the BAO contribution, which can be
done, e.g., using the analytic transfer function derived in Ref. EH98 .
Alternatively, one can use Eq. (6) to convert $P_{\rm m}$ into $\xi_{\rm
m}(r)$, directly remove the BAO peak from the correlation function, and
transform back to $P_{\rm m}$. We adopt the latter method in this work. We
refer the interested reader to Ref. Kirkby_BAOtemplate  for a detailed
comparison between different methodologies.

Since we are interested in exploring cosmologies very different from
$\Lambda$CDM, especially regarding the BAO scale and feature, we avoid fits to
polynomials at fixed scales, as done in, e.g., Ref. Kirkby_BAOtemplate .
Instead, we use a more flexible methodology described below:

  1. 1.

Obtain the total matter power spectrum $P_{\rm m}$ from a Boltzmann solver and
use Eq. (6) to obtain the corresponding correlation function, $\xi_{\rm m}$.
We use the public mcfit python package to evaluate integrals, such as the one
appearing in Eq. (6).141414https://github.com/eelregit/mcfit. In order to
avoid numerical noise, $P_{\rm m}$ needs to be evaluated at a large number of
points $N$ and for a wide range in $k$. We sample $k$ uniformly in logspace in
$N=4096$ points in the interval $k\in\left[10^{-5},20\right]$ $h/$Mpc.

  2. 2.

Take the corresponding entries to $r_{1}\in\left[60,70\right]$ Mpc/$h$ and
$r_{2}\in\left[200,300\right]$ Mpc/$h$ in the $r$ array, $i_{1}$ and $i_{2}$,
respectively, and interpolate $r^{2}\xi_{\rm m}$ evaluated in the interval
$\left[i=0,i_{1}\right]\bigcup\left[i_{2},i=N-1\right]$ using cubic splines.

  3. 3.

Evaluate the interpolation object in the original $r$ array and remove the
$r^{2}$ factor, so that a smooth correlation function $\xi_{\rm m,sm}$ without
the BAO peak is obtained.

  4. 4.

Transform $\xi_{\rm m,sm}$ back to Fourier space, i.e., obtaining $P_{\rm
m,sm}$. $O_{\rm lin}$ is obtained by computing the ratio between the total
matter power spectrum $P_{\rm m}$ and $P_{\rm m,sm}$.

  5. 5.

Iterate points 2-4 varying $r_{1}$ and $r_{2}$ to optimize the convergence of
$O_{\rm lin}=1$ at $k\lesssim 10^{-3}$ $h$/Mpc: at scales much larger than
$r_{\rm d}$ there is no effect from the BAO and the smooth and actual
clustering must be equal.

We compare the total and the smooth matter power spectrum and correlation
function in Fig. 4. Note that $r_{1}$ is smaller than the scale of the local
minimum of $\xi_{\rm m}$, and the range of $r_{2}$ is significantly larger
than $r_{\rm d}$. This is because the two minima of $\xi_{\rm m}$ are deeper
than for $\xi_{\rm m,sm}$ due to the enhanced clustering around $r_{\rm d}$.

Figure 4: Top: Total linear matter power spectrum before (solid lines) and
after (dashed lines) the removal of the BAO contribution ($P_{\rm m}$ and
$P_{\rm m,sm}$, respectively). Bottom: Same as top but in configuration space.
Factors of $k$ and $r^{2}$ are added respectively for clarity. Blue lines
correspond to $\Lambda$CDM predictions and red lines to EDE predictions (with
$f_{\rm EDE}=0.3$).

The computation of the smooth power spectrum or correlation function may be
critical for the analysis of real observations, and the performance of
different procedures must be compared. We have checked that our results are
robust to the choice of the algorithm used to extract the broadband of the
clustering, since the mock power spectrum is computed with the same modeling
that is used in the likelihood.

##  Appendix B Mock $\boldsymbol{P_{\rm g}^{\rm data}}$ computation

As explained in Section IV.3, we perform a mock BAO analysis in order to test
for the presence of a bias in the measurement of the rescaling parameters
$\alpha_{\perp}$ and $\alpha_{\parallel}$ when the measured power spectrum
does not correspond to the fiducial cosmology assumed for the computation of
the template in Eq. (9). To do so, we need to compute a measured power
spectrum for different cosmologies. In this Appendix we explain our procedure
to compute this mock $P_{\rm g}^{\rm data}$.

First, we compute $P(k,\mu)$ following Eq. (9) with the cosmological and
nuisance parameters corresponding to the target cosmological model we want to
mock. However, we need to account for the fact that this power spectrum
corresponds to a cosmology different from the assumed fiducial cosmology used
to measure it. This means that the mock observed $P(k,\mu)^{\rm data}$ is
affected by the Alcock-Paczynski effect and the isotropic dilation. Note that
in this case, $P(k,\mu)$ has been obtained with its corresponding $r_{\rm d}$
(instead of the fiducial template), so that we rescale $k$ with $q_{\perp}$
and $q_{\parallel}$ from Eq. (2). With this correction, we compute
$\bm{P}_{\rm g}^{\rm data}$ using Eq. (11). Moreover, given that the power
spectrum is measured in $\left({\rm Mpc}/h\right)^{3}$, for the fiducial value
of $h$, we need to rescale the units accordingly. Finally, we assume the
covariance would be computed from mock galaxy catalogs; hence, it would
correspond to the fiducial cosmology. Therefore, we compute the covariance by
applying Eq. (16) to the fiducial power spectrum.

In order to evaluate the bias without any confusion introduced by statistical
dispersion around the true values, we decide not to include dispersion on the
mock power spectrum around the predicted theoretical power spectrum.
Therefore, even though we consider the correct covariance, all the points of
the mock power spectrum coincide with its theoretical prediction. Nonetheless,
we describe here how to include the statistical dispersion of the mock power
spectrum around the theoretical prediction, for completeness. Note that the
covariance of the power spectrum is not diagonal (Eq. (16)), so we need to
take into account the covariance between multipoles in order to mock the
dispersion about the true values of $\bm{P}_{\rm g}^{\rm data}$. Therefore, we
diagonalize the covariance matrix and draw random values from Gaussian
distributions centered at 0 and with variances corresponding to each of the
entries of the diagonalized covariance. Then, we transform this vector back to
the original $k$ base and add it to $\bm{P}_{\rm g}^{\rm data}$. The resulting
dispersion around $\bm{P}_{\rm g}^{\rm data}$ properly accounts for the
covariance between the multipoles.

##  Appendix C Connection between CMB and BAO

Under $\Lambda$CDM, the BAO feature present in the matter power spectrum at
low redshift comes from the same dynamics that produce the acoustic structure
measured in the CMB: baryons and photons are tightly coupled in the early
Universe and undergo acoustic oscillations, which become imprinted in both the
matter power spectrum and the photon temperature fluctuations of the CMB. In
fact, the use of a $\Lambda$CDM template when analyzing the BAO feature relies
on the assumption that if $\Lambda$CDM provides a good fit to the CMB power
spectra, it also provides a good fit to the BAO feature. With this qualitative
fact in mind, it is useful to quantify this relationship in light of the
analysis we have presented in this work.

During radiation domination, and after horizon crossing, the density contrast
$\delta_{\rm c}$ of cold dark matter starts to grow logarithmically with the
scale-factor, $\delta_{\rm c}\propto\ln(a)$. As time passes, smaller-scales
modes start collapsing as $\delta_{\rm c}\propto a$ so that, after matter-
radiation equality, all modes within the horizon grow as $\delta_{\rm
c}\propto a$. Baryons, on the contrary, only fall into the dark matter
potential wells after they have recombined and got released from the radiation
pressure. Then, after decoupling, the evolution of cold dark matter and
baryons is given by Meszaros:1974tb ; Ma:1995ey

$\displaystyle\ddot{\delta}_{\rm c}+\frac{\dot{a}}{a}\dot{\delta}_{\rm c}$
$\displaystyle=\frac{3}{2}\left(\frac{\dot{a}}{a}\right)^{2}\left[R_{\rm
c}\delta_{\rm c}+(1-R_{\rm c})\delta_{\rm b}\right],$ (21)
$\displaystyle\ddot{\delta}_{\rm b}+\frac{\dot{a}}{a}\dot{\delta}_{\rm b}$
$\displaystyle=\frac{3}{2}\left(\frac{\dot{a}}{a}\right)^{2}\left[R_{\rm
c}\delta_{\rm c}+(1-R_{\rm c})\delta_{\rm b}\right],$ (22)

where $R_{\rm c}\equiv\rho_{\rm c}/\rho_{\rm m}$ is the fraction of the cold
dark matter energy density $\rho_{\rm c}$ to the total matter density
$\rho_{\rm m}=\rho_{\rm c}+\rho_{\rm b}$ of cold dark matter and baryons, and
the dot denotes a derivative with respect to conformal time, $\eta$. Note that
at decoupling the universe is matter dominated so that $\dot{a}/a=2/\eta$.

The total matter transfer function is given by the weighted sum of the cold
dark matter and baryon transfer functions:

$T_{\rm m}=R_{\rm c}\delta_{\rm c}+(1-R_{\rm c})\delta_{\rm b},$ (23)

where $\delta_{\rm c}$ and $\delta_{\rm b}$ are given by the solution of the
coupled Eqs. (21) and (22). If we assume tight-coupling between baryons and
photons, adiabatic initial conditions and instantaneous decoupling, so that
$\delta_{\rm b}$ is related with the radiation overdensity $\delta_{\gamma}$
as $\delta_{\rm b,dec}=3\delta_{\gamma,{\rm dec}}/4$ (where the subscript
‘dec’ refers to quantities evaluated at decoupling), we obtain

$T_{\rm m}=\mathcal{A}\left(\frac{\eta}{\eta_{\rm dec}}\right)^{2},$ (24)

and

$\begin{split}\mathcal{A}\equiv&\frac{1}{20}\left[4R_{\rm c}\left(3\delta_{\rm
c,dec}+\eta_{\rm dec}\dot{\delta}_{\rm c,dec}\right)\right.+\\\
&\left.+3\left(1-R_{\rm c}\right)\left(3\delta_{\gamma,{\rm dec}}+\eta_{\rm
dec}\dot{\delta}_{\gamma,{\rm dec}}\right)\right],\end{split}$ (25)

where we have neglected decaying solutions (assuming $\eta\gg\eta_{\rm dec}$)
and the effect of the cosmological constant. The linear matter power spectrum
thus takes the form

$P_{\rm m}(k)=P_{\rm prim}(k)T_{\rm m}^{2}(k,\eta)=P_{\rm m,sm}(k)O_{\rm
lin}(k),$ (26)

where $P_{\rm prim}(k)=2\pi^{2}A_{s}k^{-3}(k/k_{p})^{n_{s}-1}$ is the
primordial power spectrum (where $k_{p}$ is the pivot wave number, typically
taken to be $k_{p}=0.05\ {\rm Mpc}^{-1}$) and

$\displaystyle\frac{P_{\rm m,sm}}{P_{\rm prim}}$ $\displaystyle=$
$\displaystyle\frac{R_{\rm c}^{2}}{25}\left(3\delta_{\rm c,dec}+\eta_{\rm
dec}\dot{\delta}_{\rm c,dec}\right)^{2}\left(\frac{\eta}{\eta_{\rm
dec}}\right)^{4},$ (27) $\displaystyle O_{\rm lin}-1$ $\displaystyle\simeq$
$\displaystyle\frac{3(1-R_{\rm c})(3\delta_{\gamma,{\rm dec}}+\eta_{\rm
dec}\dot{\delta}_{\gamma,{\rm dec}})}{2R_{\rm c}(3\delta_{\rm c,dec}+\eta_{\rm
dec}\dot{\delta}_{\rm c,dec})}\,,$ (28)

where we have neglected the term proportional to $(1-R_{\rm c})^{2}$. Let us
now focus on the BAO feature, whose scale dependence is given by
$\delta_{\gamma,{\rm dec}}$ and $\dot{\delta}_{\rm\gamma,dec}$. Note also that
this approximate $O_{\rm lin}$ does not evolve with time, as discussed for the
numerically obtained $O_{\rm lin}$ described in Appendix A and shown in Fig.
1. During tight-coupling, the acoustic oscillations in the photon-baryon fluid
are damped along with a scale-dependent enhancement due to a driving force:
Hu:1996mn ; Hu:2000ti ; Dodelson:2003ft

$\displaystyle\delta_{\gamma,{\rm dec}}$ $\displaystyle\simeq$
$\displaystyle-\mathcal{D}(k)\cos(c_{\rm s}k\eta_{\rm dec}),$ (29)
$\displaystyle\dot{\delta}_{\gamma,{\rm dec}}$ $\displaystyle\simeq$
$\displaystyle\mathcal{D}(k)c_{\rm s}k\sin(c_{\rm s}k\eta_{\rm dec}),$ (30)

where $c_{\rm s}$ is the photon-baryon sound speed, and $\mathcal{D}$ encodes
the Silk damping and the scale-dependent enhancement. Note that the sound
horizon at decoupling is $r_{\rm s}=c_{\rm s}\eta_{\rm dec}$.

The acoustic features in the CMB arise from different, but related, terms. If
we assume again an instantaneous decoupling, the visibility function is a
Dirac delta function: $g(\eta)=\delta_{D}(\eta-\eta_{\rm dec})$. Under this
approximation, the CMB power spectra are given by Seljak:1994yz ;
Zaldarriaga:1995gi ; Seljak:1996is

$C_{\ell}^{XY}=(4\pi)^{2}\int k^{2}dkP_{\rm
prim}(k)\Delta_{\ell}^{X}(k)\Delta_{\ell}^{Y}(k)$ (31)

with

$\displaystyle\Delta_{\ell}^{T}(k)\approx$ $\displaystyle\left[\psi_{\rm
dec}+\frac{1}{4}\delta_{\gamma,{\rm dec}}\right]j_{\ell}(x_{0})+v_{\rm
b,dec}j^{\prime}_{\ell}(x_{0}),$ (32)
$\displaystyle\Delta_{\ell}^{E}(k)\approx$
$\displaystyle\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}\frac{4}{3}\Pi_{\rm
dec}\frac{j_{\ell}(x_{0})}{x_{0}^{2}},$ (33)

where ${\psi}$ is the Newtonian potential transfer function, $v_{\rm b}$ is
the baryon velocity perturbation transfer function, ${\Pi}$ is the
polarization transfer function, $x_{0}\equiv k(\eta_{0}-\eta_{\rm dec})$
(where $\eta_{0}$ is the conformal time today), and ′ denotes the derivative
with respect to $x_{0}$. To leading order in $\dot{\tau}/(aH)$ we have

$\displaystyle v_{\rm b,dec}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{4}\frac{\dot{\delta}_{\gamma,{\rm dec}}}{k}$ (34)
$\displaystyle{\Pi}_{\rm dec}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{4}\frac{\dot{\delta}_{\gamma,{\rm
dec}}/4}{\dot{\tau}},$ (35)

where $\dot{\tau}$ is the differential optical depth for Thomson scattering.

Combining Eq. (28) with Eqs. (29) and (30), we can see that the ratio between
the terms in the second bracket, $\lvert\eta_{\rm
dec}\dot{\delta}_{\gamma,{\rm dec}}/3\delta_{\gamma,{\rm dec}}\lvert=r_{\rm
s}k/3\simeq k/(0.02\,{\rm Mpc^{-1}})$. This shows that, in the scales of
interest for the acoustic feature, $O_{\rm lin}$ is driven by the term
depending on $\dot{\delta}_{\rm\gamma,dec}$. Therefore, comparing with Eq.
(31), the acoustic oscillations in $O_{\rm lin}$ are a factor $\pi/2$ out of
phase with respect to the CMB temperature anisotropies, which is a well-known
result (see, e.g., Ref. Eisenstein_BAOrobust ).

We can see that the CMB and BAO acoustic features are, to first approximation,
given by the photon density perturbation and its first time derivative at
decoupling. Therefore, within the context of the basic approximations made
here, we can expect that any cosmological model, even beyond $\Lambda$CDM,
providing a good fit to the CMB spectra should also provide a good fit to the
BAO feature. This argument also supports the extrapolation of the conclusions
of this work beyond the four cosmological models considered.

Nonetheless, there are some differences to note between the acoustic features
in the BAO and the CMB. One difference is that the CMB spectrum is a projected
version, and the Fourier modes are integrated over spherical Bessel functions.
Even though $\ell\simeq k(\eta_{0}-\eta_{\rm dec})$, wave numbers in the range
$\Delta k\simeq k/2$ contribute to a measured $\ell$. A more dramatic
difference is related to the amplitude of the oscillations: while CMB
oscillations are at least order $\sim 1$, BAO oscillations have an amplitude
$\sim 0.1$ (see Fig. 1). In addition to this, the nuisance parameters
necessary to robustly extract information from the BAO feature make current
measurements of the BAO less sensitive to changes (beyond a rescaling of
distances with $r_{\rm d,s}$) in the acoustic oscillations than the CMB. This
is why, even for cosmic variance limited BAO measurements over large volumes
and wide redshifts ranges, CMB priors related with the background expansion
are needed in order to obtain competitive cosmological constraints from the
shape of the BAO (see e.g., Baumann_NeffBAO_fisher  for the case of $N_{\rm
eff}$).

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