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Robustness Of Baryon Acoustic Oscillation Constraints For Early- Universe Modifications Of ΛCDM Cosmology

J. L. Bernal

Tristan L. Smith Swarthmore College, [email protected]

K. K. Boddy

M. Kamionkowski

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Recommended Citation J. L. Bernal, Tristan L. Smith, K. K. Boddy, and M. Kamionkowski. (2020). "Robustness Of Baryon Acoustic Oscillation Constraints For Early-Universe Modifications Of ΛCDM Cosmology". Physical Review D. Volume 102, Issue 12. DOI: 10.1103/PhysRevD.102.123515 https://works.swarthmore.edu/fac-physics/429

This work is brought to you for free by Swarthmore College Libraries' Works. It has been accepted for inclusion in Physics & Astronomy Faculty Works by an authorized administrator of Works. For more information, please contact [email protected]. PHYSICAL REVIEW D 102, 123515 (2020)

Robustness of baryon acoustic oscillation constraints for early-Universe modifications of ΛCDM cosmology

Jos´e Luis Bernal ,1 Tristan L. Smith,2 Kimberly K. Boddy ,3 and Marc Kamionkowski 1 1Department of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street, Baltimore, Maryland 21218, USA 2Department of Physics and Astronomy, Swarthmore College, 500 College Avenue, Swarthmore, Pennsylvania 19081, USA 3Theory Group, Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA

(Received 21 April 2020; accepted 16 November 2020; published 4 December 2020)

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 a function of redshift, normalized by the sound horizon at radiation drag. However, this methodology relies on anisotropic distance distortions of a fixed, precomputed template (obtained in a given fiducial cosmology) in order to fit the observations. Therefore, it may be possible that extensions to the consensus ΛCDM 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 that 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 (ΛCDM 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 that 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.

DOI: 10.1103/PhysRevD.102.123515

I. INTRODUCTION expansion history, they are sensitive to early and late time physics in clean, distinct ways. Baryon acoustic oscillations (BAO) appear due to the The BAO and type Ia supernovae (SNeIa) [13,14] have primordial sound waves propagating in the tightly coupled become the main complement to CMB observations to photon-baryon plasma in the early Universe until recombi- Λ nation (e.g., [1,2]). After recombination, when the interaction constrain deviations from the CDM consensus model at rate between photons and baryons becomes inefficient due to low redshift. Moreover, since BAO measurements form a Hubble expansion, the acoustic waves stop propagating and standard ruler that connects the direct and inverse distance the BAO are imprinted in the baryon distribution. Therefore, ladders [15], they have been key in addressing the Hubble – BAO are present in the temperature anisotropies of the cosmic constant (H0) tension [3,16 20]. As pointed out in microwave background (CMB), which allows a very precise Ref. [21] (and confirmed by independent analyses inference of cosmological parameters (see, e.g., [3]). In [22,23]), the H0 tension can be reframed as a mismatch addition, their features also appear in the matter (and between the anchors of the direct and the inverse cosmic subsequently the galaxy) distribution at low redshift, although distance ladders (H0 and rd, respectively), with existing with lower significance. First detected in the galaxy power data constraining deviations of the evolution of the expan- spectrum around 15 years ago [4,5], BAO have been robustly sion history at low redshift. This has led authors of measured in galaxy, quasar, and Lyman-α density distribu- Ref. [24] to theorize that any extension to ΛCDM should tions reaching percent-level precision (see, e.g., [6–9]). modify prerecombination physics rather than the low- The BAO features are characterized by a physical scale: redshift Universe if it is to alleviate the H0 tension. “ the sound horizon at radiation drag, rd. With this standard Some examples include Refs. [25–32]. ruler,” it is possible to robustly map the expansion history Standard BAO analyses rely on templates of the summary of the Universe up to the redshift of measurement [10–12]. statistic under a study computed assuming a fiducial Given that BAO measurements depend on both rd and the cosmology (normally ΛCDM). They consist of determining

2470-0010=2020=102(12)=123515(21) 123515-1 © 2020 American Physical Society BERNAL, SMITH, BODDY, and KAMIONKOWSKI PHYS. REV. D 102, 123515 (2020) the anisotropic rescaling of the distance coordinates that the to the ΛCDM prediction. We find that the standard BAO template needs in order to fit observations, marginalizing analysis returns results without significant biases, even over a plethora of nuisance parameters to account for other when considering the bad fits mock power spectra: the deviations with respect to the template. This approach allows biases in the BAO rescaling parameters are < 0.2σ except for for a tomographic, model-independent determination of the the extreme case of DNI, for which we find a bias ∼1σ. These expansion history of the Universe that can be used to results confirm the robustness of BAO analyses and their constrain different cosmological models. appropriate use for cosmological parameter inference. An This procedure has been proven to be extremely robust alternative, general approach to estimate the bias on param- and flexible for models predicting different expansion rates eter inference can be found in, e.g., Ref. [47]. at late times (see, e.g., [33–36]), and it successfully models We describe in detail our methodology for future appli- changes in rd due to early-time modifications of the cation to other models to check whether the standard BAO cosmological model. However, there may be other con- analysis is unbiased. In case a bias were to be found, we tributions to the BAO feature that are not captured by these discuss different courses of action to address the situation. rescaling and nuisance parameters, such as phase shifts Nonetheless, we also provide approximate arguments for (which can be scale dependent) or a different scale extrapolating our results to other cosmological models. dependence of the amplitude of the oscillations. These Although this work focuses on the BAO feature, there is additional contributions may bias standard BAO analyses also cosmological information contained in the broadband (in the case they are not properly modeled in the template), of the observed clustering. Redshift-space distortions at σ but they can also be targeted to constrain beyond-ΛCDM linear scales are mostly sensitive to the product f 8, where σ physics with BAO (as shown in Refs. [37–39] for the case f is the linear growth rate and 8 is the root mean square of 8 of extra relativistic species N , and in Ref. [40] for the the matter fluctuations within Mpc=h [where h ¼ eff 100 −1 −1 case of oscillations in the primordial power spectrum). H0=ð km s Mpc Þ]. The degeneracy between f σ Given the key role of BAO measurements in constraining and 8 can be broken by, e.g., using higher-order statistics, extensions to ΛCDM, it is essential to ascertain whether such as the three-point correlation function [48] or the cosmological-parameter inference from BAO analyses is bispectrum [49,50], by exploiting the clustering at smaller free of unaccounted systematic errors, especially when the scales using perturbation theory (see, e.g., [51]) or by using requirements on the robustness of the analyses will tighten phase correlations [52,53]. Additional physics related to for next-generation BAO measurements. large modifications of the scale dependence of the cluster- ing can be strongly constrained, such as suppression at In this work we explore the impact that modifications of – the BAO feature, arising from changes in the growth of the small scales (see, e.g., [54 58] and references therein) or primordial non-Gaussianities (see, e.g., [59–62]). Para- matter density perturbations, may have in a BAO analysis. meters driving features with weaker scale dependence Concretely, we consider the possibility that standard can also be constrained, but normally external data or template-fitting BAO analyses are not flexible enough to priors are required for these constraints to be competitive. characterize these modifications. If this were the case, the This paper is structured as follows: we discuss the origin reported BAO measurements obtained using a ΛCDM and nature of the cosmological information encoded in the template might be biased if the true cosmology corresponds BAO measurements in Sec. II; review the standard BAO to these models; moreover, a bias could render BAO analysis of the power spectrum in Sec. III; introduce the measurements unsuitable for constraining extended models. cosmological models considered, discuss the variations in This study is therefore timely and needed in order to the predicted BAO feature, and estimate the bias in the ensure the robustness of using BAO measurements to Λ Λ BAO fit in Sec. IV; and discuss the results and conclude in constrain models beyond CDM. We focus on CDM, a Secs. V and VI, respectively. Afterwards, we discuss the model with a different number of relativistic species Λ methodologies used to extract and isolate the BAO feature ( CDM þ Neff), and two models that claim to ease or in the Appendix A; detail how we compute the mock resolve the Hubble tension: early dark energy (EDE) [25,26] – measured power spectrum in Appendix B; and illustrate the (although see also Refs. [41 45]) and dark neutrino inter- connection between the CMB power spectra and the BAO actions (DNI) [28,46]. We perform BAO analyses using feature in the matter power spectrum at late times with a toy mock power spectrum measurements computed under these model in Appendix C. models, for both “good” and “bad” fits to CMB observa- tions. We use the term “bad fits” to refer to sets of II. BAO COSMOLOGY cosmological parameters for which one of the parameters has been chosen to have a fixed value at least ≳3σ away from In this section, we describe the cosmological information its best fit to Planck, while the rest of the parameters are contained in the BAO feature of the low-redshift clustering selected to maximize the posterior fulfilling such a require- of tracers of matter density perturbations. The analysis of ment. We also include an extreme case for DNI, the model large-scale structure clustering allows for precise determi- adding the largest difference in the BAO feature with respect nation of cosmological parameters. By separating the

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2 1=3 clustering at different scales into a smooth, featureless dilation corresponds to ðq⊥qkÞ , and the Alcock- broadband and the BAO feature, it is possible to extract Paczinski effect is given by the ratio of q⊥ and qk. robust cosmological information encoded in the BAO scale. B. BAO scale A. Alcock-Paczynski effect and isotropic dilation The Alcock-Paczysnki effect and the isotropic dilation Observations measure the positions of different tracers of are always present1 in the measurement of the clustering matter in terms of redshifts and angular positions on the statistics: it is inherent to any measurement that depends on sky, which must then be transformed to obtain three- distance scales. Nonetheless, the BAO feature is clearly dimensional clustering summary statistics (e.g., the corre- distinguishable against the broadband of the summary lation function or power spectrum) as a function of spatial statistic; it manifests as oscillations in Fourier space or a distances or the corresponding Fourier mode wave num- peak in configuration space, and large-scale clustering θ bers. Given an angular separation and a small redshift measurements have well-determined its location. δ separation z, the spatial comoving distance in the trans- To extract the BAO scale from the observed target verse direction and along the line of sight are summary statistic, standard BAO analyses employ a pre- computed template of the target summary statistic gener- δ θ c z ated assuming a given cosmology. This cosmology does r⊥ ¼ DMðzÞ ;rk ¼ ; ð1Þ HðzÞ not need to be the same as that used in Eqs. (1) and (2).As we discuss in further detail in Sec. III, using the template respectively, where DM is the comoving angular diameter allows for the extraction of rd, which is the only character- distance, H is the Hubble expansion rate, and c is the speed istic scale of matter clustering at low redshifts (at larger of light. Equation (1) can be adapted to proper distances scales than those corresponding to matter-radiation equal- substituting DM by the proper angular diameter distance ity). The BAO scale of the template might not match the 1 DA ¼ DM=ð þ zÞ. There are three main effects that alter true BAO scale; therefore, a correction on rd must be these components of the observed distances: redshift-space included when rescaling distances in order to fit the distortions [63], the Alcock-Paczynski effect [64], and the observed BAO feature with the template. The correction isotropic dilation. Redshift-space distortions are a physical th is isotropic, and the rescaling of distances becomes r⊥;k ¼ modification to r , due to the peculiar velocities of galaxies obs th obs k r α⊥ (or k ¼ k =α⊥ ), where changing the redshift of observed sources along the line of ⊥;k ;k ⊥;k ⊥;k ;k sight (hence changing their position in redshift space with fid2 fid2 respect to the real space). These distortions introduce α ðrdÞ α ðrdÞ ⊥ ¼ q⊥ ; k ¼ qk ð3Þ anisotropies in the observed clustering, which is isotropic rd rd a priori. Assuming a background expansion history (obtained provide a mapping between the observed distances (or from a fiducial cosmology) that differs from that of the true wave numbers) and those that enter our theoretical model- expansion rate of the Universe causes an artificial distance ing, denoted by “th.”2 The superscript fid2 in Eq. (3) refers distortion. The fiducial cosmology is used to compute DM to the fiducial cosmology used to generate the template for and H in Eq. (1); therefore, the recovered r⊥ and rk differ the BAO analysis. It is important to notice that the rescaling from the true distances. This distortion affects r⊥ and rk in of rd in Eq. (3) is not related to the Alcock-Paczynski effect different ways, so it is possible to decompose it into an or the isotropic dilation. Hence, the rescaling between isotropic and an anistropic component: the isotropic dila- observed distances and those entering our theoretical model tion and the Alcock-Paczynski effect. introduced in Eq. (3) is the combination of two nonphysical The Alcock-Paczynski effect and the isotropic dilation effects: the redshift-distance transformation and the ratio can be modeled by rescaling factors, obtained when between the fiducial (for the fixed template) and true rd comparing the observed distances, which assume the values. true obs fiducial cosmology, and true distances: r⊥;k ¼ r⊥;kq⊥;k true obs 1 (or k⊥;k ¼ k⊥;k=q⊥;k in Fourier space). Using Eq. (1), The only case in which the rescaling of distances in Eq. (2) is the rescaling parameters are not explicitly present in the analysis is if the clustering statistics used are functions of redshifts and angles (see, e.g., Ref. [65]). In this case, the rescaling is implicitly embedded in the modeling of D ðzÞ ðHðzÞÞfid1 the clustering statistics. M 2 q⊥ ¼ fid1 ;qk ¼ ; ð2Þ There are alternative parametrizations of these rescalings [or ðDMðzÞÞ HðzÞ α α those in Eq. (2)], obtained through combinations of ⊥ and k. Some examples focus on the isotropic and anisotropic distortions where superscript “fid1” denotes the assumed fiducial ðα; ϵÞ or on the monopole and the μ2 moment of the two-point α α cosmology used in Eq. (1). Using q⊥ and qk, the isotropic statistics ð 0; 2Þ (see, e.g., Refs. [33,66], respectively).

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Hereinafter we assume that the fiducial cosmology used Finally, there is also cosmological information contained to convert redshifts into distances in Eq. (1) is the same as not only in the location of the BAO feature but also in its the one used to compute the template of the clustering shape and amplitude (see, e.g., [37,40]). However, given statistic, as is commonly the case in BAO studies. Under that the significance of the BAO feature with respect to the this assumption, the rescaling parameters become 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 D ðzÞ=r ðHðzÞr Þfid α M d α d respect to the CMB constraints. Nonetheless, for some ⊥ ¼ fid ; k ¼ ; ð4Þ ðDMðzÞ=rdÞ HðzÞrd cases, external priors on the rescaling parameters can be used to practically fix their values and focus only on the where fid corresponds to the fiducial cosmology that has shape of the BAO. This approach has provided promising been used to both translate redshifts into distances and results to constrain Neff [38,39] and oscillations in the compute the fixed template. primordial power spectrum [40], and it offers an alternative (but not independent) approach to look for physics beyond ΛCDM, complementary to direct CMB constraints. C. Cosmological information in BAO In order to avoid biasing the information obtained from III. BAO ANALYSIS the BAO feature, the shape and amplitude of the broadband are marginalized over with the introduction of nuisance In this section, we introduce the methodology that we parameters. After marginalization, the only remaining use to extract the BAO scale in this work. We follow the cosmological information in the clustering statistics is standard analysis used to obtain BAO measurements from galaxy surveys, using the galaxy power spectrum in Fourier related to the BAO location and anisotropy, which is space. A similar analysis can be performed in configuration mostly encoded in the rescaling parameters. This is an space using the correlation function (see, e.g., [68] for the entirely geometric fit to the observations; hence, it has the computation of the templates). potential to be performed without being limited to any cosmological model without loss of generality. Specifically, A. Power spectrum template and methodology the rescaling parameters are the fit parameters, and the resulting constraints are traditionally used in global analy- The power spectrum PðkÞ and the correlation function ses to infer cosmological parameters of any cosmological ξðsÞ are equivalent estimators for two-point clustering model. statistics in Fourier and configuration space, respectively, As evident from Eq. (2), the only cosmological infor- where s is the redshift space distance and k is the associated mation the Alcock-Paczynski effect and the isotropic wave number. The Legendre multipoles of the power dilation are sensitive to is the late-time expansion rate. spectrum are given by By utilizing a fixed template in the analysis, BAO mea- Z 2l þ 1 1 surements are also sensitive to prerecombination physics PlðkÞ¼ dμPðk; μÞLlðμÞ; ð5Þ 2 −1 through rd [Eq. (4)] in an agnostic and independent way, incorporating information about both the expansion rate ≡ k and the growth of matter perturbations. While the isotropic where k j j is the module of the wave number vector and μ is the cosine of the angle between the wave number vector dilation is completely degenerate with rd, the Alcock- α α and the line of sight. The Legendre multipoles of the Paczynski effect (modeled by the ratio of ⊥ and k when a correlation function, ξl, are defined in an analogous way, fixed template is used) is independent of the BAO scale. fid and related with Pl by the Fourier transform via Since α⊥=α ¼ D H=ðD HÞ does not depend on H0, k M M Z the Alcock-Paczynski effect constrains the unnormalized 3 l k d log k ξl s i Pl k jl ks ; expansion history of the Universe, independently of the ð Þ¼ 2π2 ð Þ ð Þ ð6Þ BAO scale. Accessing early-time information about rd enables the where s ≡ jsj is the module of the redshift space distance construction of the inverse distance ladder [15] and allows and jl is the lth order spherical Bessel function. Note that for an independent constraint on the product rdH0 [67], Eq. (6) equally holds for real space distances and wave which is what makes the BAO measurements key for the numbers. In this section, we do not explicitly include the H0 tension [21]. While having sensitivity to rd is beneficial, dependence on redshift, present in practically all quantities, the fixed template may reduce the flexibility of the analysis for the sake of brevity and readability; we do, however, α and, therefore, the applicability of the constraints on ⊥;k to show the dependence on k and μ, for clarity. cosmological models that deviate from the fiducial cos- The standard BAO analysis is based on fitting a template mology in the early Universe. We explore this possibility in (precomputed under a fiducial cosmology) to the observa- Sec. IV C. tions. This template is built in such a way that the BAO

123515-4 ROBUSTNESS OF BARYON ACOUSTIC OSCILLATION … PHYS. REV. D 102, 123515 (2020) feature is identifiable and isolated. In order to isolate the The damping affects the transverse and line-of-sight direc- BAO feature, the linear matter power spectrum Pm is tions differently; hence, we introduce two separate scales Σ⊥ Σ decomposed into a smooth component Pm;sm (i.e., the and k, respectively. broadband, with no contribution from the BAO) and an The final anisotropic galaxy power spectrum, accounting oscillatory contribution Olin. In this way, the total matter for the effect of nonlinearities on the BAO features and power spectrum is given by PmðkÞ¼Pm;smðkÞOlinðkÞ. eventual density field reconstruction, can be expressed as There are different ways to extract Pm;sm from Pm, and 3 2 we describe the methodology used here in Appendix A. −k fμ2Σ2þð1−μ2ÞΣ2 g μ μ 1 − 1 2 k ⊥ Pðk; Þ¼Psmðk; Þ½ þðOlinðkÞ Þe The galaxy bias bg and a factor encoding the effect of redshift-space distortions FRSD can be applied to Pm;sm in þ Pshot; ð9Þ order to obtain the anisotropic, smoothed galaxy power −1 spectrum in redshift space where Pshot ¼ ng (where ng is the mean comoving number density of galaxies) is a scale-independent contribution μ 2 μ Psmðk; Þ¼BFRSDðk; ÞPm;smðkÞ; ð7Þ arising from the fact that we use discrete tracers of the matter density field, such as galaxies. The template for the where B is a constant absorbing bg and potential variations BAO analysis is generated with Eq. (9). on the amplitude of Pm;sm, and As shown in Eq. (9), it is clearer to express the anisotropic power spectrum as a function of k and μ, 1 F k; μ 1 βμ2R ; instead of k⊥ and kk. The rescaling of distances appearing RSDð Þ¼ð þ Þ 1 0 5 μσ 2 ð8Þ þ . ðk FoGÞ in Eq. (4) can be transformed to k and μ as [78] β where ¼ f=bg and the fingers of God small-scale kobs σ ktrue 1 μobs 2 F−2 − 1 1=2; suppression is driven by the parameter FoG, whose value ¼ α ½ þð Þ ð AP Þ is related to the halo velocity dispersion.4 ⊥ μobs The actual amplitude of the BAO feature is reduced with μtrue 1 μobs 2 −2 − 1 −1=2 ¼ ½ þð Þ ðFAP Þ ; ð10Þ respect to the linear prediction due to nonlinear collapse. In FAP addition, nonlinear clustering also introduces a subpercent ≡ α α shift in the BAO scale. However, these effects can be partially where FAP k= ⊥. reverted using density field reconstruction [70,71]. The effect Given the large scales probed, the line of sight changes of wrong assumptions regarding the fiducial cosmology and with each pointing and cannot be considered parallel to any galaxy bias on density field reconstruction was studied in Cartesian axis. Hence, it is not possible to obtain a well- Ref. [72], where a negligible shift in the BAO peak location defined μ for the observations, which makes a direct was found. However, it was also found that these small measurement Pðk; μÞ impossible. However, one can directly changes in the shape of the BAO feature and in the amplitude measure the Legendre multipoles of the anisotropic power of the quadrupole of the power spectrum might introduce spectrum using, e.g., the Yamamoto estimator [79].6 Then, small biases in the BAO fit for future surveys. The factor R in the observed power spectrum multipoles are modeled as Eq. (8) models the partial removal of redshift-space distor- Z tions produced by the density field reconstruction and takes 2l 1 1 obs þ μobs true μtrue L μobs 1 Plðk Þ¼ 2 d Pðk ; Þ lð ÞþAlðkÞ; the following values: R ¼ before reconstruction and R ¼ 2α⊥α −1 1 − − Σ 2 2 5 k exp ½ ðk reconÞ = after reconstruction. On the other hand, the nonlinear damping of the BAO is modeled ð11Þ – with an exponential suppression applied to Olin [74 77]. where Ll is the Legendre polynomial of degree l, true μtrue 3 Pðk ; Þ is computed using Eqs. (9) and (10), and a Our methodology is a generalization of the approach intro- fid 3 ðr =rdÞ term has been absorbed into the constant factor B duced in Ref. [69], accounting for the fact that rd can have (very) d different values depending on the cosmology assumed. of Psm in Eq. (7). Different polynomials AlðkÞ are added to 4The damping due to the fingers of God can also be modeled each one of the power spectrum multipoles. These poly- with a Gaussian function, providing similar results without losing nomials are added not only to marginalize over uncertainties flexibility in the fit to the observations. 5The specific functional form of R after reconstruction depends related with nonlinear clustering, but in particular to account on the reconstruction formalism used [73]. The one used in this for the possibility that the broadband of the template Pm;sm work corresponds to the “Rec-Iso” convention in Ref. [73], which does not match the actual one. These polynomials have the accounts for the removal of redshift-space distortions during form 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 6There are other compression options, such as the so-called the mock power spectra that we analyze. angular wedges [80].

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−3 −2 −1 n 2 μ AlðkÞ¼al;1k þ al;2k þ al;3k þ al;4 þ al;5k ; σ2 μ P ðk; Þ ðk; Þ¼ μ ; ð14Þ ð12Þ Nmodesðk; Þ μ where Nmodes is the number of modes per bin in k and in where n ¼ 1 and n ¼ 2 before and after density field the observed volume Vobs,givenby reconstruction, respectively [81]. k2ΔkΔμ Note that the standard BAO analysis can be performed μ Nmodesðk; Þ¼ 8π2 Vobs; ð15Þ with slightly different models of the signal that incorporate, for example, different nonlinear evolution and procedures where Δk and Δμ are the widths of the k and μ bins, to isolate the BAO feature. These slight differences are respectively. Note that Pshot is implicitly included in discussed in Ref. [82] and do not affect our results or Eq. (14), since it is part of the modeling of Pðk; μÞ [see conclusions. Eq. (9)]. In the case of shot-noise subtracted power In summary, BAO-only analyses include the following spectrum measurements, Pshot should be removed from parameters: Eq. (9) and added explicitly to Pðk; μÞ in Eq. (14). Once we have defined the covariance per k and μ bin, we can compute the covariance of the power spectrum multi- α α β a σ Σ Σ f ⊥; k;B; ; l; FoG; ⊥; kg; ð13Þ poles. This covariance can be decomposed in subcovariance matrices for each multipole, plus the subcovariance matrices between different multipoles. For multipoles l and l0, the where al are the coefficients of Al in Eq. (12). All but the subcovariance matrix under the Gaussian assumption is two first parameters α⊥ and α are nuisance parameters. k given by (see Ref. [87] for a detailed derivation) There are thus 17 parameters in the analysis of the monopole and quadrupole of the power spectrum (22 2l 1 2l0 1 C δ ð þ Þð þ Þ parameters if the hexadecapole is also included). In this ll0 ðki;kjÞ¼ ij Z 2 work we consider the monopole and quadrupole of the 1 2 postreconstruction galaxy power spectrum. The values of × dμσðki; μÞ LlðμÞLl0 ðμÞ; ð16Þ the parameters listed in Eq. (13) that we use to compute the −1 α α 1 2 β a 0 δ template are ⊥ ¼ k ¼ , B ¼ bg, ¼ f=bg, l ¼ , where ij is the Kronecker delta. Finally, it is necessary to σ 10 Σ 2 Σ 4 FoG ¼ Mpc=h, ⊥ ¼ Mpc=h, and k ¼ Mpc=h. incorporate observational effects (in both the clustering The value of bg depends on the survey considered, f summary statistic and its covariance), which are mainly depends on the redshift and cosmology assumed, and we due to the geometry of the survey itself, in the theoretical Σ Σ modeling of the observed multipoles of the power spectrum. choose the values for ⊥ and k following Ref. [74]. The specific values chosen for the fiducial parameter do not The geometry of the survey (the footprint and the fact that change our results. 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 B. Covariance and likelihood Fourier space). While modeling and properly implementing Galaxy surveys normally rely on an estimation of the the mask (see Refs. [88–90] for detailed discussions) are key variation of the power spectrum using a suite of mock for obtaining reliable conclusions from power spectrum catalogs (see, e.g., [83,84]) computed on the fiducial measurements, they do not have an effect on the focus of our cosmology to obtain the covariance of the multipoles of study: the flexibility of the analysis and its validity regarding the power spectrum. Using galaxy mocks makes it easier to cosmologies beyond ΛCDM at the level of perturbations. model selection effects, observational mask, survey geom- Moreover, the effect of the mask on the BAO analysis is etry, and other observational systematics that need to be small and more important for redshift-space-distortion taken into account to avoid a bias in the measurement [85]. measurements. Therefore, we do not account for observa- However, motivated by the ever growing number of mocks tional effects in this work. required to lower the noise of the covariance below the Taking all this into account, we can compare the statistical errors, there are proposals to obtain precise theoretical prediction of the measured power spectrum covariance matrices directly from the data (see, e.g., with respect to a fiducial cosmology with actual observa- [86]). Since in this work we aim to treat BAO analyses tions. Hence, the corresponding likelihood is in general and do not need exquisite precision on the covariance, we use an analytic approximation. Neglecting log L ∝ ðPth − PdataÞC−1ðPth − PdataÞT; ð17Þ mode coupling due to the nonlinear collapse and the observational mask, we approximate the covariance per where P denotes the concatenation of all the multipoles of k and μ bin as the power spectrum considered in the analysis, Pth and Pdata

123515-6 ROBUSTNESS OF BARYON ACOUSTIC OSCILLATION … PHYS. REV. D 102, 123515 (2020) are the theoretical prediction and the measured power size of the photon-baryon sound horizon, providing a spectrum, respectively, the superscript T refers to the resolution to the Hubble tension [25,26]. Therefore, an transpose operator, and all quantities are evaluated at the EDE resolution to the Hubble tension leads to a decrease in observed values of k [following Eq. (11)]. rd, while keeping CMB power spectra angular scales and peak heights fixed via small shifts in the standard ΛCDM IV. ESTIMATING THE BIAS ON BAO parameters. In particular, we explore the oscillating EDE MEASUREMENTS model, described in Ref. [26]. The scalar field evolves along an axionlike potential V of the form In this section we explore the impact that extensions to ΛCDM have on the BAO feature. After introducing the 4 models considered, we study the changes in Olin with VðϕÞ¼ϒ ½1 − cos ðϕ=λÞnaxion ; ð18Þ respect to the fiducial cosmology, individually varying different parameters. Afterwards, we perform mock BAO analyses as detailed in Sec. III to evaluate the flexibility and where ϒ is the normalization of the potential and λ is the validity of BAO results for the extended cosmologies. decay constant of the scalar field. The additional param- eters beyond those of ΛCDM considered for this model are c A. Cosmological models the critical redshift zEDE when the EDE becomes dynami- As a point of reference, we consider ΛCDM with one cal, the fraction fEDE of EDE in the total energy density of massive neutrino species (mν ¼ 0.06 eV) and its standard the Universe at the critical redshift, the initial scalar field Θ ≡ ϕ λ parameters: the physical baryon and cold dark matter displacement EDE i= , and the power naxion of the Ω 2 Ω 2 oscillating factor of the potential for the scalar field. The density parameters today, bh and cdmh , respectively, h n shift in rd is mainly controlled by fEDE, with a weaker the reduced Hubble constant , the spectral index s, and c the amplitude of the primordial power spectrum of scalar dependence on zEDE. The shape of the potential (which is modes A . We take the fiducial cosmology in this section to controlled by naxion) determines the rate at which the EDE s c Θ be the same used for our analysis in Sec. IV C: ΛCDM with dilutes after zEDE, and EDE mainly controls the evolution parameter values listed in Table I. In addition to ΛCDM, we of the EDE perturbations (see also [26,91]). focus on two models that have been suggested as potential The DNI model proposes an interaction between neu- χ solutions of the Hubble tension: EDE and DNI. We also trinos and a small fraction fDNI of the dark matter with consider a ΛCDM model with a different number of mass mχ in order to solve the H0 tension [28]. The scattering relativistic species than the prediction of the standard cross section σχν, taken to be independent of the neutrino 3 046 Λ ∝ σ model Neff ¼ . , CDM þ Neff, where Neff is an temperature, is parametrized as uDNI χν=mχ. In this additional parameter of this model. model, rather than decreasing the size of the sound horizon, The EDE model proposes the existence of a scalar field ϕ the new interaction inhibits the free-streaming of neutrinos, that is initially frozen (due to Hubble friction) at some field which lowers the standard phase shift φ induced by value and becomes dynamical after the Hubble parameter neutrinos in the BAO. The peak structure of the CMB drops below some critical value. If the EDE becomes power spectra can be roughly described by the position of dynamical before recombination, it leads to a decrease in the pth peak

TABLE I. 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 Ω 2 cdmh 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 χ2 for each cosmology and the corresponding best-fit nuisance parameters using iMinuit. a 100 Ω 2 Ω 2 109 c Θ ▵χ2 × bh cdmh hns × As Neff fEDE log10zEDE EDE naxion uDNI fDNI Fiducial 2.237 0.1200 0.6736 0.9649 2.100 3.046 ΛCDM 1 2.229 0.1212 0.6680 0.9608 2.077 3.046 2.0 ΛCDM 2 2.261 0.1160 0.6900 0.9709 2.140 3.046 18.8 Λ CDM þ Neff 1 2.228 0.1154 0.6614 0.9587 2.072 2.796 1.3 Λ CDM þ Neff 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 3.351 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 ahttps://iminuit.readthedocs.io/en/stable/.

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l ≃ π − φ DMðzÞ does not sufficiently capture differences between the p;peak ðp Þ ; ð19Þ fiducial and the modified cosmologies. r We show this comparison in Fig. 1, where we individually vary all the relevant parameters of the cosmologies discussed where the subscript * denotes quantities at the time of in Sec. IVA, keeping the rest of the parameters fixed. In recombination. In this model, r does not change with order to study the variations thoroughly, we show both the Λ 7 respect to CDM, so that DMðzÞ needs to compensate absolute values of O ðk=½r =rfidÞ in the upper panels, as the reduction of the phase shift in order to reproduce the lin d d well as the ratio O ðk=½r =rfidÞ=ðO ðk=½r =rfidÞÞfid to measured location of the CMB peaks. Given that the lin d d lin d d compare with the fiducial in the lower panels. Note that at physical matter, radiation, and cosmological constant low redshift, the results shown in Fig. 1 are independent of energy densities are very constrained by other CMB redshift, since all redshift dependence is contained in P , features, the required decrease in D z is achieved by m;sm Mð Þ except for the redshift evolution of the nonlinear damping of increasing H0. Since this model modifies the ΛCDM the BAO, which we do not model in this work. neutrino-induced phase shift in the BAO and the standard 2 The first two panels of Fig. 1 show O , varying Ω h and BAO analysis does not account for such effects, DNI might lin b Ω h2 under ΛCDM. We can see that r changes consid- lead to a significantly biased BAO constraint. Note that, as cdm d erably for the parameter ranges explored (since the sound indicated in Ref. [28], this model assumes massless neu- speed of the plasma and the matter content of the Universe trinos (with no implementation for massive neutrinos). change). However, after rescaling r , the only significant Therefore, we will also consider massless neutrinos when d change in O is the amplitude of the wiggles, with no assuming DNI cosmologies. lin 8 appreciable phase shift or further change of the BAO feature. We use the public Boltzmann code CLASS [92,93], as Although the amplitude of the BAO is affected by non- well as its modifications to include the EDE and DNI9 linearities [74], the nonlinearities are modeled with an models, to compute the matter power spectra and other exponential decay dependent on Σ⊥ and Σ [Eq. (9)], which cosmological quantities needed. k we marginalize over. There might be some scale dependence of the BAO amplitude that is not covered bythis modeling, but O B. Cosmological dependence of the shape of lin the fact that there is no change in the position of the BAO α α We now consider how the BAO feature changes with implies that the constraints on ⊥ and k should not be biased. Λ respect to the fiducial under these different cosmological For Olin under CDM þ Neff, shown in the third panel models with varying parameters. In particular, we are of Fig. 1, the amplitude is also slightly modified, but there concerned with changes in the evolution of perturbations is also a phase shift introduced in the BAO wiggles. This α α before recombination, which affect ⊥ and k through rd. phase shift can be seen in the skewness of the oscillations for a range of N [38], as well as in the fact that the k Once we account for modifications of rd, we can investigate eff changes in the BAO feature beyond its characteristic scale. values where the ratio of Olin (after rescaling rd) crosses 1 Changes to the background expansion history at low change with Neff. In addition, Neff changes rd significantly, due to the modification of the expansion history of the redshift affect q⊥ and qk; however, it has been shown that the standard BAO analysis is robust to such changes Universe before recombination. [34–36]. Therefore, we do not model the Alcock- The next four panels of Fig. 1 show Olin for EDE cosmo- c Θ Paczynski effect nor the isotropic dilation in this section logies, varying fEDE, zEDE, EDE,andnaxion.TheBAOfeature is most sensitive to changes in fEDE. Although the amplitude and focus on the intrinsic undistorted shape of Olin. of Olin does not change significantly, there is a small phase In order to study the BAO feature, we compare OlinðkÞ predictions for the modified and fiducial cosmologies. If shift introduced as fEDE grows. A phase shift is also noticeable only the peak position shifts away from the fiducial, while when varying the other EDE parameters, especially for naxion the overall pattern remains the same, the result is a rescaling (note that currently the whole range for naxion covered by the of k in the argument of O due to the variation of r [see color bar is within the 68% confidence level marginalized lin d constraint reported in Ref. [26]). As noted in Ref. [25], varying Eq. (4)]. In this case, all O ðk=½r =rfidÞ would be the same lin d d zc mainly affects the height of the first acoustic peak of the upon adjusting r , where we have incorporated the r EDE d d CMB power spectra as well as the ratio r =r ,wherer is dependence in the rescaling of k explicitly. If, however, the d Silk Silk the Silk damping scale; thus, it has a noticeable effect on the pattern of O changes, then the standard BAO analysis lin amplitude ratio between different peaks. Finally, the last two panels of Fig. 1 show Olin for DNI 7 Although the redshift of last scattering z and the redshift of cosmologies, varying uDNI and fDNI. The value of rd the end of radiation drag zd do not coincide, we expect minimal remains the same, but the interaction between neutrinos model dependence in the difference between r and rd. Therefore, we consider them as encapsulating redundant information. and a fraction of the dark matter modifies the fiducial phase 8http://class-code.net/. shift in the BAO feature. This change is larger for larger 9 https://github.com/subhajitghosh-phy/CLASS_DNI. values of uDNI or fDNI, evident from directly comparing

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fid FIG. 1. BAO feature Olin in the matter power spectrum for varying cosmological parameters (with k rescaled by rd=rd ), compared with the fiducial prediction (red dashed lines), and the corresponding ratio (i.e., modified over fiducial cosmology) in the lower panels. fid The insets show rd=rd for each case, and the model under consideration is given in the upper left corner of each panel. We keep all ΛCDM 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 0 2 c 3 5 observations, and additional datasets for EDE and DNI. For the panels corresponding to EDE, we use fEDE ¼ . , log10 zEDE ¼ . , ΘEDE ¼ 2.8, and naxion ¼ 3, unless otherwise indicated. For the DNI model, we use fDNI ¼ 0.02 and uDNI ¼ 5, unless otherwise indicated. DNI constraints are reported in terms of fDNIuDNI; hence they are adapted to the fixed values of uDNI and fDNI assumed in this figure. Note the change in scale of the y axis for the lower sections of each panel.

curves of Olin in the upper sections of the panels (and in, In order to do so, we perform mock BAO analyses using the e.g., Fig. 1 of Ref. [28] for the CMB power spectra). For formalism described in Sec. III, with the aim of evaluating this model, the ratio of Olin with respect to the fiducial the potential bias introduced in the determination of α⊥ and α prediction may be misleading. Although the phase shift in k. Such a bias could affect the constraints on the expansion fid Olinðk=½rd=rd Þ is toward higher k for larger values of uDNI history of the Universe and hence the inference of the or fDNI, the amplitude of the BAO is also larger, causing the cosmological parameters. Although it has been demon- oscillations of the ratio of Olin to shift toward lower k. strated [34–36] that BAO-inferred distances are extremely robust to modifications on the expansion history of the α α C. Bias in ⊥ and k Universe at late times, these distances might not be robust Now that we have established that there are cosmological against changes in the acoustic oscillations due to the models that can produce Olin patterns that do not match growth of perturbations before recombination. the fiducial prediction under a rescaling of rd (or the BAO Whether differences in the BAO pattern with respect to amplitude), we are ready to evaluate the bias on the the fiducial prediction such as those highlighted in Fig. 1 standard BAO fit that may arise under these cosmologies. introduce a potential bias on the BAO measurements

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TABLE II. 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 Ωsky of the survey, the physical volume V covered by the survey, the mean galaxy density ng, and the galaxy bias bg.

2 3 4 −3 Survey z Ωsky ½deg V ½ðGpc=hÞ 10 ng ½ðMpc=hÞ bg 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 depends on the flexibility of the BAO analysis. If a bias remaining parameters to their new best-fit values to Planck, were found for a given cosmological model, current BAO obtained from keeping the chosen parameter fixed. Λ measurements (which assume a ΛCDM fiducial) would not In addition to the standard CDM parameters and Neff, be suitable for constraining that model. we consider the extra parameters of EDE and DNI. For Λ Λ In order to perform the mock BAO analysis, we create a CDM and CDM þ Neff, we use best-fit cosmologies mock power spectrum for each specific underlying cos- obtained from Planck’s publicly available Monte Carlo mology under consideration, as if it were measured from Markov chains,11 and obtain poor fits from the same chains 69 00 −1 −1 Λ the observed galaxy distribution. Details on how we fixing H0 ¼ . km s Mpc for CDM and Neff ¼ 3 4 Λ compute the mock power spectra can be found in . for CDM þ Neff. The EDE 1 and DNI 1 cosmologies Appendix B. We perform the standard BAO analysis on are taken from the reported best fits for the considered each mock spectrum using the same fiducial ΛCDM datasets in Refs. [26,28], respectively. EDE 2 is obtained cosmology. We run Monte Carlo Markov chains using from the Monte Carlo Markov chain computed for 10 the Monte Carlo sampler emcee [94] to perform the Ref. [26], fixing fEDE to 0.2, while DNI 2 is the best fit 15 0 02 BAO fit with the likelihood in Eq. (17). to Planck data for fixed uDNI ¼ and fDNI ¼ . , found Once we obtain the marginalized constraint in the using MONTEPYTHON [95]. The DNI 2 case has been chosen α − α ⊥ k plane for each mock analysis, we can locate as a very extreme case to test the flexibility of the standard 2 the point in parameter space at which the value of the BAO analysis: its Δχ for Planck power spectra with marginalized posterior peaks to compare with the true respect to the ΛCDM mean values (i.e., our fiducial case) point. We compute Δχ2 ¼ −2Δ log L between these two is ∼338. α − α In order to mimic the power of current and next- points in the ⊥ k plane assuming a Gaussian margin- alized posterior and calculate the corresponding confidence generation galaxy surveys, we use four different sets of level for a two-dimensional χ2 distribution. This procedure survey specifications, inspired by BOSS [6] and DESI [96]. is equivalent to computing the cumulative probability of the These specifications are important, because they determine χ2 distribution up to the value of interest. Finally, we relate the effective redshift at which the power spectrum is the calculated confidence level to a number of standard measured and computed, as well as its covariance matrix. deviations σ for a Gaussian distribution. We estimate the The specifications for these surveys are listed in Table II. Marginalized constraints in the α⊥ − α plane are shown bias then to be this number of σ. k In Table I, we specify the values of the cosmological in Fig. 2, comparing the values for which the posterior parameters for both the assumed fiducial cosmology distribution peaks and the true values of these parameters needed for the BAO analysis and all of the target cosmol- (marked by circles and stars, respectively). Each panel ogies used to generate the mock power spectra. We use for shows the results for a given galaxy survey and cosmo- the cosmological parameters of our fiducial cosmology the logical model. We show cosmologies 1 in blue and mean values of the ΛCDM analysis (including temperature, cosmologies 2 in red. Note that, since each model has polarization, and lensing data) to Planck [3]. In addition, its own prediction for DM=rd and Hrd (and these quantities change with redshift), each panel shows different true we consider two different sets of cosmological parameters α α for all models: the best fit to Planck data (or Planck data values for ⊥ and k. We show results assuming finished combined with other datasets) and a poor fit. We refer to surveys (BOSS [6]) in the two left columns, while the two these two sets with the labels “1” and “2,” respectively, both right columns correspond to future surveys (DESI [96]). α α individually (e.g., in the model name) and collectively. For The true values of ⊥ and k lie well within the the latter, we choose one of the parameters to be at least 68% confidence level contours in Fig. 2, indicating that α α ≳3σ away from its nominal best-fit value and set the there is no significant bias in the inference of ⊥ and k for any of the models in the finished surveys, even when the

10https://github.com/dfm/emcee. 11http://pla.esac.esa.int/pla/.

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α − α FIG. 2. The 68% and 95% confidence level marginalized constraints in the ⊥ k plane, shown with their maximum-posterior and true values (represented as circles and stars, respectively), for different mock galaxy power spectra computed in ΛCDM, ΛCDM þ Neff , EDE, and DNI, as indicated in the legend (rows), and for BOSS LOWZ and CMASS, as well as DESI at z ¼ 0.8 and z ¼ 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 ΛCDM parameters from Planck’s analysis. Blue contours refer to the cosmologies numbered as 1 (best fit from Planck, as well as external datasets for EDE and DNI) and red contours to cosmologies numbered as 2. Cosmological parameters and survey specifications are listed in Tables I and II, respectively. fiducial model and the model under study have very appreciate this when comparing the marginalized con- different predictions for rd or Olin. Moreover, the largest straints shown in Fig. 2 for cosmologies 1 and 2. In each bias found for our extreme DNI 2 example is ∼1σ, for DESI panel, the uncertainties for each of the cases are the same. 1 survey (and ∼0.8σ for DESI 2 survey). This bias is This means that the BAO standard analysis does not α α introduced from the change in the phase shift produced by misestimate the covariance between ⊥ and k,evenif the interaction between neutrinos and dark matter, as the assumed fiducial cosmology is very far from the demonstrated in Fig. 1, and expected from the discussion cosmology corresponding to the observations. in Sec. IVA. However, note that for DNI 1, the bias in the BAO measurements is always ≲0.2σ. V. DISCUSSION Finally, in addition to biasing the best fit, inaccurate modeling can also induce a misestimation of the parameter The results shown in Sec. IV C are clear and powerful. uncertainties (see, e.g., [97]). However, we find that not The BAO standard analysis is flexible enough to capture only is there no significant bias in the best-fit values of α⊥ different growth-of-structure histories prior to recombina- α and k, but their uncertainties do not change. We can tion for the models explored in this work. Moreover, our

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α α results indicate that the inference of ⊥ and k using a computed at every point of the parameter space of the template can be unbiased even for cosmologies that differ model. However, exploring these biases for the whole significantly from the fiducial cosmology used to compute parameter space is unfeasible. We overcome this limitation the template. In most of the cases, this fiducial cosmology by studying concrete cases far from the assumed fiducial would come from the best fit to the CMB power spectra. cosmology. These cases represent the tails of the posterior, These results, as well as the discussion included in where residual small biases in the BAO constraints would Sec. II, provide a strong motivation to keep using template- not modify the bulk of the distribution nor introduce a fitting methodologies to perform BAO-only analysis: this significant bias in the parameter inference. approach is robust, flexible, and general enough so that it On the other hand, it is expected that the bias in α⊥ and α allows agnostic studies of cosmology. Therefore, the results k decreases as we move in the parameter space from the of standard BAO analyses can be used to infer cosmologi- region corresponding to these cosmologies toward the cal parameters without introducing any bias in the sub- fiducial cosmology. Therefore, an unbiased analysis of sequent constraints. Moreover, compressing the BAO the extreme cases that we consider further supports the use α α α α information in terms of ⊥ and k reduces the probability of ⊥ and k constraints for a joint parameter inference of of introducing observational systematics and reduces com- cosmological models beyond ΛCDM with other cosmo- putational costs on cosmological parameter inference, logical probes, such as the CMB power spectra. This means which proves key as the number of observations grows. that our results indicate that previously reported BAO α α In this work we have focused on the BAO fit to the measurements in the form of ⊥ and k constraints are galaxy power spectrum. However, other tracers can be used valid for constraining not only late-time deviations of the to infer the matter power spectrum; regardless of which expansion history of the Universe but also early-Universe tracer is used, BAO imprints are present in the perturbations extensions to ΛCDM, without the need to refit the galaxy of all of them. Therefore, our study can be extrapolated to power spectrum. the power spectrum of the fluctuations of the number of Our findings ease the worry regarding the validity of quasars, the Lyman-α forest, line-intensity mapping, etc., reported BAO measurements assuming a ΛCDM template as well as their cross-correlations. The particularities of the for models beyond ΛCDM with large deviations at the analysis applied to each specific tracer are related with perturbation level. This concern has prompted some studies observational systematics and not with the underlying to omit BAO measurements from the set of cosmological data cosmology; hence, they do not affect the extrapolation used to constrain cosmological models (see, e.g., Ref. [28]). of the conclusions of this work. We argue that BAO measurements can and should be Finally, although we have focused on four specific included, especially given their importance in constraining cosmological models, our results can be extrapolated to proposed solutions to the H0 problem [21,23,24]. other cosmological models. As explored in Appendix C, In particular, let us consider how the best-fit DNI model the BAO feature present on the matter density distribution obtained in Ref. [28] without including BAO measurements at the time of recombination (indirectly probed by CMB fits the BAO observations. In Fig. 3, we show the expansion observations to great precision) survives unmodified until history as predicted by the best-fit ΛCDM cosmology to the present. This is true modulo nonlinear collapse of Planck 2018 and the best-fit DNI model to data from Planck perturbations, which still preserves some of the features 2015 [100], WiggleZ dark energy survey [101], and SH0ES encoded in the BAO, as shown in Ref. [37]. [16] (the dataset considered in Ref. [28]). We also show Therefore, we conclude that using a template computed existing measurements from BOSS [6] and eBOSS [7–9], for a good fit to the CMB observations should be suitable demonstrating that ΛCDM provides a much better fit. The for a BAO analysis. This assessment might not hold, DNI prediction implies a χ2 ¼ 5.9 larger than the ΛCDM however, if the evolution of dark matter and baryon Λ prediction for the combined likelihood of BOSS [6], 2dFGS overdensities is influenced by beyond- CDM physics post [102], and SDSS DR7 MGS [103] (the BAO dataset recombination, such as baryon-dark matter interactions considered in Planck analyses). We conclude that BAO (e.g., see Refs. [98,99] for a description of the modified measurements disfavor the reported best-fit DNI model. We Boltzmann equations). Moreover, general relativity is expect that BAO measurements favor DNI models with assumed in density-field reconstruction (see, e.g., [70]); lower H0 than the reported best fit (and corresponding lower hence only BAO measurements obtained before applying u and f ), closer to the ΛCDM constraint. density-field reconstruction shall be used to constrain DNI DNI This expectation is common to any model that modifies cosmological models that modify gravity. prerecombination physics without altering the ΛCDM prediction of r . This is due to the strong, model-indepen- A. Constraining cosmological models with BAO d dent constraint set by BAO on rdh [67]. This result hints α − α Cosmological parameter inference using ⊥ k con- that only introducing a phase shift in the acoustic pertur- straints would be completely accurate only if the BAO bations without modifying rd is not enough, and that a rd measurements are unbiased for every power spectrum needs to be lower to solve the H0 tension [21–24].

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of the measurements of the rescaling parameters, i and j are the entries of this covariance and the α vectors, and S refers to each of the surveys or independent measurements included in the analysis. Consider the DNI cosmologies from Table I, 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 Ω M today and H0. We obtain each Fisher matrix trans- −1 Ω − forming the corresponding ðCovSÞ to the M H0 space. We find biases of ∼−0.1σ (∼0.1σ) and ∼−1.4σ (∼−0.3σ) Ω for M and H0, respectively, for the DNI 2 (1) cosmology. When only the BOSS surveys are considered, these biases reduce to ∼−0.01σ (∼−0.03σ) and ∼−0.2σ (∼−0.01σ). All the biases are compared with the corresponding uncer- tainties 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 ϑ when combining additional cosmological probes can be estimated in a similar way following Ref. [47]. Nonetheless, when different probes are combined, small systematic errors may lead to significant biases in the joint parameter inference. The study per- formed 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 ΛCDM. This error budget may vary from case to case, FIG. 3. Hubble expansion rate over ð1 þ zÞ3=2 (top) and depending on the cosmological model under study and the comoving angular diameter distance over ð1 þ zÞ3=2 (bottom) actual observations used, and might be important for the as a function of redshift, weighted by the ratio between the actual analysis of future BAO measurements. sound horizon at radiation drag and its fiducial value (ΛCDM Λ best fit to Planck). We show predictions for CDM (blue lines) B. Amending biases on standard BAO analyses and DNI (green lines), as well as existing measurements from BOSS [6] and eBOSS [7–9]. Note that the error bars shown here If there is a situation in which a significant bias on α⊥ or α do not include the covariance between measurements. k is indeed found when applying the analysis presented in this work, the power spectrum must be reanalyzed to We note, however, that cosmological parameter infer- measure the BAO scale. There are a few different options ence could be systematically affected by coherent small about how to proceed. One option is to find a template assuming a different biases in the determination of α⊥ and α at several redshifts. k fiducial cosmology that minimizes the bias seen in the We explore this possibility adapting the estimation of the analysis and refit the BAO with this template. In this case, systematic bias in parameter inference due to inaccurate α α ⊥ and k would be given by Eq. (3). This new fiducial approximations presented in Ref. [47] to our needs. The Λ systematic shift in the cosmological parameters ϑ when the cosmology will be most likely beyond CDM. However, measured rescaling parameters αm are biased with respect the impact of the feature not captured by a rescaling may to their true values αtr can be estimated as depend on cosmological parameters. In this case, a fixed template without modeling such a feature would still fail to α α X −1X remove the bias in ⊥ and k. −1 m tr Δϑ F ∇ϑα ; Cov α −α ; 20 Hence, a second option consists of adding freedom to the ¼ S S ð S iÞð SÞij ð j S;jÞ ð Þ S;i;j analysis. In case the extra contribution to Olin can be robustly modeled, it is possible to include it in the BAO where Δϑ ¼ ϑm − ϑtr (with ϑm denoting the parameters analysis with one or more new nuisance parameters that inferred from the observations without accounting for the control its impact. Using this approach, however, would α α bias in the analysis), F is the Fisher matrix of the weaken the constraints on ⊥ and k, given the addition of cosmological parameters, Cov is the covariance matrix extra nuisance parameters.

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Finally, if a reliable model describing the extra feature parameters, which encode the cosmological information of cannot be found, the analysis can be performed changing interest. In addition, differences in the shape between the the template at each step of the MCMC. This procedure is measured power spectrum and the precomputed template related to alternative methodologies to extract cosmological (sourced by the choice of a wrong fiducial cosmology or information from the BAO feature that do not rely on a uncertainties in small-scale clustering) are marginalized precomputed template (see, e.g., Refs. [104,105]). These over using a plethora of nuisance parameters. As extensions methods do not aim to obtain a model-independent meas- of ΛCDM become more complex in the search of new urement of the expansion history of the Universe, nor can physics or in the attempt to resolve cosmological tensions, they extract agnostic independent information of early-time this approach might not be flexible enough to cover these physics through rd, since the template readjusts during the cosmologies, rendering BAO measurement unsuitable for fitting procedure. In these cases, a prior is needed to break studying such models. Ω 2 Ω 2 the degeneracy between at least cdmh and bh . This In this work, we have assessed the flexibility of the prior may either come from CMB observations and be standard BAO analysis under different cosmological mod- Λ Λ directly applied to rd or be obtained from primordial els: CDM, CDM þ Neff, EDE, and DNI. We have deuterium and helium abundances assuming standard big performed mock analyses of galaxy power spectra, using Ω 2 Λ bang nucleosynthesis and be applied to bh and Neff. the same CDM template for all of them. We find that, for the models explored in this work, there is no bias in the BAO α α C. BAO cosmology beyond ⊥ and k analyses for existing data. Moreover, the maximum bias Even if there is no bias in the BAO measurements, there found for future surveys is ∼1σ for an extreme DNI model can still be cosmological signatures that are not captured by which provides a very bad fit to Planck data. Therefore, our the modeling of the rescaling of the template as described in findings reinforce the model independence and robustness Λ Sec. III. This is the case for CDM þ Neff, for instance: of the standard BAO analysis, and they support using α α varying Neff changes the phase shift of the BAO in a way reported BAO measurements in terms of ⊥ and k (obtained α α that cannot be reproduced by a rescaling with ⊥ or k. using a ΛCDM template) to constrain these models. This Additional freedom is needed in order to consistently will be further supported by the implementation of blinding include this effect in the BAO analysis. This was explored analyses in future galaxy surveys [106]. in Ref. [38], where an extra parameter was included as a The models considered in this work have been chosen multiplicative factor of a k-dependent function that mod- as a representative sample of models that aim to solve the eled the amount of phase shift. H0 tension through modifications prior to recombination. Once this additional freedom is included in the analysis, Nonetheless, our results can be extrapolated to other instead of considering the extra parameter as a nuisance and models that do not impact the clustering of dark matter marginalizing over it, it can also be treated as the parameter and baryons with new physics after recombination. of interest. In this case, one would marginalize over the rest Moreover, futuristic surveys may reach a level of of the parameters to constrain the amplitude of the extra precision for which the low significance of the biases feature introduced in Olin. This is the case for the phase found in this work is enough to affect the measurement. shift induced by Neff in Ref. [39], where a phase shift was If there is a model with a modified perturbation history detected at 95% confidence level after applying CMB that changes the BAO feature beyond rescaling factors, priors on α. Since most of these effects are expected to be we advocate for the methodology described in this work α − α isotropic (affecting only Olin and not the Alcock-Paczynski to ascertain whether reported constraints in the ⊥ k effect or the redshift-space distortions), it may be preferable plane are valid for cosmological parameter inference. In to perform an isotropic BAO analysis in order to reduce case a bias in the BAO fit is found, we have suggested degeneracies between parameters. possible ways forward to add flexibility to the modeling of the power spectrum and perform an unbiased BAO VI. CONCLUSIONS measurement. BAO provide a robust probe of the expansion history of the Next-generation galaxy surveys such as DESI [96], Universe at low redshift, as well as a calibrator of the sound Euclid [107], and SKA [108] will improve upon existing horizon at radiation drag. Recently, the importance of BAO BAO measurements through higher precision observations measurements has been highlighted again with a crucial role and reaching higher redshifts. Additionally, future line- in the resolution of the H0 tension, which can be reframed as a intensity mapping experiments promise to extend BAO mismatch between the two anchors of the cosmic distance measurements up to z ∼ 9 with competitive precision ladder: H0 and rd [21]. These are precisely the two quantities [109,110]. Checking the validity of BAO analyses for which set the normalization of the expansion history of the exotic models is of the utmost importance in order to Universe as constrained by BAO measurements. exploit the promising outlook for measurements in the near Standard BAO analyses are robust and model indepen- future, especially given the importance of BAO as a dent, since they rely on a template fitting using rescaling cosmological probe.

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ACKNOWLEDGMENTS We thank Licia Verde, H´ector Gil-Marín, and Vivian Poulin for useful discussions, and Graeme Addison and Alvise Raccanelli for comments on the last stages of the manuscript. J. L. B. is supported by the Allan C. and Dorothy H. Davis Fellowship. T. L. S. is supported by the Research Corporation and NASA ATP Grant No. 80NSSC18K0728.

APPENDIX A: EXTRACTION OF THE SMOOTH POWER SPECTRUM Pm;sm The idea behind the extraction of the smooth matter power spectrum Pm;sm from the total matter power spec- trum Pm is conceptually simple: it is based on removing the wiggles produced by the BAO. However, rather than performing a brute-force smoothing of Pm or interpolating using only the zero points between the wiggles, one can choose more stable and efficient options. There are several procedures to extract Pm;sm, mainly divided in two broad groups. One option relies on the computation of the matter FIG. 4. Top: Total linear matter power spectrum before (solid power spectrum without including the BAO contribution, lines) and after (dashed lines) the removal of the BAO contri- bution (P and P , respectively). Bottom: Same as top but in which can be done, e.g., using the analytic transfer function m m;sm configuration space. Factors of k and r2 are added, respectively, derived in Ref. [111]. Alternatively, one can use Eq. (6) to for clarity. Blue lines correspond to ΛCDM predictions and red convert P into ξ ðrÞ, directly remove the BAO peak from 0 3 m m lines to EDE predictions (with fEDE ¼ . ). the correlation function, and transform back to Pm.We adopt the latter method in this work. We refer the interested ξ reader to Ref. [69] for a detailed comparison between (4) Transform m;sm back to Fourier space, i.e., different methodologies. obtaining Pm;sm. Olin is obtained by computing Since we are interested in exploring cosmologies very the ratio between the total matter power spectrum different from ΛCDM, especially regarding the BAO scale Pm and Pm;sm. and feature, we avoid fits to polynomials at fixed scales, as (5) Iterate points 2–4 varying r1 and r2 to optimize the 1 ≲ 10−3 done in, e.g., Ref. [69]. Instead, we use a more flexible convergence of Olin ¼ at k h=Mpc: at methodology described below: scales much larger than rd there is no effect from (1) Obtain the total matter power spectrum Pm from a the BAO and the smooth and actual clustering must Boltzmann solver and use Eq. (6) to obtain the be equal. ξ corresponding correlation function, m. We use the We compare the total and the smooth matter power public MCFIT PYTHON package to evaluate integrals, spectrum and correlation function in Fig. 4. Note that r1 12 ξ such as the one appearing in Eq. (6). In order to is smaller than the scale of the local minimum of m, and avoid numerical noise, Pm needs to be evaluated at a the range of r2 is significantly larger than rd. This is ξ ξ large number of points N and for a wide range in k. because the two minima of m are deeper than for m;sm due 4096 We sample k uniformly in logspace in N ¼ to the enhanced clustering around rd. points in the interval k ∈ ½10−5; 20 h=Mpc. The computation of the smooth power spectrum or (2) Take the corresponding entries to r1 ∈ correlation function may be critical for the analysis of real ½60; 70 Mpc=h and r2 ∈ ½200; 300 Mpc=h in the observations, and the performance of different procedures 2ξ r array, i1 and i2, respectively, and interpolate r m must be compared. We have checked that our results are evaluated in the interval ½i ¼ 0;i1 ∪ ½i2;i¼ N − 1 robust to the choice of the algorithm used to extract the using cubic splines. broadband of the clustering, since the mock power spec- (3) Evaluate the interpolation object in the original r trum is computed with the same modeling that is used in the array and remove the r2 factor, so that a smooth likelihood. ξ correlation function m;sm without the BAO peak is Pdata obtained. APPENDIX B: MOCK g COMPUTATION As explained in Sec. IV C, we perform a mock BAO 12https://github.com/eelregit/mcfit. analysis in order to test for the presence of a bias in the

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α α measurement of the rescaling parameters ⊥ and k when the photon temperature fluctuations of the CMB. In fact, the measured power spectrum does not correspond to the the use of a ΛCDM template when analyzing the BAO fiducial cosmology assumed for the computation of the feature relies on the assumption that if ΛCDM provides a template in Eq. (9). To do so, we need to compute a good fit to the CMB power spectra, it also provides a good measured power spectrum for different cosmologies. In this fit to the BAO feature. With this qualitative fact in mind, it Appendix we explain our procedure to compute this is useful to quantify this relationship in light of the analysis data mock Pg . we have presented in this work. First, we compute Pðk; μÞ following Eq. (9) with the During radiation domination, and after horizon crossing, δ cosmological and nuisance parameters corresponding to the the density contrast c of cold dark matter starts to grow δ ∝ target cosmological model we want to mock. However, we logarithmically with the scale factor, c lnðaÞ. As time δ ∝ need to account for the fact that this power spectrum passes, smaller-scale modes start collapsing as c a so corresponds to a cosmology different from the assumed that, after matter-radiation equality, all modes within the δ ∝ fiducial cosmology used to measure it. This means that the horizon grow as c a. Baryons, on the contrary, only fall mock observed Pðk; μÞdata is affected by the Alcock- into the dark matter potential wells after they have Paczynski effect and the isotropic dilation. Note that in recombined and got released from the radiation pressure. this case, Pðk; μÞ has been obtained with its corresponding Then, after decoupling, the evolution of cold dark matter rd (instead of with the fiducial template), so that we rescale and baryons is given by [112,113] k with q⊥ and q from Eq. (2). With this correction, we k _ 3 _ 2 Pdata δ̈ a δ_ a δ 1 − δ compute g using Eq. (11). Moreover, given that the c þ c ¼ ½Rc c þð RcÞ b; ðC1Þ 3 a 2 a power spectrum is measured in ðMpc=hÞ , for the fiducial value of h, we need to rescale the units accordingly. Finally, a_ 3 a_ 2 δ̈ þ δ_ ¼ ½R δ þð1 − R Þδ ; ðC2Þ we assume the covariance would be computed from mock b a b 2 a c c c b galaxy catalogs; hence, it would correspond to the fiducial cosmology. Therefore, we compute the covariance by ≡ ρ ρ where Rc c= m is the fraction of the cold dark matter applying Eq. (16) to the fiducial power spectrum. ρ ρ ρ ρ energy density c to the total matter density m ¼ c þ b In order to evaluate the bias without any confusion of cold dark matter and baryons, and the dot denotes a introduced by statistical dispersion around the true values, derivative with respect to conformal time, η. Note that at we decide not to include dispersion on the mock power decoupling the universe is matter dominated so that spectrum around the predicted theoretical power spectrum. a=a_ ¼ 2=η. Therefore, even though we consider the correct covariance, The total matter transfer function is given by the all the points of the mock power spectrum coincide with its weighted sum of the cold dark matter and baryon transfer theoretical prediction. Nonetheless, we describe here how functions: to include the statistical dispersion of the mock power δ 1 − δ spectrum around the theoretical prediction, for complete- Tm ¼ Rc c þð RcÞ b; ðC3Þ ness. Note that the covariance of the power spectrum is not δ δ diagonal [Eq. (16)], so we need to take into account the where c and b are given by the solution of the coupled covariance between multipoles in order to mock the Eqs. (C1) and (C2). If we assume tight coupling between Pdata dispersion about the true values of g . Therefore, we baryons and photons, adiabatic initial conditions, and δ diagonalize the covariance matrix and draw random values instantaneous decoupling, so that b is related with the δ δ 3δ 4 from Gaussian distributions centered at 0 and with var- radiation overdensity γ as b;dec ¼ γ;dec= (where the iances corresponding to each of the entries of the diagon- subscript “dec” refers to quantities evaluated at decou- alized covariance. Then, we transform this vector back to pling), we obtain the original k base and add it to Pdata. The resulting g η 2 dispersion around Pdata properly accounts for the covari- A g Tm ¼ η ðC4Þ ance between the multipoles. dec and APPENDIX C: CONNECTION BETWEEN CMB AND BAO 1 A ≡ 4 3δ η δ_ ½ Rcð c;dec þ dec c;decÞ Under ΛCDM, the BAO feature present in the matter 20 3 1 − 3δ η δ_ power spectrum at low redshift comes from the same þ ð RcÞð γ;dec þ dec γ;decÞ; ðC5Þ dynamics that produce the acoustic structure measured in the CMB: baryons and photons are tightly coupled in the where we have neglected decaying solutions (assuming η ≫ η early Universe and undergo acoustic oscillations, which dec) and the effect of the cosmological constant. become imprinted in both the matter power spectrum and The linear matter power spectrum thus takes the form

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2 η 0 PmðkÞ¼PprimðkÞTmðk; Þ¼Pm;smðkÞOlinðkÞ; ðC6Þ is the conformal time today), and denotes the derivative with respect to x0. To leading order in τ_=ðaHÞ we have 2 −3 −1 2π ns where PprimðkÞ¼ Ask ðk=kpÞ is the primordial 3 δ_ power spectrum (where kp is the pivot wave number, ≃ γ;dec −1 vb;dec 4 ; ðC14Þ typically taken to be kp ¼ 0.05 Mpc ) and k 2 4 3 δ_ 4 P R η γ;dec= m;sm c 3δ η δ_ 2 Π ≃ ; ðC15Þ ¼ 25 ð c;dec þ dec c;decÞ η ; ðC7Þ dec 4 τ_ Pprim dec _ where τ_ is the differential optical depth for Thomson 3ð1 − R Þð3δγ þ η δγ Þ O − 1 ≃ c ;dec dec ;dec ; ðC8Þ scattering. lin 2 3δ η δ_ Rcð c;dec þ dec c;decÞ Combining Eq. (C8) with Eqs. (C9) and (C10), we can see the ratio between the terms in the second bracket, _ −1 where we have neglected the term proportional to jη δγ =3δγ j¼r k=3 ≃ k=ð0.02 Mpc Þ. This shows 1 − 2 dec ;dec ;dec s ð RcÞ . Let us now focus on the BAO feature, whose that, in the scales of interest for the acoustic feature, O is δ δ_ lin scale dependence is given by γ;dec and γ;dec. Note also that δ_ driven by the term depending on γ;dec. Therefore, compar- this approximate Olin does not evolve with time, as ing with Eq. (C11), the acoustic oscillations in Olin are a discussed for the numerically obtained Olin described in factor of π=2 out of phase with respect to the CMB Appendix A and shown in Fig. 1. During tight coupling, the temperature anisotropies, which is a well-known result acoustic oscillations in the photon-baryon fluid are damped (see, e.g., Ref. [74]). along with a scale-dependent enhancement due to a driving We can see that the CMB and BAO acoustic features are, – force [114 116], to first approximation, given by the photon density per- turbation and its first time derivative at decoupling. δ ≃−D η γ;dec ðkÞ cosðcsk decÞ; ðC9Þ Therefore, within the context of the basic approximations made here, we can expect that any cosmological model, δ_ ≃ D η γ;dec ðkÞcsk sinðcsk decÞ; ðC10Þ even beyond ΛCDM, providing a good fit to the CMB spectra should also provide a good fit to the BAO feature. D where cs is the photon-baryon sound speed and encodes This argument also supports the extrapolation of the the Silk damping and the scale-dependent enhancement. conclusions of this work beyond the four cosmological η Note that the sound horizon at decoupling is rs ¼ cs dec. models considered. The acoustic features in the CMB arise from different, Nonetheless, there are some differences to note but related, terms. If we assume again an instantaneous between the acoustic features in the BAO and the decoupling, the visibility function is a Dirac delta function: CMB. One difference is that the CMB spectrum is a η δ η − η gð Þ¼ Dð decÞ. Under this approximation, the CMB projected version, and the Fourier modes are integrated – power spectra are given by [117 119] over spherical Bessel functions. Even though Z l ≃ η − η Δ ≃ 2 kð 0 decÞ, wave numbers in the range k k= XY 4π 2 2 ΔX ΔY l Cl ¼ð Þ k dkPprimðkÞ l ðkÞ lðkÞðC11Þ contribute to a measured . A more dramatic difference is related to the amplitude of the oscillations: while CMB oscillations are at least order ∼1, BAO oscillations have with an amplitude ∼0.1 (see Fig. 1). In addition to this, the 1 nuisance parameters necessary to robustly extract infor- ΔT ≈ ψ δ 0 lðkÞ dec þ 4 γ;dec jlðx0Þþvb;decjlðx0Þ; ðC12Þ mation from the BAO feature make current measure- ments of the BAO less sensitive to changes (beyond a sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rescaling of distances with rd;s) in the acoustic oscil- l 2 ! 4 jl x0 ΔE ≈ ð þ Þ Π ð Þ lations than the CMB. This is why, even for cosmic l ðkÞ dec 2 ; ðC13Þ ðl − 2Þ! 3 x0 variance limited BAO measurements over large volumes and wide redshifts ranges, CMB priors related with the ψ where is the Newtonian potential transfer function, vb is background expansion are needed in order to obtain the baryon velocity perturbation transfer function, Π is the competitive cosmological constraints from the shape of ≡ η − η η polarization transfer function, x0 kð 0 decÞ (where 0 the BAO (see, e.g., [38] for the case of Neff).

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