Evidence for baryon acoustic oscillations from galaxy–ellipticity correlations

The baryon acoustic oscillation feature in the clustering of galaxies or quasars provides a ‘standard ruler’ for distance measurements in cosmology. In this work, we report a 2–3σ signal of the baryon acoustic oscillation dip feature in the galaxy density–ellipticity cross-correlation functions using the spectroscopic sample of the Baryon Oscillation Spectroscopic Survey CMASS, combined with the deep Dark Energy Spectroscopic Instrument Legacy Imaging Surveys for precise galaxy shape measurements. We measure the galaxy–ellipticity correlation functions and model them using the linear alignment model. We constrain the distance DV/rd to redshift 0.57 to a precision of 3–5%, depending on the details of modelling. The galaxy–ellipticity measurement reduces the uncertainty of distance measurement by ~10% on top of that derived from the galaxy–galaxy correlation. More importantly, for future large and deep galaxy surveys, the independent galaxy–ellipticity measurements can help sort out the systematics in the baryon acoustic oscillation studies. Baryon acoustic oscillations from the early Universe are imprinted in the large-scale structure, providing a cosmic expansion ruler. A comparison of the shapes and positions of galaxies can yield an independent measure of these primordial oscillations.

3 ∼ 5%, depending on the details of modeling.The GI measurement reduces the uncertainty of distance measurement by ∼ 10% on top of that derived from the galaxy-galaxy (GG) correlation.More importantly, for future large and deep galaxy surveys, the independent GI measurements can help sort out the systematics in the BAO studies.
Measuring the expansion history of the Universe is one of the key goals in cosmology.The best constraints now are from the measurements of the distance-redshift relation over a wide range of redshifts 1 .The Baryon Acoustic Oscillations (BAO) feature in the clustering of galaxies is a "standard ruler" for robust distance measurements 2,3 .BAO arise from tight coupling of photons and baryons in the early Universe.Sound waves travel through this medium and give rise to a characteristic scale in the density perturbations, corresponding to the propagation distance of the waves before the recombination.With large galaxy surveys, using BAO, distances have been measured to percent level at various redshifts [4][5][6][7][8] .
In the aforementioned studies, only the spatial distributions of galaxies are used while the shapes or orientations of galaxies are ignored.The intrinsic alignment (IA) of galaxies is usually treated as a contaminant in weak lensing analysis 9,10 .However, as we will demonstrate using actual observations in this work, IA is actually a promising cosmological probe and contains valuable information.The galaxy-ellipticity (GI) and ellipticity-ellipticity (II) intrinsic alignments were first detected in Luminous Red Galaxies (LRG) [11][12][13] .Studies showed that the IA of galaxies can be related to the gravitational tidal field using the linear alignment (LA) model [14][15][16] .According to the LA model, a BAO feature in both GI and II correlations shows up as a dip rather than a peak [17][18][19] as seen in the GG correlations.Furthermore, the entire 2D and anisotropic pattern of GI and II cor-relations may provide additional information to the BAO.The results were tested and confirmed in the N -body simulations 20 .Taruya & Okumura 21 performed a forecast of cosmological constraints for IA statistics using the LA model, and found that IA can provide a similar level of constraints on cosmological parameters as the galaxy spatial distributions.
In this Letter, we report a first measurement of BAO feature using IA statistics, namely GI, and confirm that IA can provide additional information to galaxy-galaxy (GG) correlations.In addition to reducing statistical uncertainties of the distance measurements, GI can also provide a test of systematics when compared with the BAO measurements from GG.Details of this analysis including the datasets, measurements and modeling are presented in Methods.Our fiducial results, as shown in the main text, are derived using data vectors in the range of 50-200 h −1 Mpc with a polynomial marginalised over.This range is chosen to ensure that the polynomial captures the entire shape and avoid the turnover around 20 h −1 Mpc.To be complete, we also show results with other choices of the fitting range and with or without marginalising over the polynomial in Supplementary Figure 4, 5 and Supplementary Table 1.We observe a 2 − 3σ signal of BAO in all of these variations.
The measurements and modeling of the isotropic GI and GG correlation functions are shown in Fig. 1.The GI measurements show an apparent dip around the BAO scale at ∼ 100 h −1 Mpc from both the pre-(panel a) and post-reconstructed samples (panel b).The fitting results are reasonable on all scales, namely, the reduced χ 2 is 0.97 for pre-reconstruction and 0.87 for post-reconstruction.
To show the significance of the BAO detection, we display the ∆χ 2 ≡ χ 2 − χ 2 min surfaces in Fig. 2 (see also Supplementary Figure 1), where χ 2 min is the χ 2 for the best-fitting model.We compare ∆χ 2 for the no-wiggle model with the BAO model and find a 3σ detection of the BAO feature in both the pre-and post reconstructed samples.
The constraint on α, which represents the deviation from the fiducial cosmology (see Methods), is 1.050 +0.030 −0.028 and 1.057 +0.035 −0.036 using the pre-and post-reconstructed samples, respectively.Both results are in good agreement (within 2σ) with the fiducial Planck18 22 results (TT,TE,EE+lowE+lensing+ BAO), which assumes a ΛCDM cosmology.We find that the constraint is not improved after reconstruction for the GI correlations, which may be due to the fact that we only reconstruct the density field but keep the shape field unchanged.The result may be further improved in principle, if the shape field is also reconstructed, which is left for a future study.
The post-reconstructed GG (panel c) and combined (panel d) measurements and modeling are also shown in Fig. 1.The GG correlation alone measures α to be 0.986 +0.013 −0.013 and the combined GG+GI derives the constraint to be 0.997 +0.012 −0.012 , demonstrating that the GI measurement gives rise to a ∼ 10% improvement in terms of the uncertainty on α.And more importantly, as we mentioned above, the next generation surveys can tighten the GG BAO constraints by a factor of 2-3 1,23 and systematic errors become more and more important.If the GI measurements can be improved at the same level, comparisons between a sub-percent (< 0.5%) GG measurements and a 1% GI measurements can provide a check of the systematic bias in the measurements.Our GG results are consistent with the results reported by the BOSS team 6 within the 68% confidence level uncertainty, although the numbers are slightly different due to a few effects, including fitting ranges, details of radial bins and error estimations.
In Fig. 3, we convert the constraints on α to distance D V /r d measured at redshift z = 0.57 13.91 +0.17 −0.17 using the pre-reconstructed GI, post-reconstructed GI, GG and GI+GG, respectively.
All these results are consistent with Planck18 within 2σ level.
In the work, we obtain a 2 ∼ 3σ measurement of the BAO dip feature in galaxy-ellipticity correlations, although the constraints on the distance from GI are only around 1/3 of that from GG, much weaker than predicted in Taruya & Okumura 21 using the linear alignment model.The reason may be that the galaxy-halo misalignment 24 can reduce the intrinsic alignment signals and weaken the BAO constraints, which may not be considered appropriately by Taruya & Okumura 21 .According to Okumura & Jing 24 , taking into account the misalignment, the GI signals can be reduced by 2-3, which is consistent with our results.Moreover, since realistic mock catalogs for galaxy shapes are unavailable yet, the covariance matrices in this study are estimated using the jackknife resampling method.Employing more reliable error estimation techniques could potentially improve the accuracy of the results, and is left for a future study.Nevertheless, the results are already promising.With the next generation spectroscopic and photometric surveys including the Dark Energy Spectroscopic Instrument (DESI) 23 and the Legacy Survey of Space and Time (LSST) 25 , we will have larger galaxy samples and better shape measurements.We expect that the intrinsic alignment statistics can provide much tighter constraints on cosmology from BAO and other probes 26,27 .

Methods
Statistics of the intrinsic alignment The shape of galaxies can be characterized by a two-component ellipticity, which is defined as follows: where q represents the minor-to-major axial ratio of the projected shape, and θ denotes the angle between the major axis projected onto the celestial sphere and the projected separation vector pointing towards a specific object.
The GI correlations, denoted as the cross-correlation functions between density and ellipticity fields, can be expressed as 15,18 ξ gi (r Here, r = x 1 − x 2 , and i = {+, ×}. In this work, we focus on the GI correlation ξ g+ since the signal of ξ g× vanishes due to parity considerations.It is worth noting that the IA statistics exhibit anisotropy even in real space due to the utilization of projected shapes of galaxies, and the presence of redshift space distortion (RSD) 28 can introduce additional anisotropies in ξ g+ (r).Therefore, we define the multipole moments of the correlation functions as 29 Here, X represents one of the correlation functions, P ℓ denotes the Legendre polynomials, and µ corresponds to the directional cosine between r and the line-of-sight direction.
The linear alignment model On large scales, the linear alignment (LA) model is frequently employed in studies of intrinsic alignments 14,15,18 .This model assumes a linear relationship between the ellipticity fields of galaxies and halos and the gravitational tidal field.
where Ψ p represents the gravitational potential, G denotes the gravitational constant, and C 1 characterizes the strength of IA.Although the observed ellipticity field is density-weighted, namely and it can be neglected at the BAO scale.In the Fourier space, Equation 4 can be expressed as: Then, ξ g+ (r) can be represented by the matter power spectrum P δδ 18, 30 : where b g is the linear galaxy bias and j 2 is the second-order spherical Bessel function.
The RSD effect 28 can also be considered in ξ g+ (r) at large scales 18 .However, in this work, we do not consider the RSD effect and only focus on the monopole component of ξ g+ (r) given the sensitivity of current data.
We plan to measure the entire 2D ξ g+ (r) with future large galaxy surveys, which may contain much more information.To test the LA model, Okumura et al. 20 measured the IA statistics in N-body simulations and found that the results agree well with the predictions from the LA model on large scales.Thus, it is reasonable to use the above formula of ξ g+,0 (r) for BAO studies.
Fitting the BAO scale We fit the BAO features in GG correlations following the SDSS-III BOSS DR12 analysis 6,31 .
To model the BAO features in GI correlations, we adopt the similar methodology used in isotropic galaxy correlations (GG) studies 5,32 .In spherically averaged two-point measurements, the BAO position is fixed by the sound horizon at the baryon-drag epoch r d and provide a measurement of 4 where D A (z) is the angular diameter distance and H(z) is the Hubble parameter.The correlation functions are measured under an assumed fiducial cosmological model to convert angles and redshifts into distances.The deviation of the fiducial cosmology from the true one can be measured by comparing the BAO scale in clustering measurements to its position in a template constructed using the fiducial cosmology.The deviation is characterized by where the subscripts "fid" denote the quantities from the fiducial cosmology.
The template of ξ g+,0 is generated using the linear power spectrum, P lin , from the CLASS code 33 .
In GG BAO peak fitting, a linear power spectrum with damped BAO is usually used to account for the non-linear effect, where P nw is the fitting formula of the no-wiggle power spectrum 3 and Σ nl is the damping scale.
In this analysis, we set Σ nl = 0 as our fiducial model for GI, and we also show the results with Σ nl as a free parameter in Supplementary Table 1.
Using the template, our model for GI correlation is given by where B accounts for all the effects that only affect the amplitude of the correlation such as IA strength, galaxy bias and shape responsivity (see Equation 16).As in the GG analysis, we further add an additional polynomial in our model to marginalize over the broad band shape: Thus, with the observed GI correlation ξ obs g+,0 (s) and the covariance matrix C, we can assume a likelihood function L ∝ exp (−χ 2 /2), with where C −1 is the inverse of C, i, j indicate the data points at different radial bins, N JK and N bin are the total number of sub-samples and radial bins, and (N JK − N bin − 2)/(N JK − 1) is the Hartlap correction factor 34 to get the unbiased covariance matrix.The covariance matrices are estimated using the jackknife resampling from the observation data: where ξ n i is the measurement in the nth sub-sample at the ith radial bin and ξi is the mean jackknife correlation function at the ith radial bin.We use the Markov Chain Monte Carlo (MCMC) sampler emcee 35 to perform a maximum likelihood analysis.N JK is chosen by gradually increasing it until the constrains on α are stable (Supplementary Table 1).

Sample selection
We use the data from the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS DR12 sample 6,36,37 .The CMASS sample covers an effective area of 9329 deg 2 and provides spectra for over 0.8 million galaxies.Galaxies are selected with a number of magnitude and colour cuts to get an approximately constant stellar mass.We use the CMASSLOWZTOT LSS catalog in BOSS DR12 and adopt a reshift cut of 0.43 < z < 0.70 to select the CMASS sample with an effective redshift z eff = 0.57.
Reconstruction methods can improve the significance of the detection of the BAO feature, and reduce the uncertainty in BAO scale measurements, by correcting for the density field smoothing effect associated to large-scale bulk flows [38][39][40] .We also use the post-reconstructed catalogs from BOSS DR12 and we refer the details of the reconstruction methods to their papers 6,40 .
To get high quality images for the CMASS galaxies, we cross match them with the DESI Legacy Imaging Surveys DR9 data 41 (https://www.legacysurvey.org/dr9/files/#survey-dr9-region-specobj-dr16-fits), which covers the full CMASS footprint and contains all the CMASS sources.The Legacy Surveys can reach a r band PSF depth fainter than 23.5 mag, which is 2-3 deeper than the SDSS photometry survey used for target selection and is more than adequate to study the orientations of the massive CMASS galaxies.The Legacy Surveys images are processed using Tractor 42 , a forward-modeling approach to perform source extraction on pixel-level data.We use shape e1 and shape e2 in the Legacy Surveys DR9 catalogs (https://www.legacysurvey.org/dr9/catalogs/#ellipticities) as the shape measurements for each CMASS galaxy.These two quantities are then converted to the ellipticity defined in Equation 1.Following Okumura & Jing 24 , we assume that all the galaxies have q = 0, which is equivalent to assuming that a galaxy is a line along its major axis.This assumption only affects the amplitude of the GI correlations, and the measurements of the position angles (PA) are more accurate than the whole galaxy shapes.
The whole CMASS sample is used to trace the density field.While for the tracers of the ellipticity field, we further select galaxies with Sérsic index 43 n > 2 , since only elliptical galaxies show strong shape alignments, and q < 0.8 for reliable PA measurements.In principle, we should also exclude satellite galaxies.However, since selecting centrals in redshift space is arbitrary and most of the CMASS LRGs are already centrals, we do not consider the central-satellite separation.Selections using n and q remove nearly half (from 816, 779 to 425, 823) of the CMASS galaxies.We show the results using the whole sample in Supplementary Table 1 and confirm that the morphology and q selection can really improve the measurements and tighten the constraints.
Estimators To estimate the GI correlations, we generate two random samples R s and R for the tracers of ellipticity and density fields respectively.Following Reid et.al. 37 , redshifts are assigned to randoms to make sure that the galaxy and random catalogues have exactly the same redshift distribution.We adopt the generalized Landy-Szalay estimator 44,45 ξ g+,0 (s where R s R is the normalized random-random pairs.S + D is the sum of the + component of ellipticity in all pairs: where the ellipticity of jth galaxy in the ellipticity tracers is defined relative to the direction to the ith galaxy in the density tracers, and R = 1 − ⟨γ 2 + ⟩ is the shape responsivity 46 .R equals to 0.5 under our q = 0 assumption.S + R is calculated in a similar way using the random catalog.
We also measure the GI correlation functions for the reconstructed catalogs.The ellipticities of galaxies are assumed unchanged in the reconstruction process.The estimator becomes where E, T represent the reconstructed data and random sample, and R and R s are the original random samples.In above calculations, we adopt the FKP weights 47 (w FKP ) and weights for correcting the redshift failure (w zf ), fibre collisions (w cp ) and image systematics (w sys ) for the density field tracers 37 : w tot = w FKP w sys (w cp + w zf − 1), while no weight is used for the ellipticity field tracers.

Measurements and modeling
We measure ξ g+,0 (s) for both pre-and post-reconstruction catalogs in 50 < s < 200 h −1 Mpc with a bin width of 5 h −1 Mpc.We calculate their covariance matrices using the jackknife resampling with N JK = 400, and model the GI correlation functions using Equation 11 with a Planck18 22 fiducial cosmology at z = 0.57.In Supplementary Table 1, we show that the pre-and post-reconstruction GI results are relatively stable if N JK ≥ 400, verifying that N JK = 400 is a reasonable choice.We measure and model the post-reconstruction isotropic GG correlation functions with the same radial bins and error estimation schedule (N JK = 400).
We also model the GG and GI correlation together with a 60 × 60 covariance matrix that includes the GG-GI cross-covariance to get the combined results.pre-and post-reconstruction GI measurements, which proves that our BAO detection and distance constraints are relatively robust to the details of modeling.Results with the whole CMASS sample as the tracers of the ellipticity field are also included in Supplementary Table 1, confirming that our sample selection can improve the precision of GI BAO measurements.

Here, C 1
(a) ≡ a 2 C 1 ρ(a)/ D(a), where ρ represents the mean mass density of the Universe, D(a) ∝ D(a)/a, and D(a) corresponds to the linear growth factor, with a denoting the scale factor.

Figure 1 Figure
Figure 1 Measurements and the modeling of GI and GG correlation functions.Panel a: pre-reconstruction GI correlations; Panel b: post-reconstruction GI correlations; Panel c: post-reconstruction GG correlation; Panel d: post-reconstruction combined modeling (GI multiplied by 4 for better illustration).Dots with error bars show the mean and standard error of the mean of clustering measurements.Errors are from the diagonal elements of the jackknife covariance matrices estimated using 400 subsamples.Lines with shadows are the best-fit models and 68% confidence level regions derived from the marginalized posterior distributions.

Figure 3 Supplementary Figure 2 :
Figure 3 Constraints of D V /r d from GG and GI correlation functions with N JK = 400.A combined post-reconstructed GG+GI constraint is also provided.The central values are medians, and the errors are the 16 & 84 percentiles.The vertical orange line shows the fiducial Planck18 results.

Supplementary Figure 1 := 1 Supplementary Figure 2 :
Comparison of the best-fit BAO and non-BAO models for the pre (left) and post (right) reconstruction GI correlations with fiducial models.Dots with error bars show the mean and standard error of the mean of clustering measurements.Errors are from the diagonal elements of the jackknife covariance matrices estimated using 400 subsamples.We compare the best-fit BAO and non-BAO models for the pre and post-reconstruction GI correlations in Supplementary Figure1.The BAO model fits the measurements much better than the non-BAO model. 1 arXiv:2306.09407v2[astro-ph.CO] 27 Jul 2023 Constraints from the pre-reconstruction GI correlation functions using MCMC sampling with fiducial model.The central values are medians, and the errors are the 16 & 84 percentiles after other parameters are marginalized over.

Table 1 :
+0.000 0.000 Using the whole CMASS sample as the tracers of the ellipticity field without n and q selections.Constrains of α from different correlation functions, jackknife resampling scales, models and fitting ranges.pre-andpost-reconstructionGImeasurements, which proves that our BAO detection and distance constraints are relatively robust to the details of modeling.Results with the whole CMASS sample as the tracers of the ellipticity field are also included in Supplementary Table1, confirming that our sample selection can improve the precision of GI BAO measurements.
a 'poly' indicates using polynomial to fit the broad band shape and 'Σ nl ' means setting Σ nl as a free parameter.b