Issue 
A&A
Volume 629, September 2019



Article Number  A86  
Number of page(s)  18  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201935641  
Published online  10 September 2019 
Baryon acoustic oscillations from the crosscorrelation of Lyα absorption and quasars in eBOSS DR14
^{1}
Aix Marseille Univ., CNRS, CNES, LAM, Marseille, France
email: michael.blomqvist@lam.fr
^{2}
Department of Physics and Astronomy, University of Utah, 115 S. 1400 E., Salt Lake City, UT 84112, USA
^{3}
Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de Hautes Energies, LPNHE, 4 Place Jussieu, 75252 Paris, France
^{4}
IRFU, CEA, Université ParisSaclay, 91191 GifsurYvette, France
^{5}
Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth PO1 3FX, UK
^{6}
University College London, Gower St, Kings Cross, WC1E 6BT London, UK
^{7}
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA
^{8}
Department of Physics and Astronomy, University of Waterloo, 200 University Ave. W., Waterloo, ON N2L 3G1, Canada
^{9}
Department of Astronomy and Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA
^{10}
Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA
^{11}
Brookhaven National Laboratory, 2 Center Road, Upton, NY 11973, USA
Received:
8
April
2019
Accepted:
22
July
2019
We present a measurement of the baryon acoustic oscillation (BAO) scale at redshift z = 2.35 from the threedimensional correlation of Lymanα (Lyα) forest absorption and quasars. The study uses 266 590 quasars in the redshift range 1.77 < z < 3.5 from the Sloan Digital Sky Survey (SDSS) Data Release 14 (DR14). The sample includes the first two years of observations by the SDSSIV extended Baryon Oscillation Spectroscopic Survey (eBOSS), providing new quasars and reobservations of BOSS quasars for improved statistical precision. Statistics are further improved by including Lyα absorption occurring in the Lyβ wavelength band of the spectra. From the measured BAO peak position along and across the line of sight, we determined the Hubble distance D_{H} and the comoving angular diameter distance D_{M} relative to the sound horizon at the drag epoch r_{d}: D_{H}(z = 2.35)/r_{d} = 9.20 ± 0.36 and D_{M}(z = 2.35)/r_{d} = 36.3 ± 1.8. These results are consistent at 1.5σ with the prediction of the bestfit spatiallyflat cosmological model with the cosmological constant reported for the Planck (2016) analysis of cosmic microwave background anisotropies. Combined with the Lyα autocorrelation measurement presented in a companion paper, the BAO measurements at z = 2.34 are within 1.7σ of the predictions of this model.
Key words: largescale structure of Universe / dark energy / cosmological parameters / cosmology: observations / quasars: absorption lines
© M. Blomqvist et al. 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The baryon acoustic oscillation (BAO) peak in the cosmological matter correlation function at a distance corresponding to the sound horizon, r_{d} ∼ 100 h^{−1} Mpc, has been seen at several redshifts using a variety of tracers. Following the original measurements (Eisenstein et al. 2005; Cole et al. 2005), the most precise results have been obtained using bright galaxies in the redshift range 0.35 < z < 0.65 (Anderson et al. 2014a,b, 2012; Alam et al. 2017) from the Baryon Oscillation Spectroscopy Survey (BOSS; Dawson et al. 2013) of the Sloan Digital Sky SurveyIII (SDSSIII; Eisenstein et al. 2011). Other measurements using galaxies cover the range 0.1 < z < 0.8 (Percival et al. 2007, 2010; Beutler et al. 2011; Blake et al. 2011; Padmanabhan et al. 2012; Mehta et al. 2012; Chuang & Wang 2012; Xu et al. 2013; Ross et al. 2015; Bautista et al. 2018). At higher redshift, the peak has been seen in the correlation function of quasars at a mean redshift z ∼ 1.5 (Ata et al. 2018; GilMarín et al. 2018; Hou et al. 2018; Zarrouk et al. 2018) and in the fluxtransmission correlation function in Lymanα (Lyα) forests at z ∼ 2.3 (Busca et al. 2013; Slosar et al. 2013; Kirkby et al. 2013; Delubac et al. 2015; Bautista et al. 2017) and in the forest crosscorrelation with quasars (FontRibera et al. 2014; du Mas des Bourboux et al. 2017). These observations all yield measurements of comoving angulardiameter distances and Hubble distances at the corresponding redshift, D_{M}(z)/r_{d} and D_{H}(z)/r_{d} = c/(H(z)r_{d}), relative to the sound horizon.
BAO measurements have found an important role in testing the robustness of the spatiallyflat cosmology with cold dark matter and the cosmological constant (ΛCDM) that is consistent with observed cosmic microwave background (CMB) anisotropies (Planck Collaboration XIII 2016). While the parameters of this model are precisely determined by the CMB data by itself, more general models are not constrained as well. Most significantly, adding BAO data improves constraints on curvature (Planck Collaboration XIII 2016). The addition of BAO and type Ia supernova (SN Ia) data (Betoule et al. 2014) generalizes the “CMB” measurement of H_{0}, which assumes flatness, to give an “inverseladder” measurement of H_{0} (Aubourg et al. 2015) that can be compared with distanceladder measurements (Riess et al. 2016, 2018a,b). Here, the inverseladder method uses the CMBdetermined value of r_{d} to define BAOdetermined absolute distances to intermediate redshifts, z ∼ 0.5, which can then be used to calibrate SN Ia luminosities. The usual distance ladder calibrates the SN Ia luminosity using Cepheid luminosities, themselves calibrated through geometrical distance determinations.
A third use of BAO data is to determine ΛCDM parameters in a CMBindependent way. The Lyα forest auto and crosscorrelations that BOSS has pioneered are critical when gathering such measurements. It is striking that the oΛCDM parameters (Ω_{M}, Ω_{Λ}) determined by this method are in good agreement with the CMB values determined by assuming flat ΛCDM (Aubourg et al. 2015).
The individual BAO measurements of D_{M}(z)/r_{d} and D_{H}(z)/r_{d} are generally in good agreement with the CMB flat ΛCDM model. The largest single discrepancy, 1.8 standard deviations, is that of the BOSS (SDSS Data Release 12) measurement of the Lyα forest–quasar crosscorrelation of du Mas des Bourboux et al. (2017, hereafter dMdB17). In this paper, we update this analysis with new quasars and forests from the SDSS Data Release 14 (DR14; Abolfathi et al. 2018; Pâris et al. 2018) obtained in the extended Baryon Oscillation Spectroscopy Survey (eBOSS) program (Dawson et al. 2016) of SDSSIV (Blanton et al. 2017). This data set has been previously used to measure the crosscorrelation between quasars and the flux in the “CIV forest” due to absorption by triplyionized carbon (Blomqvist et al. 2018).
Besides the addition of new quasars and forests, our analysis differs in a few ways with that of dMdB17. Most importantly, we expand the wavelength range of the forest from the nominal Lyα forest, 104.0 < λ_{rf} < 120.0 nm, to include Lyα absorption (λ_{α} = 121.567 nm) in the Lyβ region of the spectra, 97.4 < λ_{rf} < 102.0 nm, thus increasing the statistical power of the sample. The procedure for fitting the correlation function is also slightly modified by including relativistic corrections (Bonvin et al. 2014; Iršič et al. 2016). Furthermore, we divide the data to report BAO measurements for two redshift bins. We have not developed new sets of mock spectra beyond those used in dMdB17. We refer to Sect. 6 of dMdB17 for the analysis of those mocks and the tests used to justify the analysis procedure.
The organization of this paper follows closely that of dMdB17. Section 2 describes the DR14 data set used in this study. Section 3 summarizes the measurement of the fluxtransmission field. Section 4 describes the measurement of the crosscorrelation of the transmission field with quasars and the associated covariance matrix. We also derive the “distortion matrix” that describes how the measured crosscorrelation is related to the underlying physical crosscorrelation. Section 5 describes our theoretical model of the crosscorrelation. Section 6 presents the fits to the observed correlation function and Sect. 7 combines these results with those from the Lyα autocorrelation function presented in a companion paper (de Sainte Agathe et al. 2019). Section 8 summarizes the constraints on cosmological parameters derived from these results and those from de Sainte Agathe et al. (2019). Our conclusions are presented in Sect. 9. The measurements presented in this paper were made using the publicly available Python package picca^{1} developed by our team.
2. Data sample and reduction
The quasars and forests used in this study are drawn from SDSS DR14. This release includes data from DR12 taken in the first two generations SDSSI/II, in the BOSS program of SDSSIII and in the eBOSS pilot program SEQUELS (Myers et al. 2015). These data were used in the measurement of the quasar–forest crosscorrelation of dMdB17. Here, we use in addition data from the first two years of the eBOSS program and the completed SEQUELS.
The quasar target selection for BOSS, summarized in Ross et al. (2012), combines different targeting methods described in Yèche et al. (2010), Kirkpatrick et al. (2011), and Bovy et al. (2011). The methods employed for eBOSS quasar target selection are described in Myers et al. (2015) and PalanqueDelabrouille et al. (2016).
The catalog of identified quasars, DR14Q (Pâris et al. 2018), includes 266 590 quasars^{2} in the redshift range 1.77 < z_{q} < 3.5. The distribution on the sky of these quasars is shown in Fig. 1 and the redshift distribution in Fig. 2.
Fig. 1. Sky distribution for sample of 266 590 tracer quasars (1.77 < z_{q} < 3.5) from DR14Q in J2000 equatorial coordinates. The solid black curve is the Galactic plane. The highdensity regions are the eBOSS and SEQUELS observations (for the northern regions of the two Galactic hemispheres) and SDSSstripe 82 (for declination δ ∼ 0). The discontiguous small areas contain only SDSS DR7 quasars. 
Fig. 2. Normalized redshift distributions for tracer quasars (black) and Lyα forest absorption pixels of Lyα region (blue) and Lyβ region (red). The histograms include 266 590 tracer quasars, 30.2 × 10^{6} pixels in the Lyα region, and 4.0 × 10^{6} pixels in the Lyβ region. The vertical dashed lines show the mean value of each distribution: (tracer quasars), 2.37 (in Lyα), 2.26 (in Lyβ). 
All spectra used for this analysis were obtained using the BOSS spectrograph (Smee et al. 2013) on the 2.5 m SDSS telescope (Gunn et al. 2006) at Apache Point Observatory (APO). The spectrograph covers observed wavelengths 360.0 ≲ λ ≲ 1040.0 nm, with a resolving power R ≡ λ/Δλ_{FWHM} increasing from ∼1300 to ∼2600 across the wavelength range. The data were processed by the eBOSS pipeline, the same (but a marginally updated version) as that used for the crosscorrelation measurement of dMdB17. The pipeline performs wavelength calibration, flux calibration and sky subtraction of the spectra. The individual exposures (typically four of 15 min) of a given object are combined into a coadded spectrum that is rebinned onto pixels on a uniform grid with Δ log_{10}(λ) = 10^{−4} (velocity width Δv ≈ 69 km s^{−1}). The pipeline additionally provides an automatic classification into object type (galaxy, quasar or star) and a redshift estimate by fitting a model spectrum (Bolton et al. 2012).
Visual inspection of quasar spectra was an important procedure during the first three generations of SDSS to correct for misclassifications of object type and inaccurate redshift determinations by the pipeline (Schneider et al. 2010; Pâris et al. 2017). Starting in SDSSIV, most of the objects are securely classified by the pipeline, with less than 10% of the spectra requiring visual inspection (Dawson et al. 2016). The visualinspection redshifts, when available, are taken as the definitive quasar redshifts, while the remaining quasars have redshifts estimated by the pipeline.
The crosscorrelation analysis presented here involves the selection of three quasar samples from DR14Q: tracer quasars (for which we only need the redshifts and positions on the sky), quasars providing Lyα forest absorption in the Lyα region, and quasars providing Lyα forest absorption in the Lyβ region. The selected sample of tracer quasars contains 266 590 quasars in the range 1.77 < z_{q} < 3.5. It includes 13 406 SDSS DR7 quasars (Schneider et al. 2010) and 18 418 broad absorption line (BAL) quasars, the latter identified as having a CIV balnicity index (Weymann et al. 1991) BI_CIV> 0 in DR14Q. Quasars with redshifts less than 1.77 are excluded because they are necessarily separated from observable forest pixels (see below) by more than 200 h^{−1} Mpc, the maximum distance where the correlation function is measured. The upper limit of z_{q} = 3.5 is adopted because of the low number of higherredshift quasars that both limits their usefulness for correlation measurements and make them subject to contamination due to redshift errors of the much more numerous lowredshift quasars (Busca & Balland 2018). Such contaminations would be expected to add noise (but not signal) to the crosscorrelation.
The summary of the Lyα forest data covering the Lyα or Lyβ region of the quasar spectrum is given in Table 1. Both samples exclude SDSS DR7 quasars and BAL quasars. The Lyα sample is derived from a super set consisting of 194,166 quasars in the redshift range 2.05 < z_{q} < 3.5, whereas the Lyβ sample is taken from a super set containing 76,650 quasars with 2.55 < z_{q} < 3.5. The lower redshift limits are a consequence of the forests exiting the wavelength coverage of the spectrograph for quasars with z_{q} < 2 and z_{q} < 2.53, respectively. Spectra with the same object identification THING_ID (reobserved quasars) are coadded using inversevariance weighting. For the selected forest samples, 17% of the quasars have duplicate spectra (less than 2% have more than one reobservation) taken with the BOSS spectrograph.
Definition of Lyα and Lyβ regions of quasar spectrum in which we measured Lyα forest absorption.
The forest spectra are prepared for analysis by discarding pixels which were flagged as problematic in the flux calibration or sky subtraction by the pipeline. We mask pixels around bright sky lines using the condition 10^{4}log_{10}(λ/λ_{sky}) ≤ 1.5, where λ_{sky} is the wavelength at the pixel center of the sky line where the pipeline sky subtraction is found to be inaccurate. Finally, we double the mask width to remove pixels around the observed CaII H&K lines arising from absorption by the interstellar medium of the Milky Way.
Forests featuring identified damped Lyα systems (DLAs) are given a special treatment. We use an updated (DR14) version of the DLA catalog of DR9 (Noterdaeme et al. 2012). The DLA detection and estimation of the neutralhydrogen column density N_{HI} was based on correlating observed spectra with synthetic spectra. The effective threshold for DLA detection depends on the signaltonoise ratio (and therefore on redshift) but is typically log_{10}N_{HI} ≈ 20.3 for spectra with S/N > 3 for which the efficiency and purity are ≈95%. For the purposes of the measurement of the correlation function, all pixels in the DLA where the transmission is less than 20% are masked and the absorption in the wings is corrected using a Voigt profile following the procedure of Lee et al. (2013). The effect on the correlation function of undetected DLAs or more generally of highcolumndensity (HCD) systems with log_{10}N_{HI} > 17.2 are modeled in the theoretical power spectrum, as described in Sect. 5.3.
To facilitate the computation of the crosscorrelation, we follow the approach in Bautista et al. (2017) to combine three adjacent pipeline pixels into wider “analysis pixels” defined as the inversevarianceweighted flux average. Requiring a minimum of 20 analysis pixels in each spectrum discards 2447 (6155) forests for the Lyα (Lyβ) region. Lastly, 3087 (1882) forests failed the continuumfitting procedure (see Sect. 3) for the Lyα (Lyβ) region by having negative continua due to their low spectral signaltonoise ratios. The final samples include 188 632 forests for the Lyα region and 68 613 forests for the Lyβ region. Figure 2 shows the redshift distributions for the tracer quasars and the Lyα absorption pixels. Our samples can be compared to those of dMdB17, which included 234 367 quasars (217 780 with 1.8 < z_{q} < 3.5) and 168 889 forests (157 845 with 2.0 < z_{q} < 3.5) over a wider redshift range.
3. The Lyα forest fluxtransmission field
The transmitted flux fraction F in a pixel of the forest region of quasar q is defined as the ratio of the observed flux density f_{q} with the continuum flux C_{q} (the flux density that would be observed in the absence of absorption). We will be studying the transmission relative to the mean value at the observed wavelength , and refer to this quantity as the “deltafield”:
We employ a similar method to the one established by previous Lyα forest BAO analyses (Busca et al. 2013; Delubac et al. 2015) in which the deltafield is derived by estimating the product for each quasar. Each spectrum is modeled assuming a uniform forest spectral template which is multiplied by a quasardependent linear function, setting the overall amplitude and slope, to account for the diversity of quasar luminosity and spectral shape:
where a_{q} and b_{q} are free parameters fit to the observed flux of the quasar. The forest spectral template is derived from the data as a weighted mean normalized flux, obtained by stacking the spectra in the quasar restframe. The continuum fitting procedure is handled separately for the Lyα and Lyβ regions.
The total variance of the deltafield is modeled as
where the noise variance . The first term represents the pipeline estimate of the flux variance, corrected by a function η(λ) that accounts for possible misestimation. The second term gives the contribution due to the largescale structure (LSS) and acts as a lower limit on the variance at high signaltonoise ratio. Lastly, the third term absorbs additional variance from quasar diversity apparent at high signaltonoise ratio. In bins of and observed wavelength, we measure the variance of the deltafield and fit for the values of η, and ϵ as a function of observed wavelength. These three functions are different for the Lyα and Lyβ regions. The procedure of stacking the spectra, fitting the continua and measuring the variance of δ is iterated, until the three functions converge. We find that five iterations is sufficient. Figure 3 presents an example spectrum and the bestfit model for the Lyα and Lyβ regions.
Fig. 3. Example spectrum of DR14Q quasar identified by (Plate, MJD, FiberID)=(7305, 56 991, 570) at z_{q} = 3.0. The blue line indicates the bestfit model for the Lyα region covering the restframe wavelength interval 104.0 < λ_{rf} < 120.0 nm. The red line indicates the same for the Lyβ region over the range 97.4 < λ_{rf} < 102.0 nm. The Lyα and Lyβ emission lines are located at λ_{α} = 121.567 nm and λ_{β} = 102.572 nm in the quasar restframe. The spectrum has not been rebinned into analysis pixels in this figure. 
As detailed in Bautista et al. (2017), the deltafield can be redefined in two steps to make exact the biases introduced by the continuum fitting procedure. In the first step, we define
where the overbars refer to weighted averages over individual forests. Next, we transform the by subtracting the weighted average at each observed wavelength:
4. The Lyα forest–quasar crosscorrelation
The threedimensional positions of the quasars and the Lyα forest deltafield are determined by their redshifts and angular positions on the sky. We transform the observed angular and redshift separations (Δθ, Δz) of the quasar–Lyα absorption pixel pairs into Cartesian coordinates (r_{⊥}, r_{∥}) assuming a spatially flat fiducial cosmology. The comoving separations along the line of sight r_{∥} (parallel direction) and transverse to the line of sight r_{⊥} (perpendicular direction) are calculated as
where D_{α} ≡ D_{c}(z_{α}) and D_{q} ≡ D_{c}(z_{q}) are the comoving distances to the Lyα absorption pixel and the quasar, respectively. Line of sight separations r_{∥} > 0 (< 0) thus correspond to background (foreground) absorption with respect to the tracer quasar position. In this paper, we will also refer to the coordinates (r, μ), where and μ = r_{∥}/r, the cosine of the angle of the vector r from the line of sight. The pair redshift is defined as z_{pair} = (z_{α} + z_{q})/2. A histogram of the pair redshifts is displayed in Fig. 4. We do not include pairs involving a quasar and pixels from its own forest in the crosscorrelation analysis, because the correlation of such pairs vanishes due to the continuum fit and deltafield redefinition (Eq. (4)).
Fig. 4. Redshift distribution of 9.7 × 10^{9} correlation pairs. The dashed vertical black line indicates the effective redshift of the BAO measurement, z_{eff} = 2.35, calculated as the weighted mean of the pair redshifts for separations in the range 80 < r < 120 h^{−1} Mpc. 
The fiducial cosmology used in the analysis is a flat ΛCDM model with parameter values taken from the Planck (2016) result for the TT+lowP combination (Planck Collaboration XIII 2016) described in Table 2. It is the same fiducial cosmology employed by dMdB17.
Parameters of flat ΛCDM fiducial cosmological model (Planck Collaboration XIII 2016).
4.1. Crosscorrelation
We estimate the crosscorrelation at a separation bin A, ξ_{A}, as the weighted mean of the deltafield in pairs of pixel i and quasar k at a separation within the bin A (FontRibera et al. 2012):
The weights w_{i} are defined as the inverse of the total pixel variance (see Eq. (3)), multiplied by redshift evolution factors for the forest and quasar, so as to approximately minimize the relative error on (Busca et al. 2013):
where γ_{α} = 2.9 (McDonald et al. 2006) and γ_{q} = 1.44 (du Mas des Bourboux et al. 2019). The validity of the correlation estimator, as well as the accuracy of the distortion matrix (Sect. 4.2) and covariance matrix estimation (Sect. 4.3) were tested and confirmed on simulated data in dMdB17.
Our separation grid consists of 100 bins of 4 h^{−1} Mpc for separations r_{∥} ∈ [ − 200, 200] h^{−1} Mpc in the parallel direction and 50 bins of 4 h^{−1} Mpc for separations r_{⊥} ∈ [0, 200] h^{−1} Mpc in the perpendicular direction; the total number of bins is N_{bin} = 5000. Each bin is defined by the weighted mean (r_{⊥}, r_{∥}) of the quasarpixel pairs of that bin, and its redshift by the weighted mean pair redshift. The mean redshifts range from z = 2.29 to z = 2.40. The effective redshift of the crosscorrelation measurement is defined to be the inversevarianceweighted mean of the redshifts of the bins with separations in the range 80 < r < 120 h^{−1} Mpc around the BAO scale. Its value is z_{eff} = 2.35.
Because the Lyβ transition is sufficiently separated in wavelength from the Lyα transition, corresponding to large physical separations > 441 h^{−1} Mpc for the wavelength range of the analysis, we neglect the contamination from Lyβ absorption interpreted as Lyα absorption. The total number of pairs of the crosscorrelation measurement is 9.7 × 10^{9}. The Lyα absorption in the Lyβ region contributes 1.2 × 10^{9} pairs (13%) and reduces the mean variance of the correlation function by 9% compared to the Lyα regiononly measurement. Our crosscorrelation measurement has 39% lower mean variance than the measurement of dMdB17.
4.2. Distortion matrix
The procedure used to estimate the deltafield (Sect. 3) suppresses fluctuations of characteristic scales corresponding to the forest length, since the estimate of the product (Eq. (1)) would typically erase such a fluctuation. The result is a suppression of the power spectrum in the radial direction on large scales (low k_{∥}). As illustrated in Fig. 11 of dMdB17, this induces a significant but smooth distortion of the correlation function on all relevant scales while leaving the BAO peak visually intact. As first noted in Slosar et al. (2011) and further investigated in Blomqvist et al. (2015), the distortion effect can be modeled in Fourier space as a multiplicative function of the radial component k_{∥} on the Lyα forest transmission power spectrum.
Here, we use the method introduced by Bautista et al. (2017) for the Lyα autocorrelation and adapted to the crosscorrelation by dMdB17 which allows one to encode the effect of this distortion on the correlation function in a distortion matrix. This approach, extensively validated in these publications using simulated data, uses the fact that that Eqs. (4) and (5) are linear in δ. This fact allows one to describe the measured correlation function for a separation bin A as a linear combination of the true correlation function for bins A′:
The distortion matrix D_{AA′} depends only on the geometry of the survey, the lengths of the forests and the pixel weights,
where the projection matrix
and δ^{K} is the Kronecker delta. The indices i and j in Eq. (11) refer to pixels from the same forest, k refers to a quasar, and the sums run over all pixelquasar pairs that contribute to the separation bins A and A′. The diagonal elements dominate the distortion matrix and are close to unity, D_{AA} ≈ 0.97, whereas the offdiagonal elements are small, D_{AA′} ≲ 0.03. We use the distortion matrix when performing fits of the measured crosscorrelation function (see Eq. (16)).
4.3. Covariance matrix
We estimate the covariance matrix of the crosscorrelation from the data by using the subsampling technique introduced by Busca et al. (2013) and adapted to the crosscorrelation by dMdB17. We divide the DR14 footprint of Fig. 1 into subsamples and measure the covariance from the variability across the subsamples. Such estimates of the covariance matrix are unbiased, but the noise due to the finite number of subsamples leads to biases in the inverse of the covariance (Joachimi et al. 2014). As was done in dMdB17, we smooth the noise by assuming, to good approximation, that the covariance between separation bins A and B depends only on the absolute difference .
We define the subsamples through a HEALPix (Górski et al. 2005) pixelization of the sky. A quasarabsorption pixel pair is assigned to a subsample s if the forest that contains the absorption belongs to that HEALPix pixel. We use HEALPix parameter nside=32, resulting in 3262 subsamples. Using fewer but larger HEALPix pixels (nside = 16; 876 subsamples) has no significant impact on the covariance matrix or the BAO peak position measurement.
The (noisy) covariance matrix is calculated as
where the sum runs over all subsamples and W_{A} is the sum of the pair weights w belonging to bin A,
From the covariance, we calculate the correlation matrix:
The smoothing procedure is applied to this correlation matrix by averaging as a function of (Δr_{∥}, Δr_{⊥}). The final covariance used in the fits is obtained by multiplying the smoothed correlation matrix by the diagonal elements of the original covariance matrix. Figure 5 displays the smoothed correlation matrix as a function of Δr_{∥} for the three lowest values of Δr_{⊥}.
Fig. 5. Smoothed correlation matrix from subsampling as a function of Δr_{∥} = r_{∥, A} − r_{∥, B}. The curves are for constant Δr_{⊥} = r_{⊥, A} − r_{⊥, B} for the three lowest values Δr_{⊥} = [0,4,8] h^{−1} Mpc. The right panel shows an expansion of the region Δr_{∥} < 140 h^{−1} Mpc. 
5. Model of the crosscorrelation
We fit the measured crosscorrelation function, , in the (r_{⊥}, r_{∥}) bin A, to a cosmological correlation function :
where D_{AA′} is the distortion matrix (Eq. (11)). The broadband term, , is an optional function used to test for imperfections in the model and for systematic errors. The set of parameters for the model is summarized in Table 3. The model is calculated at the weighted mean (r_{⊥}, r_{∥}) and redshift of each bin of the correlation function. Because of the relatively narrow redshift distribution of the bins (Δz = 0.11), most model parameters can be assumed as redshift independent to good accuracy.
Parameters of crosscorrelation model.
The cosmological crosscorrelation function is the sum of several contributions
The first term represents the standard correlation between quasars, q, and Lyα absorption in the IGM. It is the most important part of the correlation function and, used by itself, would lead to an accurate determination of the BAO peak position (see results in Sect. 6).
The remaining terms in Eq. (17) represent subdominant effects but contribute toward improving the fit of the correlation function outside the BAO peak. The second term is the sum over correlations from metal absorbers in the IGM. The third term represents Lyα absorption by high column density systems (HCDs). The fourth term is the correlation from the effect of a quasar’s radiation on a neighboring forest (“transverse proximity effect”). The fifth term is a relativistic correction leading to oddℓ multipoles in the correlation function, and the final term includes other sources of oddℓ multipoles (Bonvin et al. 2014). These terms will be described in detail below.
5.1. Quasar–Lyα correlation term
The quasar–Lyα crosscorrelation, ξ^{qα}, is the dominant contribution to the cosmological crosscorrelation. It is assumed to be a biased version of the total matter autocorrelation of the appropriate flat ΛCDM model, separated into a smooth component and a peak component to free the position of the BAO peak:
where A_{peak} is the BAO peak amplitude. The anisotropic shift of the observed BAO peak position relative to the peak position of the fiducial cosmological model from Table 2 is described by the lineofsight and transverse scale parameters
The nominal correlation function, ξ^{qα}(r_{⊥}, r_{∥}, α_{⊥} = α_{∥} = 1), is the Fourier transform of the quasar–Lyα crosspower spectrum:
where k = (k_{‖}, k_{⊥}) is the wavenumber of modulus k with components k_{∥} along the line of sight and k_{⊥} across, and μ_{k} = k_{∥}/k is the cosine of the angle of the wavenumber from the line of sight. As described in detail below, P_{QL} is the (quasi) linear matter spectrum, d_{q} and d_{Lyα} are the standard lineartheory factors describing the tracer bias and redshiftspace distortion (Kaiser 1987), V_{NL} describes further nonlinear corrections not included in P_{QL}, and G(k) gives the effects of (r_{⊥}, r_{∥}) binning on the measurement.
The first term in ((20)) provides for the aforementioned decoupling of the peak component (Eq. (18)):
where the smooth component, P_{sm}, is derived from the linear power spectrum, P_{L}(k, z), via the sideband technique (Kirkby et al. 2013) and P_{peak} = P_{L} − P_{sm}. The redshiftdependent linear power spectrum is obtained from CAMB (Lewis et al. 2000) with the fiducial cosmology.
The correction for nonlinear broadening of the BAO peak is parameterized by Σ = (Σ_{‖}, Σ_{⊥}), with Σ_{⊥} = 3.26 h^{−1} Mpc and
where is the linear growth rate of structure.
The second term in (20) describes the quasar bias and redshiftspace distortion
Because the fit of the crosscorrelation is only sensitive to the product of the quasar and Lyα biases, we set b_{q} ≡ b_{q}(z_{eff}) = 3.77 and assume a redshift dependence of the quasar bias given by (Croom et al. 2005)
The quasar redshiftspace distortion, assumed to be redshift independent, is
Setting f = 0.969 for our fiducial cosmology yields β_{q} = 0.257.
The third term in (20) is the Lyα forest factor,
We assume that the transmission bias evolves with redshift as
with γ_{α} = 2.9 (McDonald et al. 2006), while β_{α} is assumed to be redshift independent. We choose to fit for β_{α} and the velocity gradient bias of the Lyα forest:
Beyond our standard treatment of the Lyα transmission bias, we also consider the effect of fluctuations of ionizing UV radiation which lead to a scaledependence of b_{α} (Pontzen 2014; Gontcho A Gontcho et al. 2014):
where W(x)=arctan(x)/x (following the parameterization of Gontcho A Gontcho et al. 2014). Our standard fit does not include the effect of UV fluctuations due to its minor contribution to the fit quality. A fit that includes the UV modeling is presented in Table A.1 for which we fix the UV photon mean free path λ_{UV} = 300 h^{−1} Mpc (Rudie et al. 2013) and (Gontcho A Gontcho et al. 2014), and fit for b_{Γ}, as was done in dMdB17.
The effect of quasar nonlinear velocities and statistical redshift errors on the power spectrum is modeled as a Lorentz damping (Percival & White 2009),
where σ_{v} is a free parameter.
The last term in (20), G(k), accounts for smoothing due to the binning of the measurement of the correlation function (Bautista et al. 2017). We use
where R_{∥} and R_{⊥} are the scales of the smoothing. In the transverse direction, this form is not exact, but we have verified that it generates a sufficiently accurate correlation function. We fix both to the bin width, R_{∥} = R_{⊥} = 4 h^{−1} Mpc.
Systematic errors in the quasar redshift estimates lead to a shift of the crosscorrelation along the line of sight which is accounted for in the fit using the free parameter
5.2. Quasar–metal correlation terms
Absorption by metals in the intergalactic medium (e.g., Pieri et al. 2014) with similar restframe wavelengths to Lyα yields a subdominant contribution to the measured crosscorrelation. Assuming that these contaminant absorptions have redshifts corresponding to Lyα absorption results in an apparent shift of the quasar–metal crosscorrelations along the line of sight in the observed crosscorrelation. Following Blomqvist et al. (2018), metal correlations are modeled as
where
is a “metal distortion matrix” that allows us to calculate the shifted quasar–metal crosscorrelation function for a given nonshifted quasar–metal crosscorrelation function. The condition (i, k)∈A refers to pixel distances calculated using z_{α}, but (i, k)∈B refers to pixel distances calculated using z_{m}. For each metal absorption line, the (nonshifted) quasar–metal correlation is modeled using (20) with d_{α} replaced by
The metal absorption lines included in the fit are listed in Table 4. Because the redshiftspace distortion parameter of each metal is poorly determined in the fit, we fix β_{m} = 0.5, the value derived for DLA host halos (FontRibera et al. 2012; PérezRàfols et al. 2018). Transmission biases are assumed to evolve with redshift as a powerlaw with exponent γ_{m} = 1, similar to the measured evolution of the CIV bias (Blomqvist et al. 2018), but our results are not sensitive to this choice.
Most important metal absorptions of intergalactic medium that imprint correlations observed in Lyα–quasar crosscorrelation for r_{∥} ∈ [ − 200, 200] h^{−1} Mpc.
5.3. Other correlation terms
The presence of HCDs in the absorption spectra modifies the expected correlation function. The flux transmission of spectra with identified DLAs are estimated by masking the strong absorption regions (transmission less than 20%) and correcting the wings using a Voigt profile following the procedure of Lee et al. (2013). If this procedure worked perfectly, we would expect no strong modification of the power spectrum. However, it does not operate for HCDs below the nominal threshold of log N_{HI} ≈ 20, and even above this threshold the detection efficiency depends on the signaltonoise ratio of the spectrum. These imperfections modify the expected power spectrum.
We model the correlations due to absorption by unidentified HCD systems by adding to the power spectrum a term with the same form as the usual Lyα correlations (Eq. (20)) but with d_{α} replaced by
where the bias and the redshiftspace distortion β_{HCD} are free parameters in the fit. The function F_{HCD}(L_{HCD}k_{∥}) describes the suppression of power at large k_{∥} due to unidentified HCDs of typical extent L_{HCD}. The studies of mock data sets by Bautista et al. (2017) tried several functional forms and F = sinc(L_{HCD}k_{∥}) was adopted by them and by dMdB17, though other forms gave similar results. Following the more detailed studies of Rogers et al. (2018), we choose to use the form F = exp(−L_{HCD}k_{∥}).
Our DLAidentification procedure requires their width (wavelength interval for absorption greater than 20% ) to be above ∼2.0 nm, corresponding to ∼14 h^{−1} Mpc. Following the study of Rogers et al. (2018), the corresponding unidentified HCD systems are wellmodeled with L_{HCD} = 10 h^{−1} Mpc and we fix L_{HCD} to this value in the fits. We have verified that varying this parameter over the range 5 < L_{HCD} < 15 h^{−1} Mpc does not change the fit position of the BAO peak. Due to degeneracies, we add a Gaussian prior on β_{HCD} of mean 0.5 and standard deviation 0.2.
The term in Eq. (17) representing the transverse proximity effect takes the form (FontRibera et al. 2013):
This form supposes isotropic emission from the quasars. We fix λ_{UV} = 300 h^{−1} Mpc (Rudie et al. 2013) and fit for the amplitude .
In addition to accounting for asymmetries in the crosscorrelation introduced by metal absorptions, continuumfitting distortion and systematic redshift errors, the standard fit includes modeling of relativistic effects (Bonvin et al. 2014). The relativistic correction in (17) is the sum of two components describing a dipole and an octupole,
where L_{1} and L_{3} are the Legendre polynomial of degree 1 and 3 respectively, A_{rel1} and A_{rel3} are the amplitudes, and
where j_{ℓ} is the spherical Bessel function. The relativistic dipole is expected to be the dominant contribution of oddℓ asymmetry and our standard fit therefore neglects the relativistic octupole (A_{rel3} = 0).
Dipole and octupole asymmetries also arise in the “standard” correlation function due to the evolution of the tracer biases and growth factor, as well as from the wideangle correction (Bonvin et al. 2014):
where
Here, the two amplitudes A_{asy0} and A_{asy2} determine the dipole contribution, while A_{asy3} is the octupole amplitude. The ξ^{asy} term is neglected in the standard fit, but we check the robustness of the BAO measurement with respect to the oddℓ multipoles in Table A.1.
5.4. Broadband function
The optional ξ^{bb} term of (16) is a “broadband function” that is a slowly varying function of (r_{∥}, r_{⊥}):
where L_{j} is the Legendre polynomial of degree j. Its purpose is to account for unknown physical, instrumental or analytical effects missing in the model that could potentially impact the BAO measurement. The standard fit features no broadband function. The result of adding a broadband function of the form (i_{min}, i_{max}, j_{min}, j_{max}) = (0, 2, 0, 6) is presented in Table A.1.
6. Fits of the crosscorrelation
Our standard fit of the crosscorrelation function uses the 14 parameters in the first group of Table 3. The fit includes 3180 data bins in the range 10 < r < 180 h^{−1} Mpc. The bestfit values are presented in the column “Lyα–quasar” of Table 5. Figure 6 shows the best fit for four ranges of μ and Fig. 7 for the two lowest r_{⊥} bins.
Fig. 6. Crosscorrelation function averaged in four ranges of μ = r_{∥}/r. The red curves show the bestfit model of the standard fit obtained for the fitting range 10 < r < 180 h^{−1} Mpc. The curves have been extrapolated outside this range. 
Fig. 7. Crosscorrelation function as a function of r_{∥} for two lowest values r_{⊥} = [2, 6] h^{−1} Mpc. The red curves indicate the bestfit model of the standard fit obtained for the fitting range 10 < r < 180 h^{−1} Mpc. The curves have been extrapolated outside this range. The imprints of quasar–metal correlations are visible as peaks indicated by the dashed black lines at r_{∥} ≈ −21 h^{−1} Mpc (SiIII(120.7)), r_{∥} ≈ −53 h^{−1} Mpc (SiII(119.0)), r_{∥} ≈ −59 h^{−1} Mpc (SiII(119.3)), and r_{∥} ≈ +103 h^{−1} Mpc (SiII(126.0)). 
Fit results for crosscorrelation, autocorrelation of de Sainte Agathe et al. (2019), and combined fit.
Constraints on the BAO parameters (α_{⊥}, α_{∥}) are presented in Fig. 8. Following the method introduced and described in detail in dMdB17, we estimate the relation between and confidence levels for the BAO parameters using a large number of simulated correlation functions generated from the bestfit model and the covariance matrix measured with the data. The results of the study, summarized in Table 6, indicate that the (68.27,95.45%) confidence levels for (α_{⊥}, α_{∥}) correspond to Δχ^{2} = (2.51, 6.67) (instead of the nominal values Δχ^{2} = (2.3, 6.18)). These levels are shown as contours in Fig. 8. The bestfit values and confidence level (68.27,95.45%) ranges are:
Fig. 8. Constraints on (α_{∥}, α_{⊥}) for crosscorrelation (red) and combination with autocorrelation (black). Contours correspond to confidence levels of (68.27%,95.45%). The black point at (α_{∥}, α_{⊥})=(1, 1) indicates the prediction of the Planck (2016) bestfit flat ΛCDM cosmology. The effective redshift of the combined fit is z_{eff} = 2.34 where the fiducial distance ratios are (D_{M}/r_{d}, D_{H}/r_{d})=(39.26, 8.58). 
Values of Δχ^{2} corresponding to CL = (68.27, 95.45%).
corresponding to
These results are consistent at 1.5 standard deviations with the prediction of the Planck (2016) bestfit flat ΛCDM model. Using a model without the BAO peak (A_{peak} = 0) degrades the quality of the fit by Δχ^{2} = 22.48.
Our BAO constraints can be compared with the DR12 measurement of dMdB17 at a slightly higher redshift: D_{M}(2.40)/r_{d} = 35.7 ± 1.7 and D_{H}(2.40)=9.01 ± 0.36 corresponding to α_{⊥} = 0.898 ± 0.042 and α_{∥} = 1.077 ± 0.042, relative to the same Planck model. The results (43) and (44) thus represent a movement of ∼0.3σ toward the Planckinspired model through a shift in α_{⊥}. As a crosscheck of the results, we apply our analysis to the DR12 data set of dMdB17, without including the absorption in the Lyβ region. The bestfit values are α_{⊥} = 0.889 ± 0.040 and α_{∥} = 1.080 ± 0.039 (errors correspond to Δχ^{2} = 1), in good agreement with the measurement of dMdB17. This result indicates that the movement toward the fiducial model in DR14 is driven by the data.
Model predictions for D_{M}/r_{d} and D_{H}/r_{d} depend both on prerecombination physics, which determine r_{d}, and on latetime physics, which determine D_{M} and D_{H}. Taking the ratio, yielding the Alcock–Paczyński parameter F_{AP} = D_{M}/D_{H} (Alcock & Paczynski 1979), isolates the latetime effects which, in the ΛCDM model depend only on (Ω_{m}, Ω_{Λ}). We find
where the Δχ^{2} curve is shown in Fig. 9 and we have adopted that the (68.27,95.45%) confidence levels correspond to Δχ^{2} = (1.13, 4.74) (instead of the nominal values Δχ^{2} = (1, 4)). This result is 1.8 standard deviations from the prediction of the Planckinspired model, F_{AP}(z = 2.35) = 4.60.
Fig. 9. Constraints on Alcock–Paczyński parameter F_{AP} for crosscorrelation (red) and combination with autocorrelation (black). Confidence levels of (68.27%,95.45%) are indicated with the horizontal dotted lines for the crosscorrelation and dashed lines for the combined fit. The prediction of the Planck (2016) bestfit flat ΛCDM cosmology is indicated with the vertical dotted line at F_{AP}(z = 2.35)=4.60 for the crosscorrelation and dashed line at F_{AP}(z = 2.34)=4.57 for the combined fit. 
The fit values of the Lyα bias parameters, b_{ηα} = −0.267 ± 0.014 and β_{α} = 2.28 ± 0.31 are consistent with those found by dMdB17, b_{ηα} = −0.23 ± 0.02 and β_{α} = 1.90 ± 0.34. These parameters can also be determined from the Lyα autocorrelation and our value of β_{α} is consistent with that found with the autocorrelation function, β_{α} = 1.93 ± 0.10 (de Sainte Agathe et al. 2019). However, these values are not in good agreement with the value β_{α} = 1.656 ± 0.086 found earlier by Bautista et al. (2017). The auto and crosscorrelations values of b_{ηα} also differ by ∼20%: −0.267 ± 0.014 for the cross correlation and −0.211 ± 0.004 for the autocorrelation. Furthermore, the bias parameters are not in good agreement with recent simulations (ArinyoiPrats et al. 2015) which predict β_{α} ≈ 1.4 and b_{ηα} in the range 0.14 to 0.20. Since our quoted uncertainties on the bias parameters (not on BAO parameters) come from approximating the likelihood as Gaussian, they might be underestimated in the presence of nontrivial correlations between the parameters. A dedicated study would be necessary to further investigate the consistency between the measured and predicted values. Fortunately, the bias parameters describe mostly the smooth component of the correlation function and do not significantly influence the BAO parameters (α_{⊥}, α_{∥}), as indicated by the nonstandard fits discussed below and summarized in Table A.1.
The fit of the crosscorrelation prefers a vanishing contribution from the quasar–HCD correlation term (b_{HCD} ≈ 0). This preference is in contrast to the Lyα autocorrelation of de Sainte Agathe et al. (2019) where the HCD model is a crucial element to obtain a good fit (but does not affect the BAO peak position measurement). The bestfit radial coordinate shift Δr_{∥} is consistent with zero systematic redshift error, but the parameter is strongly correlated with the amplitude of the relativistic dipole. Setting A_{rel1} = 0 in the fit yields Δr_{∥} = −0.92 ± 0.12, in good agreement with the value reported in dMdB17. The best fit suggests marginal support for a nonzero value of A_{rel1}, and the combined fit increases the significance of this result. However, even with sufficient statistical significance, its correlation with Δr_{∥} (as well as other potential systematic errors on A_{rel1}) prevents claims of a discovery of relativistic effects. The parameters σ_{v} and have bestfit values in agreement with the result of dMdB17.
Among the metals, only SiIII(120.7) has a bias parameter significantly different from zero (> 4σ) to show evidence for largescale correlations with quasars. The imprints of the metal correlations are visible in the lineofsight direction in Fig. 7.
Besides the standard approach, we have also performed nonstandard analyses, described in Appendix A, to search for unexpected systematic errors in the BAO peakposition measurement. The results for the nonstandard fits of the crosscorrelation are summarized in Table A.1. No significant changes of the bestfit values of (α_{⊥}, α_{∥}) are observed. We have also divided the data to perform fits of the crosscorrelation for a low and a highredshift bin as described in Appendix B. These fits are summarized in Table B.1 and yield consistent bestfit BAO parameters for the two bins.
7. Combination with the Lyα autocorrelation
We combine our measurement of the Lyα–quasar crosscorrelation with the DR14 Lyα autocorrelation of de Sainte Agathe et al. (2019) by performing a combined fit of the correlation functions. For the autocorrelation, we use the combination Lyα(Lyα) × Lyα(Lyα) + Lyα(Lyα) × Lyα(Lyβ) described in de Sainte Agathe et al. (2019). Because the covariance between the auto and crosscorrelation is sufficiently small to be ignored, as studied in Delubac et al. (2015) and dMdB17, we treat their errors as independent. The combined fit uses the standard fit model of each analysis. In addition to the 14 free parameters in Table 3, we let free the redshiftspace distortion of quasars, β_{q}, and the autocorrelation fit introduces three additional bias parameters (b_{CIV(154.9)}, , ), for a total of 18 free parameters. The effective redshift of the combined fit is z_{eff} = 2.34.
The bestfit results are presented in the column “combined” of Table 5. Figure 8 displays the constraints on (α_{⊥}, α_{∥}) from the combined measurement as black contours indicating the (68.27,95.45%) confidence levels (corresponding to Δχ^{2} = (2.47, 6.71); see Table 6). The combined constraints on the BAO parameters are:
corresponding to
These results are within 1.7 standard deviations of the prediction of the Planck (2016) bestfit flat ΛCDM model. This movement of ∼0.6σ toward the Planck prediction compared to the DR12 combinedfit result of dMdB17 is a consequence of the auto and crosscorrelation results individually moving toward the fiducial model.
Figure 9 shows the Δχ^{2} curve for the Alcock–Paczyński parameter from the combined fit, for which the (68.27,95.45%) confidence levels correspond to Δχ^{2} = (1.11, 4.39). The combined constraint is
within 2.1 standard deviations of the value F_{AP}(z = 2.34)=4.57 expected in the Planckinspired model.
8. Implications for cosmological parameters
The combinedfit measurement of (D_{M}/r_{d}, D_{H}/r_{d}) at z = 2.34 presented here is within 1.7 standard deviations of the predictions of the flat ΛCDM model favored by the measurement of CMB anisotropies (Planck Collaboration XIII 2016). This result thus does not constitute statistically significant evidence for new physics or unidentified systematic errors in the measurement. Figure 10 illustrates the agreement with the Planck prediction for the ensemble of BAO measurements.
Fig. 10. Measurements of D_{M}/r_{d}, D_{H}/r_{d} and D_{V}/r_{d} at various redshifts: 6dFGS (Beutler et al. 2011), SDSS MGS (Ross et al. 2015), BOSS galaxies (Alam et al. 2017), eBOSS Galaxies (Bautista et al. 2018), eBOSS quasars (Ata et al. 2018), eBOSS Lyα–Lyα (de Sainte Agathe et al. 2019), and eBOSS Lyα–quasars (this work). For clarity, the Lyα–Lyα results at z = 2.34 and the Lyα–quasar results at z = 2.35 have been separated slightly in the horizontal direction. Error bars represent 1σ uncertainties. 
Independent of CMB data and without assuming flatness, the BAO data by themselves constrain the parameters (Ω_{m}, Ω_{Λ}, H_{0}r_{d}) of the (o)ΛCDM model. Using the combined fit (Eqs. (50) and (51)), the galaxy data of Alam et al. (2017), Beutler et al. (2011) Ross et al. (2015) and Bautista et al. (2018) and the quasar data of Ata et al. (2018) yields
corresponding to Ω_{k} = 0.032 ± 0.117. The best fit gives (c/H_{0})/r_{d} = 29.78 ± 0.56 corresponding to hr_{d} = (0.683 ± 0.013)×147.33 Mpc. The CMB inspired flat ΛCDM model has χ^{2} = 13.76 for 12 degrees of freedom and is within one standard deviation of the best fit, as illustrated in Fig. 11.
Fig. 11. One and two standard deviation constraints on (Ω_{m}, Ω_{Λ}). The red contours use BAO measurements of D_{M}/r_{d} and D_{H}/r_{d} of this work, of de Sainte Agathe et al. (2019) and Alam et al. (2017), and the measurements of D_{V}/r_{d} of Beutler et al. (2011), Ross et al. (2015), Ata et al. (2018), and Bautista et al. (2018). The blue contours do not use the Lyα autocorrelation measurement of de Sainte Agathe et al. (2019). The green contours show the constraints from SNIa Pantheon sample (Scolnic et al. 2018). The black point indicates the values for the Planck (2016) bestfit flat ΛCDM cosmology. 
A value of H_{0} can be obtained either by using the CMB measurement of r_{d} or by using the PrimordialNucleosynethsis value of Ω_{b}h^{2} to constrain r_{d}. Adopting the value 100Ω_{b}h^{2} = 2.260 ± 0.034 derived from the deuterium abundance measurement of Cooke et al. (2018) and assuming flat ΛCDM, we derive the constraints on (H_{0}, Ω_{m}h^{2}) shown in Fig. 12 with
Fig. 12. One and two standard deviation constraints on H_{0} and Ω_{m}h^{2} derived from BAO data used in Fig. 11 and from BigBang Nucleosynthesis. This figure assumes a flat universe and a Gaussian prior 100Ω_{b}h^{2} = 2.260 ± 0.034 derived from the deuterium abundance measurement of Cooke et al. (2018). 
or h < 0.706 at 95% CL The limit degrades to h < 0.724 (95% CL) if one adopts a more conservative uncertainty on the baryon density: 100Ω_{b}h^{2} = 2.26 ± 0.20. Nevertheless, as previously noted (Aubourg et al. 2015; Addison et al. 2018), the combination of BAO and nucleosynthsis provides a CMBfree confirmation of the tension with the distanceladder determinations of H_{0} (Riess et al. 2016, 2018a,b).
9. Conclusions
Using the entirety of BOSS and the first two years of eBOSS observations from SDSS DR14, this paper has presented a measurement of the crosscorrelation of quasars and the Lyα flux transmission at redshift 2.35. In addition to the new and reobserved quasars provided in DR14, we have improved statistics further by extending the Lyα forest to include Lyα absorption in the Lyβ region of the spectra.
The position of the BAO peak is 1.5σ from the flat ΛCDM model favored by CMB anisotropy measurements (Planck Collaboration XIII 2016). We emphasize that the measured peak position shows no significant variation when adding astrophysical elements to the fit model. The basic Lyαonly model on its own provides an accurate determination of the peak position, while still yielding an acceptable fit to the data. Compared to the BAO measurement for the DR12 data set reported by dMdB17, our result represents a movement of ∼0.3σ toward the Planckcosmology prediction through a shift in the transverse BAO parameter α_{⊥}. This change is driven by the data and not by differences in the analyses. The inclusion of Lyα absorption in the Lyβ region has no impact on the bestfit value of α_{⊥}. Combined with the Lyα–fluxtransmission autocorrelation measurement presented in a companion paper (de Sainte Agathe et al. 2019), the BAO peak at z = 2.34 is 1.7σ from the expected value.
The ensemble of BAO measurements is in good agreement with the CMBinspired flat ΛCDM model. By themselves, the BAO data provide a good confirmation of this model. The use of SNIa to measure cosmological distances (Scolnic et al. 2018) provides independent measurements of the model parameters. As can be seen in Fig. 11 they are in agreement with the BAO measurements.
The BAO measurements presented here will be improved by the upcoming DESI (DESI Collaboration 2016) and WEAVEQSO (Pieri et al. 2016) projects both by increasing the number of quasars and improving the spectral resolution.
The bestfit results and the χ^{2} scans for the crosscorrelation by itself and the combination with the autocorrelation are publicly available^{3}.
Package for Igm CosmologicalCorrelations Analyses (picca) is available at https://github.com/igmhub/picca/
Acknowledgments
We thank Pasquier Noterdaeme for providing the DLA catalog for eBOSS DR14 quasars. This work was supported by the A*MIDEX project (ANR11IDEX000102) funded by the “Investissements d’Avenir” French Government program, managed by the French National Research Agency (ANR), and by ANR under contract ANR14ACHN0021. Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS acknowledges support and resources from the Center for HighPerformance Computing at the University of Utah. The SDSS web site is http://www.sdss.org/. SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, HarvardSmithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), MaxPlanckInstitut für Astronomie (MPIA Heidelberg), MaxPlanckInstitut für Astrophysik (MPA Garching), MaxPlanckInstitut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatório Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University. A.F.R. was supported by an STFC Ernest Rutherford Fellowship, grant reference ST/N003853/1, and by STFC Consolidated Grant no ST/R000476/1.
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Appendix A: Nonstandard fits of the crosscorrelation
The results of performing nonstandard fits of the crosscorrelation are summarized in Table A.1. The first group reports results obtained by successively adding elements to the model, starting with a model with only the standard Lyα correlation function and ending with the complete model of Table 5. Adding elements does not significantly change the bestfit values of (α_{⊥}, α_{∥}) while gradually improving the quality of the fit.
Results of nonstandard fits.
The second group of fits in Table A.1 adds elements to the standard fit in the form of fluctuations of the UV background radiation (Eq. (29)), the A_{rel3} term (Eq. (38)) and the other oddℓ terms from Eq. (40), or the broadband function (Eq. (42)). For these fits, we set ξ^{qHCD} = 0 to facilitate the parameter error estimation with no impact on the best fits. No significant changes of the BAO parameters are observed.
The third group of fits concern nonstandard data samples that either omits the correlation pairs from the Lyβ region (“no Lyβ”) or leaves the DLAs uncorrected in the spectra (“keep DLAs”). Even for these modified data samples the bestfit values of (α_{⊥}, α_{∥}) do not deviate significantly from those of our standard analysis.
Figure A.1 shows the measured crosscorrelation for four ranges of μ and three of the fits listed in Table A.1: the standard fit used to measure the BAO parameters, the basic Lyαonly fit, and the fit with the broadband function.
Fig. A.1. Same as Fig. 6 but showing three models fit to data. Red curves indicate the standard fit, blue curves the basic Lyαonly model, and green curves the standard fit (with ξ^{qHCD} = 0) with the addition of the broadband function (Eq. (42)) of the form (i_{min}, i_{max}, j_{min}, j_{max})=(0, 2, 0, 6). The curves have been extrapolated outside the fitting range 10 < r < 180 h^{−1} Mpc. 
Appendix B: Redshift split
The statistical limitations of the present data set are such that it is not possible to usefully measure the expected redshiftvariation of D_{M}(z)/r_{d} and D_{H}(z)/r_{d}. However, to search for unexpected effects, we perform an analysis that independently treats a low and a highredshift bin.
A quasar and entire forest pair is assigned to either bin depending on their mean redshift:
where z_{i, max} is the pixel with the highest absorption redshift in the forest. As the data split is defined, individual forests and quasars can contribute to both redshift bins. The limiting value of z_{m} is chosen so as to approximately equalize the correlation signaltonoise ratio (as determined by the bestfit fiducial correlation model) on BAO scales for the two redshift bins. This approach ensures that the redshift bins have similar statistical power for determining the BAO peak position. We set the limit at z_{m} = 2.48. After identifying which quasar–forest pairs contribute to each redshift bin, we rederive the delta fields for each bin separately to ensure that the mean deltas vanish. The effective redshifts are z_{eff} = 2.21 and z_{eff} = 2.58 for the lowz and highz bin, respectively. The pair redshift distribution for the lowz bin extends up to z = 2.48 (by definition) and its overlap with the distribution for the highz bin is Δz ≈ 0.25. Correlations between the redshift bins are at the per cent level.
The result of the data split is summarized in Table B.1. Figure B.1 shows the correlation functions and the bestfit models for four ranges of μ. The bestfit values of (α_{⊥}, α_{∥}) for the two bins are consistent, with similar BAO errors of ∼6%. The bias parameter b_{ηα} changes between the two redshifts by a factor 1.57 ± 0.15 consistent with the expected factor (3.58/3.21)^{2.9} = 1.37. The parameter β_{α} increases by a factor 1.6 ± 0.4, within two standard deviations of the predicted decrease of 6% from simulations of ArinyoiPrats et al. (2015).
Fit results for two redshift bins.
Fig. B.1. Crosscorrelation function averaged in four ranges of μ = r_{∥}/r for the fitting range 10 < r < 180 h^{−1} Mpc. The blue points are the data for the lowz bin (z_{m} < 2.48) and the blue curve the bestfit model. The red points are the data for the highz bin (z_{m} > 2.48) and the red curve the bestfit model. 
All Tables
Definition of Lyα and Lyβ regions of quasar spectrum in which we measured Lyα forest absorption.
Parameters of flat ΛCDM fiducial cosmological model (Planck Collaboration XIII 2016).
Most important metal absorptions of intergalactic medium that imprint correlations observed in Lyα–quasar crosscorrelation for r_{∥} ∈ [ − 200, 200] h^{−1} Mpc.
Fit results for crosscorrelation, autocorrelation of de Sainte Agathe et al. (2019), and combined fit.
All Figures
Fig. 1. Sky distribution for sample of 266 590 tracer quasars (1.77 < z_{q} < 3.5) from DR14Q in J2000 equatorial coordinates. The solid black curve is the Galactic plane. The highdensity regions are the eBOSS and SEQUELS observations (for the northern regions of the two Galactic hemispheres) and SDSSstripe 82 (for declination δ ∼ 0). The discontiguous small areas contain only SDSS DR7 quasars. 

In the text 
Fig. 2. Normalized redshift distributions for tracer quasars (black) and Lyα forest absorption pixels of Lyα region (blue) and Lyβ region (red). The histograms include 266 590 tracer quasars, 30.2 × 10^{6} pixels in the Lyα region, and 4.0 × 10^{6} pixels in the Lyβ region. The vertical dashed lines show the mean value of each distribution: (tracer quasars), 2.37 (in Lyα), 2.26 (in Lyβ). 

In the text 
Fig. 3. Example spectrum of DR14Q quasar identified by (Plate, MJD, FiberID)=(7305, 56 991, 570) at z_{q} = 3.0. The blue line indicates the bestfit model for the Lyα region covering the restframe wavelength interval 104.0 < λ_{rf} < 120.0 nm. The red line indicates the same for the Lyβ region over the range 97.4 < λ_{rf} < 102.0 nm. The Lyα and Lyβ emission lines are located at λ_{α} = 121.567 nm and λ_{β} = 102.572 nm in the quasar restframe. The spectrum has not been rebinned into analysis pixels in this figure. 

In the text 
Fig. 4. Redshift distribution of 9.7 × 10^{9} correlation pairs. The dashed vertical black line indicates the effective redshift of the BAO measurement, z_{eff} = 2.35, calculated as the weighted mean of the pair redshifts for separations in the range 80 < r < 120 h^{−1} Mpc. 

In the text 
Fig. 5. Smoothed correlation matrix from subsampling as a function of Δr_{∥} = r_{∥, A} − r_{∥, B}. The curves are for constant Δr_{⊥} = r_{⊥, A} − r_{⊥, B} for the three lowest values Δr_{⊥} = [0,4,8] h^{−1} Mpc. The right panel shows an expansion of the region Δr_{∥} < 140 h^{−1} Mpc. 

In the text 
Fig. 6. Crosscorrelation function averaged in four ranges of μ = r_{∥}/r. The red curves show the bestfit model of the standard fit obtained for the fitting range 10 < r < 180 h^{−1} Mpc. The curves have been extrapolated outside this range. 

In the text 
Fig. 7. Crosscorrelation function as a function of r_{∥} for two lowest values r_{⊥} = [2, 6] h^{−1} Mpc. The red curves indicate the bestfit model of the standard fit obtained for the fitting range 10 < r < 180 h^{−1} Mpc. The curves have been extrapolated outside this range. The imprints of quasar–metal correlations are visible as peaks indicated by the dashed black lines at r_{∥} ≈ −21 h^{−1} Mpc (SiIII(120.7)), r_{∥} ≈ −53 h^{−1} Mpc (SiII(119.0)), r_{∥} ≈ −59 h^{−1} Mpc (SiII(119.3)), and r_{∥} ≈ +103 h^{−1} Mpc (SiII(126.0)). 

In the text 
Fig. 8. Constraints on (α_{∥}, α_{⊥}) for crosscorrelation (red) and combination with autocorrelation (black). Contours correspond to confidence levels of (68.27%,95.45%). The black point at (α_{∥}, α_{⊥})=(1, 1) indicates the prediction of the Planck (2016) bestfit flat ΛCDM cosmology. The effective redshift of the combined fit is z_{eff} = 2.34 where the fiducial distance ratios are (D_{M}/r_{d}, D_{H}/r_{d})=(39.26, 8.58). 

In the text 
Fig. 9. Constraints on Alcock–Paczyński parameter F_{AP} for crosscorrelation (red) and combination with autocorrelation (black). Confidence levels of (68.27%,95.45%) are indicated with the horizontal dotted lines for the crosscorrelation and dashed lines for the combined fit. The prediction of the Planck (2016) bestfit flat ΛCDM cosmology is indicated with the vertical dotted line at F_{AP}(z = 2.35)=4.60 for the crosscorrelation and dashed line at F_{AP}(z = 2.34)=4.57 for the combined fit. 

In the text 
Fig. 10. Measurements of D_{M}/r_{d}, D_{H}/r_{d} and D_{V}/r_{d} at various redshifts: 6dFGS (Beutler et al. 2011), SDSS MGS (Ross et al. 2015), BOSS galaxies (Alam et al. 2017), eBOSS Galaxies (Bautista et al. 2018), eBOSS quasars (Ata et al. 2018), eBOSS Lyα–Lyα (de Sainte Agathe et al. 2019), and eBOSS Lyα–quasars (this work). For clarity, the Lyα–Lyα results at z = 2.34 and the Lyα–quasar results at z = 2.35 have been separated slightly in the horizontal direction. Error bars represent 1σ uncertainties. 

In the text 
Fig. 11. One and two standard deviation constraints on (Ω_{m}, Ω_{Λ}). The red contours use BAO measurements of D_{M}/r_{d} and D_{H}/r_{d} of this work, of de Sainte Agathe et al. (2019) and Alam et al. (2017), and the measurements of D_{V}/r_{d} of Beutler et al. (2011), Ross et al. (2015), Ata et al. (2018), and Bautista et al. (2018). The blue contours do not use the Lyα autocorrelation measurement of de Sainte Agathe et al. (2019). The green contours show the constraints from SNIa Pantheon sample (Scolnic et al. 2018). The black point indicates the values for the Planck (2016) bestfit flat ΛCDM cosmology. 

In the text 
Fig. 12. One and two standard deviation constraints on H_{0} and Ω_{m}h^{2} derived from BAO data used in Fig. 11 and from BigBang Nucleosynthesis. This figure assumes a flat universe and a Gaussian prior 100Ω_{b}h^{2} = 2.260 ± 0.034 derived from the deuterium abundance measurement of Cooke et al. (2018). 

In the text 
Fig. A.1. Same as Fig. 6 but showing three models fit to data. Red curves indicate the standard fit, blue curves the basic Lyαonly model, and green curves the standard fit (with ξ^{qHCD} = 0) with the addition of the broadband function (Eq. (42)) of the form (i_{min}, i_{max}, j_{min}, j_{max})=(0, 2, 0, 6). The curves have been extrapolated outside the fitting range 10 < r < 180 h^{−1} Mpc. 

In the text 
Fig. B.1. Crosscorrelation function averaged in four ranges of μ = r_{∥}/r for the fitting range 10 < r < 180 h^{−1} Mpc. The blue points are the data for the lowz bin (z_{m} < 2.48) and the blue curve the bestfit model. The red points are the data for the highz bin (z_{m} > 2.48) and the red curve the bestfit model. 

In the text 
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