The large-scale Quasar-Lyman α Forest Cross-Correlation from BOSS
Andreu Font-Ribera, Eduard Arnau, Jordi Miralda-Escudé, Emmanuel Rollinde, J. Brinkmann, Joel R. Brownstein, Khee-Gan Lee, Adam D. Myers, Nathalie Palanque-Delabrouille, Isabelle Pâris, Patrick Petitjean, James Rich, Nicholas P. Ross, Donald P. Schneider, Martin White
aa r X i v : . [ a s t r o - ph . C O ] S e p Prepared for submission to JCAP
The large-scale Quasar-Lyman α Forest Cross-Correlation from BOSS
Andreu Font-Ribera a,b , Eduard Arnau c , Jordi Miralda-Escud´e d,c ,Emmanuel Rollinde e , J. Brinkmann f , Joel R. Brownstein g ,Khee-Gan Lee h , Adam D. Myers i , NathaliePalanque-Delabrouille j , Isabelle Pˆaris k , Patrick Petitjean e ,James Rich j , Nicholas P. Ross b , Donald P. Schneider l,m andMartin White b,n a Institute of Theoretical Physics, University of Zurich, 8057 Zurich, Switzerland b Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA, USA c Institut de Ci`encies del Cosmos (IEEC/UB), Barcelona, Catalonia d Instituci´o Catalana de Recerca i Estudis Avan¸cats, Catalonia e Universit´e Paris 6 et CNRS, Institut d’Astrophysique de Paris, 98bis blvd. Arago, 75014Paris, France f Apache Point Observatory, P.O. Box 59, Sunspot, NM 88349, USA g Department of Physics and Astronomy, University of Utah, 115 S 1400 E, Salt Lake City,UT 84112, USA h Max-Planck-Institut f¨ur Astronomie, K¨onigstuhl 17, D-69117 Heidelberg, Germany i Department of Physics and Astronomy, University of Wyoming, Laramie, WY 82071, USA j CEA, Centre de Saclay, IRFU, 91191 Gif-sur-Yvette, France k Departamento de Astronom´ıa, Universidad de Chile, Casilla 36-D, Santiago, Chile l Department of Astronomy and Astrophysics, The Pennsylvania State University, UniversityPark, PA 16802, USA m Institute for Gravitation and the Cosmos, The Pennsylvania State University, UniversityPark, PA 16802, USA n Departments of Physics and Astronomy, 601 Campbell Hall, University of California Berke-ley, CA 94720, USA-mail: [email protected]
Abstract.
We measure the large-scale cross-correlation of quasars with the Ly α forestabsorption in redshift space, using ∼ r of 80 h − Mpc. For r > h − Mpc,we show that the cross-correlation is well fitted by the linear theory prediction for the meanoverdensity around a quasar host halo in the standard ΛCDM model, with the redshiftdistortions indicative of gravitational evolution detected at high confidence. Using previousdeterminations of the Ly α forest bias factor obtained from the Ly α autocorrelation, we inferthe quasar bias factor to be b q = 3 . +0 . − . at a mean redshift z = 2 .
38, in agreement withprevious measurements from the quasar auto-correlation. We also obtain a new estimate ofthe Ly α forest redshift distortion factor, β F = 1 . ± .
15, slightly larger than but consistentwith the previous measurement from the Ly α forest autocorrelation. The simple linear modelwe use fails at separations r < h − Mpc, and we show that this may reasonably be dueto the enhanced ionization due to radiation from the quasars. We also provide the expectedcorrection that the mass overdensity around the quasar implies for measurements of theionizing radiation background from the line-of-sight proximity effect.
Keywords: large-scale structure: redshift surveys — large-scale structure: Lyman alphaforest – 1 – ontents α sample 4 As the most optically luminous objects known in the universe, quasars are used as lamppostsat high redshift to obtain absorption spectra of the intervening intergalactic medium, as wellas tracers of large-scale structure. Their absorption spectra blueward of the Ly α emissionline reveal the Ly α forest, reflecting the structure in the hydrogen gas density in the inter-galactic medium as it evolves through gravitational collapse around dark matter halos inwhich galaxies form (e.g., [1],[2], [3], [4]).The large-scale clustering of quasars was measured in the 2dF survey (e.g. [5], [6], [7])and in the Sloan Digital Sky Survey (SDSS, [8], [9], [10]). Both the Ly α forest and the quasarclustering can be used as tracers of the underlying large-scale mass fluctuations, which arethought to have an origin in the initial conditions of the universe. In the linear regime, theobserved quasar correlation function should be equal to the mass autocorrelation times thesquare of the mean bias factor of the quasar host halos. Recent results from the analyses of– 1 –ata from large-scale surveys have indicated a bias factor that increases with redshift andhas a value b q = 3 . ± . z = 2 .
4, and is nearly independent of quasar luminosity (see[11], [12]).Quasar clustering can also be probed by means of the cross-correlation with other trac-ers. The quasar cross-correlation with galaxies was measured by [13], [14] and [15]. Thesestudies found that the quasar bias factor has a value near unity, comparable to typical star-forming galaxies, at redshift z .
1, but the small samples at higher redshift already indicateda larger clustering amplitude. Quasars can also be cross-correlated with absorption systemsfound in the spectra of other quasars. This can be done with the hydrogen Ly α forest, witha high abundance of absorption features, and also with more sparse metal line systems suchas the CIV lines, which was recently accomplished by [16].The cross-correlation of quasars with the Ly α forest absorption was first searched foralong the same line of sight of each individual quasar, looking for the impact of the quasarionizing radiation reducing the Ly α absorption, which has been designated as “proximityeffect” or “inverse effect” ([17], [18], [19]). The ionizing radiation emitted by a quasar isadded to the intensity of the cosmic ionizing background, and the higher than average in-tensity in the quasar vicinity implies an increased degree of ionization of the intergalacticmedium, and therefore a decreased absorption by the Ly α forest. Studies of the proximityeffect have therefore expected a lower than average absorption of the Ly α forest near thequasar Ly α emission line compared to the absorption at the same redshift seen in quasars athigher redshift, using this to infer the intensity of the cosmic ionizing background (see [20]for a more recent study). These investigations have generally not included in the analysisthe fact that the intergalactic gas should have higher density near a quasar, because of thepositive correlation of the quasar host halo with the mass density. In reality, the observa-tions of the quasar-Ly α absorption cross-correlation should reveal the combined effect of themean overdensity and the additional ionizing intensity around quasars, which substantiallycomplicates the theoretical interpretation.In general, the quasar-Ly α cross-correlation can be measured in redshift space fromLy α forest lines of sight near another quasar, as a function of the perpendicular and parallelcomponents of the separation, σ and π , respectively. The effect of a quasar on the Ly α forestin the spectrum of another nearby quasar was investigated in several papers that examinedindividual quasar pairs, generally separated by a few arc minutes ([21], [22], [23], [24]). Theseobservations generally found no evidence for any decrease of Ly α absorption near quasars dueto the excess ionizing radiation. In fact, both in the case of quasar pairs at small separation,and in the case of using a larger number of pairs at wider separations, it has been found thatthe Ly α absorption is stronger near quasars, rather than weaker ([25], [26], [27], [28],[29]),[30]. This result has been attributed to the mean overdensity near a quasar, combined withthe reduction of the ionizing intensity from the quasar due to both anisotropic emission andtime variability of the quasars, as discussed by [23], [26] and [27]. These works obtainedupper limits to the luminosity of the quasars emitted in the perpendicular direction witha time delay, although no detailed analysis was done to attempt to model the effect of theoverdensity and provide a robust interpretation of the data.This question can now be investigated with the large sample of quasar spectra providedby the Baryon Oscillation Spectroscopic Survey (BOSS) of the SDSS-III Collaboration ([31],[32]). The DR9 Catalogue of quasars ([33]) already contains 87822 quasars, with more than– 2 –0000 of them at z >
2, distributed over 3275 square degrees of the sky. The extensivearea covered and the large number of quasars makes this sample particularly useful to studythe quasar-Ly α cross-correlation at large scales (i.e., the typical nearest neighbor projectedseparation of ∼ h − Mpc and larger).At these separations, the cross-correlation function induced by the mass overdensityaround quasars can be predicted from linear theory, and should be proportional to the productof the bias factors of the quasars and the Ly α forest and show the expected redshift distortions([34], [35]). These bias factors can be independently determined by observations of theautocorrelations of the Ly α forest and the quasars ([4], [12]), and should agree with themeasured amplitude of the cross-correlation.This paper presents the quasar-Ly α cross correlation measured from the quasars inthe DR9 catalogue of BOSS. The data sample is presented in section 2, and the method ofanalysis in section 3. Linear theory is used to model the mean cross-correlation, and the fitsto the observational results are presented in section 4, showing that the overdensity effectalone adequately matches the results at comoving separations r > h − Mpc for reasonablevalues of the bias factors. Section 5 discusses the expected effect of the ionizing radiationof the quasars on the cross-correlation for the quasar luminosities in our sample, and themanner that this may impact the results in section 4. Our conclusions are summarized insection 6.Throughout this paper we use the flat ΛCDM cosmology, with Ω m = 0 . b = 0 . h = 0 . n s = 0 .
963 and σ = 0 .
8, consistent with the cosmological parameters from theWMAP mission [36].
The data used in this paper is from the publicly available 9th Data Release (DR9, [37]) of theSDSS-III Collaboration ([31], [38], [39], [40], [41], [42], [43]), comprising the first two yearsof observations of the Baryon Oscillation Spectroscopic Survey (BOSS, [32]). The quasarsample is a subsample of the catalogue described in [33], while the Ly α absorption sightlinesmake use of the products described in [44].We first describe the cuts we apply to the catalogue to select our quasar sample, and thenthe set of quasar lines of sight that we use for the Ly α absorption field to be cross-correlatedwith the quasars. A total of 87822 quasars are present in the DR9 quasar catalogue ([33]). These quasars weretargeted for spectroscopy using a complex target selection procedure presented in [45] thatcombines a series of methods described in [46], [47], and [48]. The French Participation Group(FPG) of the SDSS-III Collaboration verified each of these objects by visually inspecting thespectra. A number of estimates of the quasar redshift based on different methods are providedin this catalogue. We generally use the redshift obtained using the Principal ComponentAnalysis method (Z PCA in DR9Q), but in Section 4.6 we examine our results for the quasar-Ly α cross-correlation when using other estimates for the quasar redshifts.– 3 – N u m be r o f qua s a r s z 0 1000 2000 3000 4000 5000 6000-30 -29 -28 -27 -26 -25 -24 -23 N u m be r o f qua s a r s Mi Figure 1 . Left panel: Distribution of the 61342 quasar redshifts in our sample. Right panel: i -bandabsolute magnitude distribution. In order to have a well defined redshift interval in our sample, we use only the 61366quasars with a redshift in the range 2 < z q < .
5. Most of the 26456 quasars discarded bythis criteria are at a redshift that is too low to be cross-correlated with the observed Ly α absorption spectra, and the few at z q > . i -band absolute magnitude, − < M i < − i -band absolute magnitude distributions of our quasar sampleare shown in figure 1. α sample Not all quasars present in the catalogue are useful and free of systematic effects for using theLy α forest spectrum: they may be affected by broad absorption lines, they may have too lowa redshift so that their Ly α forest pixels lie in the noisy blue end of the BOSS spectrograph,or their continuum may be too difficult to model. For these reasons, we select the samequasars as in [49], a previous study of the cross-correlation of the Ly α forest with DampedLy α Systems (DLAs) using the DLA catalogue of [50]. This reduces the number of availablelines of sight to 52449. The selection criteria are that the quasar redshift lies in the range2 . < z < .
5, that the BAL FLAG VI flag is not set in the catalogue of [33], and thatthe median signal-to-noise ratio per pixel (of width ∼
69 km s − ) in the quasar rest-framewavelength range 1220 ˚A ≤ λ r ≤ S/N > . α forest region as the rest-frame wavelength range 1041 ˚A ≤ λ r ≤ The method we use to measure the quasar-Ly α cross-correlation is the same one as forthe cross-correlation of Damped Lyman Alpha systems (DLAs) with the Ly α absorption,described in [49]. Here, a brief summary of the method is presented, discussing in detail onlythe issues that are special for our analysis of the quasar cross-correlation and any differenceswith respect to the method used for DLAs.The observed flux in each pixel of the quasar spectra is f i = C i ¯ F ( z i ) [1 + δ F i ] + N i ,where C i is the quasar continuum (equal to the flux that would be observed in the absence ofabsorption), ¯ F ( z i ) is the mean transmitted fraction, δ F i is the Ly α transmission fluctuation,and N i is the observational noise, which is assumed to be uncorrelated in all pixel pairs.To estimate δ F i and its cross-correlation we must first model the continua of the quasars.This is done using the PCA technique described in [52], but without applying the
Mean FluxRegulation described in this paper, which can suppress the large-scale power in the Ly α forestin ways that are difficult to model. In the same way as in [49], we generally apply instead the Mean Transmission Correction (hereafter MTC), which enforces the mean transmission ineach quasar spectrum to be equal to the value measured in independent observations by [53],using equations (3.5) and (3.6) in [49]. This correction is useful to remove the broadbandnoise caused by spectrophotometric errors, but we will also show results when no correctionto the quasar continua is applied. We will see in section 4 that including the MTC increasesthe accuracy of the measured quasar bias in our parameterized model by ∼ F i is measured here in 150 bins of ∆ z = 0 .
01 over therange 1 . < z < . α forest data is used outside this range). These redshift bins arethree times smaller than the ones used in [49] for the same purpose. The measured ¯ F ( z i )has fluctuations when using small redshift bins owing to systematic errors in the calibratingreference stars ([54], [44]), and we found that the fine redshift bins are necessary to correctlyeliminate the effect of these fluctuations. In general, the larger number of available quasarscompared to DLAs allows for a more accurate measurement of the cross-correlation for thequasars, and therefore greater care needs to be taken in the analysis for quasars.The cross-correlation is computed with the simple estimator (see Appendix B of [49] fora discussion of the approximations involved and the differences with an optimal estimator):ˆ ξ A = P i ∈ A w i δ F i P i ∈ A w i , (3.1)where the summation is done over all quasars and over all Ly α pixels that are within a bin( A ) of the separation from each quasar in the perpendicular ( σ ) and parallel ( π ) directions, δ F i is the estimated value of the transmission fluctuation from the observed flux and thecontinuum model, and the weights w i are computed independently at each pixel from thenoise N i provided for the DR9 data [38], assuming a model for the intrinsic Ly α absorptionvariance that is added in quadrature to the noise (equation 3.10 in [49]).– 5 – set of 9 bins in σ and 18 bins in π are used to measure the cross-correlation,which are the same ones as in [49] except that we add the extra bin at large separations60 h − Mpc < σ < h − Mpc, and the same for both signs of π (the cross-correlation ismeasured without assuming symmetry with respect to a sign change of the parallel separa-tion π ). This procedure yields a total of 162 bins. The weighted average values of ( π, σ ) of allthe contributing pixel pairs to every bin A are computed together with the cross-correlationfrom equation (3.1), using the same weights. These averages are generally close but notexactly equal to the central values of each bin. The models to be fitted are evaluated at theseweighted averages of ( σ, π ). A single redshift bin is generally used for the mean redshift z of the Ly α forest pixel and the quasar, which is required to be in the range 2 . < z < . α forestintrinsic autocorrelation. 2) Our quasar sample is divided into twelve subsamples in adjacentareas of the sky, and the cross-correlation is computed separately in each subsample to inferbootstrap errors. The method used and the subsample areas are the same as in [49]. Thecovariance matrix is computed including only pixel pairs up to a transverse separation σ < h − Mpc in order to make the computer time required for the calculation more manageable.Most of the contributions to the covariance comes from pixel pairs in the same line of sight( σ = 0).We use the same linear theory model described in Section 3.6 of [49] (their equation 3.16),with bias parameters b q and b F , and redshift distortion parameters β q and β F , for the quasarsand the Ly α forest, respectively. The quasar redshift distortion parameter obeys the relation β q = f (Ω) /b q , and for the Ly α forest we impose the condition b F (1 + β F ) = − .
336 from theobservational result of [4] obtained from the measured Ly α autocorrelation at z = 2 .
25. Wealso impose that β F and b q are constant with redshift and b F ∝ (1 + z ) . , as discussed in [4].However, whereas the effect of redshift errors for DLAs could be neglected, this is not thecase for quasars. We therefore add two extra free parameters to the model: a dispersion ǫ z and a mean offset ∆ z of the quasar redshift error, assuming these errors to be Gaussian. Thetheoretical model for the linear cross-correlation is smoothed with a Gaussian in the paralleldirection with this dispersion and offset.Fits to this model are generally done using only separation bins at r = ( σ + π ) / > h − Mpc in order to avoid the near region that is possibly affected by radiation and non-linear effects, although some fits will also be shown using all bins at r > h − Mpc. Thetheoretical prediction is corrected for the MTC using the equations explained in AppendixA of [49].
The results of the cross-correlation of the Ly α transmission with quasars, with the methodpresented in section 3, are shown in figure 2 for each bin in the transverse separation σ . Theerror bars are the diagonal elements of the computed covariance matrix, which we have found– 6 – σ < 4 -0.15-0.12-0.09-0.06-0.03 0 0.03-80 -60 -40 -20 0 20 40 60 804 < σ < 7 -0.08-0.06-0.04-0.02 0 0.02-80 -60 -40 -20 0 20 40 60 807 < σ < 10-0.05-0.04-0.03-0.02-0.01 0 0.01-80 -60 -40 -20 0 20 40 60 80 ξ ( π , σ )
10 < σ < 15 -0.03-0.02-0.01 0 0.01-80 -60 -40 -20 0 20 40 60 8015 < σ < 20 -0.016-0.012-0.008-0.004 0 0.004-80 -60 -40 -20 0 20 40 60 8020 < σ < 30-0.008-0.006-0.004-0.002 0 0.002 0.004-80 -60 -40 -20 0 20 40 60 8030 < σ < 40 -0.006-0.004-0.002 0 0.002-80 -60 -40 -20 0 20 40 60 80 π (Mpc/h)40 < σ < 60 -0.003-0.002-0.001 0 0.001-80 -60 -40 -20 0 20 40 60 8060 < σ < 80 Figure 2 . Measured cross-correlation in the indicated bins of perpendicular separation σ , as afunction of the parallel separation π . The data points in green have a total separation r = ( σ + π ) / < h − Mpc, and are not used in most of our fits. Solid (dashed) dark (blue) lines show thebest fit model for the fiducial analysis, when using bins with separations down to r = 15 h − Mpc( r = 7 h − Mpc). to be consistent with the bootstrap errors from the scatter of the cross-correlation functionmeasured in subsamples. Two model fits to the data are also plotted in figure 2: the solid(black) line uses only bins with separation r = ( σ + π ) / > h − Mpc, and the dashed(blue) line uses all the bins with r > h − Mpc. The data points in green have a separation r < h − Mpc and are not used in our main analysis. The same results are shown as contourplots in the left panel of figure 3, with our best fit fiducial model, the one using only the r > h − Mpc bins, shown in the right panel.Our fiducial model fit applies the MTC to the data and includes the correspondingcorrection to the theory, and uses the whole sample of 61342 quasars described in section 2with the PCA quasar redshifts. The model has four free parameters, as described at the endof section 3: β F (the Ly α redshift space distortion parameter), b q (the quasar bias factor), ǫ z (the rms of the quasar redshift error distribution), and ∆ z (the mean offset of the quasarredshift), while the Ly α forest bias parameter b F is computed for each value of β F using thewell-constrained quantity b F (1 + β F ) = − .
336 at z = 2 .
25 from the Ly α autocorrelationresult of [4]. The first row in table 1 gives the parameters for the best fit result (in the sense– 7 –
10 20 30 40 50 60 σ [Mpc/h] −30−20−100102030 π [ M p c / h ] Measured QSO Lyα cross correlation −0.100−0.086−0.072−0.058−0.044−0.030−0.016−0.0020.012 0 10 20 30 40 50 60 σ [Mpc/h] −30−20−100102030 π [ M p c / h ] Theoretical QSO Lyα cross correlation −0.100−0.086−0.072−0.058−0.044−0.030−0.016−0.0020.012
Figure 3 . Two dimensional contours of the measured cross-correlation (left panel), compared tothe best fit theoretical models for r > h − Mpc (right panel). The black circle corresponds to r = 15 h − Mpc. of minimum value of χ ) using the covariance matrix as described earlier, with the value of χ and the number of degrees of freedom given in the last column. Errors correspond to thecontours of ∆ χ = 1, after marginalizing over all other parameters. In the case of ǫ z , onlyan upper limit is provided when ǫ z = 0 is within the ∆ χ = 1 contour. The parametersfor other models that will be described below are given in the additional rows of the table.The variables related to quasar redshift errors, ǫ z and ∆ z , are expressed in units of km s − ,reflecting the directly-measured separation along the line of sight.Our basic result, seen in the figures and table 1, is that the simple linear theory model forthe cross-correlation of quasars as tracers of the mass distribution and the Ly α forest providesan excellent fit to all the data at large scales. Moreover, the predicted redshift distortions arean excellent match to the observed cross-correlation, as seen in figure 3, confirming the large-scale mass inflow toward the quasar host halos expected from the gravitational evolutionof density perturbations. The quasar bias factor required to match the cross-correlation is b q = 3 . +0 . − . , in excellent agreement with the independently-determined bias factor fromthe quasar auto-correlation, b q = 3 . ± .
3, from [12]. The redshift distortion parameter ofthe Ly α forest is found to be β F = 1 . +0 . − . , also in good agreement with the measurement of[4]. Finally, we find that the best match of the quasar redshift error distribution requires asignificant mean offset of ∆ z = −
160 km s − , with the negative sign indicating that the PCAquasar redshifts are on average too small (so the cross-correlation seen in figure 3 is shiftedto a redshift higher than that of the quasar), and a surprisingly small error dispersion of ǫ z < (370 , − at the χ = (1 ,
4) confidence levels. Note that ǫ z is the combination ofthe observational error of the quasar redshift and the intrinsic velocity dispersion of quasarswith respect to their host halo. We need to keep in mind, however, that this upper limit isobtained using only the pixels at r > h − Mpc, and that the value of ǫ z has degeneracieswith slight modifications of our simple four-parameter fiducial model.This degeneracy is well illustrated in our model by the χ contours for the parameters β F and ǫ z , shown in figure 4 (right panel). The quadrupole moment of the cross-correlationis determined mostly by these two parameters. The quadrupole moment increases with β F – 8 – F b q ǫ z ( km s − ) ∆ z ( km s − ) χ (d.o.f)FIDUCIAL 1 . +0 . − . . +0 . − . < − +38 −
116 (130) r > h − Mpc 1 . +0 . − . . +0 . − . +44 − − +20 −
164 (152)NOCOR 3 . +0 . − . +0 . − . − − +33 −
142 (130)NOMTC 0 . +0 . − . . +0 . − . < − +34 −
112 (130)LOW-Z 1 . +0 . − . . +0 . − . +137 − − +70 −
124 (130)MID-Z 1 . +0 . − . . +0 . − . < − +63 −
131 (130)HIGH-Z 1 . +0 . − . . +0 . − . < − +70 −
134 (130)LOW-L 1 . +0 . − . . +0 . − . < − +58 −
128 (130)MID-L 1 . +0 . − . . +0 . − . < − +56 −
113 (130)HIGH-L 1 . +0 . − . . +0 . − . +170 − − +46 −
118 (130)Z VISUAL 1 . +0 . − . . +0 . − . +110 − − +28 −
142 (130)Z PIPELINE 1 . +0 . − . . +0 . − . +86 − − +43 −
114 (130)Z CIV 1 . +0 . − . . +0 . − . +72 − − +28 −
137 (130)Z CIII 1 . +0 . − . . +0 . − . +87 − − +48 −
137 (130)Z MgII 1 . +0 . − . . +0 . − . +110 − − +38 −
126 (130)
Table 1 . Best fit parameters and χ for the different analyses: FIDUCIAL (with the MTC andthe corrected theory, using r > h − Mpc), r > h − Mpc (extending to smaller scales), NOCOR(MTC, uncorrected theory), NOMTC (PCA-only continuum fitting, uncorrected theory), data splitin redshift bins (LOW-Z for 2 < z < .
25, MID-Z for 2 . < z < .
5, HIGH-Z for 2 . < z < . − . < M i < −
23, MID-L for − . 2, and HIGH-L for − < M i < − . 1) and finally different quasar redshift estimates(Z VISUAL,Z PIPELINE,Z CIV, Z CIII,Z MgII). Uncertainties correspond to values with ∆ χ = 1,with upper limits for ǫ z when ∆ χ < ǫ z = 0. b q β F ǫ z [km/s / 100] β F Figure 4 . Contours of χ in the two-parameter plane of the Ly α forest redshift distortion parameterversus: quasar bias (left), and quasar redshift error dispersion (right). The number of degrees offreedom is 130. and decreases with ǫ z , but the effect of ǫ z is obviously important only at small radius (thedependence of the quadrupole moment on the quasar bias factor is relatively small becauseof the requirement β q = f (Ω) /b q , which implies a small value for β q ). A modification of our– 9 –odel at small radius owing to the radiation effects of the quasars or non-linearities maymodify the best fit values of ǫ z and ∆ z . The left panel of figure 4 also demonstrates that thequasar bias b q tends to slightly increase with increasing β F for a given analysis.The value of χ = 116 for our fiducial model fit for 130 degrees of freedom, and theagreement of the fitted parameters with other independent determinations, shows that theobserved cross-correlation is sufficiently well reproduced without the need to include the effectof the quasar ionizing radiation. Clearly, the linear overdensity around the quasar host halois the dominant effect when measuring the quasar-Ly α cross-correlation away from the lineof sight, on large scales and for the luminosities of the BOSS quasars.The second row of table 1 gives the best fit parameters when all the bins at separationsdown to r > h − Mpc are used. The best fit shifts to a larger redshift distortion parameterof the Ly α forest, a lower quasar bias factor, and a larger quasar redshift error dispersion. The χ worsens significantly, with an increase of 48 when adding only 22 degrees of freedom. Thisresult suggests that our 4-parameter model does not include all the important physical effectswhen analyzing the entire range of separations in figure 2. Non-linearities and radiation effectsare likely to play a role at small separations, and a more complex analysis will be requiredto discern this. In general, we have tested that the bootstrap errors computed as described in section 3 arein agreement with the errors derived from the covariance matrix. We mention here as anexample this error comparison for the quasar bias factor, when keeping the other parametersfixed to their best fit value of our fiducial analysis. The covariance matrix yields errors fromthe ∆ χ = 1 contour of b q = 3 . +0 . − . , and the result of 100 bootstrap realizations fromthe 12 subsamples of our data set (see [49]) is b q = 3 . ± . χ = 1. In figure 5 we present the value of the quasar bias that is obtained when using only binswithin narrow rings of the separation r . For this analysis, we fix the other parameters totheir best fit value when using the range 15 h − Mpc < r < h − Mpc, which are β F = 1 . ǫ z = 2 . h − Mpc, ∆ z = − . h − Mpc. The horizontal lines in the figure show the best fitand uncertainties when using all separations above r > h − Mpc, i.e., all the points lyingto the right of the dotted vertical line.The constancy of the bias factor for r > h − Mpc is again a success of the simple lineartheory model for large scales, meaning that the radial dependence of the cross-correlationagrees with the prediction of the standard ΛCDM model of structure formation. The smallervalue of b q at smaller separation confirms our previous conclusion that other effects are likelyto be important at r < h − Mpc.The amplitude of the cross-correlation is proportional to b F b q σ , this is the actualquantity we are measuring, and the value for b q plotted in figure 5 assumes that b F (1 + β F ) =– 10 – b q (r) r (Mpc/h) Figure 5 . Fitted QSO bias b q in several bins of the separation r , when fixing the other parametersto their best fit value in the fiducial model, β F = 1 . ǫ z = 2 . h − Mpc, ∆ z = − . h − Mpc.The horizontal lines show the best fit and uncertainties when using all separations larger than r > h − Mpc, the scale that is marked with a vertical dotted line in the figure. − . 336 at z = 2 . 25 (corrected to the mean quasar redshift ¯ z q = 2 . 38, where b F (1 + β F ) = − . σ = 0 . Our fiducial model uses the Mean Transmission Correction (MTC) as part of the continuumfitting, and corrects the theoretical model accordingly by using the analytical expressionderived in appendix A of [49]. As a test of the importance of this correction, table 1 givesthe results for two additional models: the NOCOR case (fourth row) treats the observationsin the same way as the fiducial model (applying the MTC to the data), but does not correctthe theoretical model. The result is a considerably worse fit (∆ χ = 26), confirming thevalidity and the need for the theoretical correction. In the NOMTC case, the data forthe transmission fluctuation is obtained with the direct use of the PCA continua, withoutapplying the MTC. The uncorrected theory then fits the data properly, but the errors of thebest fit parameters are considerably worse, supporting our reasons to use the MTC. An evensmaller upper limit for ǫ z is obtained for the NOMTC case, indicating that this upper limitis questionable because of its degeneracy with other model variations.Figure 6 shows the effect of the MTC on the measured cross-correlation, for the sixthbin of σ (20 h − Mpc < σ < h − Mpc), as well as the correction we apply to the theoreticalmodel. As seen in the plot, the theoretical correction clearly captures the difference in theanalyses. The errorbars in the NOMTC analysis are considerably larger, due to the largespectro-photometric errors present in BOSS quasars. Since these errors have a coherenteffect on all pixels of a given spectrum, the errorbars in the NOMTC analysis are stronglycorrelated. – 11 – ξ ( π , σ ) π (Mpc/h)20 < σ < 30 FIDUCIALNOMTCcorrected theoryuncorrected theory Figure 6 . Measured cross-correlation in the sixth bin in σ (20 h − Mpc < σ < h − Mpc). Red(green) data points correspond to the analysis with (without) the MTC step in the continuum fitting.The black solid (blue dashed) line shows the best fit model for the fiducial analysis with (without)the MTC correction. We test here for the dependence of our measured quasar bias on the redshift and quasarluminosity. In table 1 we show the results obtained when splitting the quasar sample in threeredshift bins: LOW-Z (2.00 < z < < z < < z < b q , although ourerrors are large and the redshift range that is probed is limited.In the same table we show the results obtained when splitting the sample in threeluminosity bins: LOW-L for − . < M i < − . 0, MID-L for − . < M i < − . 2, andHIGH-L for − . < M i < − . 1. Again, the changes are not significant, and unfortunatelythe quasar redshift error distribution may vary with the quasar luminosity, making it difficultto search for any physical dependence of the cross-correlation with quasar luminosity becauseof the parameter degeneracies. There are six different quasar redshift estimators specified in the DR9 quasar catalog ([33]).In the main part of this study we use the PCA redshift estimator (Z PCA in DR9Q), butwe also show in table 1 the results obtained when using the other ones: the visual inspectionredshift (Z VI), the estimator from the BOSS pipeline (Z PIPELINE), and three estimatorsthat use a single metal absorption line (Z CII,Z CIV,Z MgII).Table 1 shows the best fit parameters for each of the quasar redshift estimators, andthe quasar bias obtained is consistent among them. However, the redshift error distributionvaries considerably for the different estimators. Our fits suggest that all the estimatorssystematically underestimate the quasar redshift by several hundreds of km s − , except for– 12 –he one based on the MgII line which is basically consistent with ∆ z = 0, but has a largerdispersion ǫ z .The presence of a mean redshift displacement can also be tested using narrow emissionlines of the quasar host galaxies, namely OII and OIII. The mean redshifts of the MgII andOII lines were compared by [55] in a large sample of low redshift SDSS quasars, who foundthat the difference had to be smaller than 30 km s − . This comparison, however, would bedifficult to carry out for the higher redshift quasars of our sample, where the OII line isshifted to the infrared.We caution, however, that these values of ∆ z and ǫ z are likely to change once oursimple four-parameter model is improved with more parameters to include the quasar ionizingradiation effects. The systematic negative value of ∆ z might be adjusting a real asymmetry ofthe cross-correlation introduced by the reduction of the effective quasar luminosity with thetime delay. The effect of the time-delayed quasar ionizing radiation on the cross-correlationshould obviously have a different radial dependence than a simple shift of the quasar redshifts,but some degree of degeneracy can be expected. At the same time, it seems difficult to believethat the true dispersion ǫ z , arising from both observational errors and the intrinsic velocitydispersion of quasars within halos, is smaller than ∼ 500 km s − , and its value will alsoprobably be modified in a more complex model that better reflects the underlying physicaleffects. As we have seen in the previous section, the large-scale form of the quasar-Ly α cross-correlation is consistent with the linear overdensity expected around the quasar host ha-los. This result may be surprising because of the previous detection of the proximity effectwhen measuring this cross-correlation along the line of sight, caused by the increased ion-ization of the intergalactic medium induced by the quasar ionizing radiation, although theBOSS quasars are of lower luminosity than most of the quasars on which the proximity effectwas measured and therefore the expected additional ionization is much weaker in our casecompared to previous studies. In this section we present a simple estimate of the expected ra-diation effect on the full three-dimensional cross-correlation with our quasar sample. A moredetailed analysis involving fitting of a more complex model that includes both the radiationand overdensity effects is left for a future study.The radiation of a quasar of luminosity L ν increases the photoionization rate of gas ata proper distance d by the amountΓ q = 14 πd h Z ∞ ν HI dν L ν σ HI ( ν ) ν , (5.1)where ν HI is the hydrogen Lyman limit frequency, σ HI is the photoionization cross section,and h is the Planck constant. We neglect here the absorption by Lyman limit systems andthe redshifting of the radiation, which reduces the quasar intensity when d is comparableto the absorption mean free path or the local horizon. Let the relative fluctuation of thephotoionization rate relative to its average value Γ be δ Γ . In general, δ Γ can be affected bymany sources. Neglecting the effects of ionizing source clustering (which increases the mean– 13 –onizing flux near a quasar beyond that emitted by the quasar itself), assuming a quasarspectrum L ν ∝ ν − α , and approximating σ HI ( ν ) ∝ ν − at ν > ν HI , the average value of δ Γ near the quasar is δ Γ = Γ q Γ ≃ L ν HI σ HI ( ν HI )4 πd h Γ (3 + α ) . (5.2)The impact of the perturbation δ Γ on the Ly α forest can be calculated using the ap-proximation that the gas is purely photoionized (neglecting any contribution from collisionalionization) and that the neutral fraction is everywhere much smaller than unity. In thiscase, the optical depth at any point in the spectrum of the Ly α forest is inversely pro-portional to the photoionization rate, so the fractional perturbation in the optical depth is δ τ = 1 / (1 + δ Γ ) − ≃ − δ Γ , where the last approximate equality assumes δ Γ ≪ 1. Now,let F = e − τ be the Ly α transmission fraction when the photoionization rate has the uni-form value Γ , with a distribution P ( F ). The transmission in the presence of radiationfluctuations is F = e − τ (1 − δ Γ ) , and the mean transmission is¯ F = Z dF P ( F ) e − τ (1 − δ Γ ) (5.3) ≃ Z dF P ( F ) F (1 + τ δ Γ ) ≃ ¯ F − δ Γ Z dF P ( F ) F log( F ) . We can now define the radiation bias factor of the Ly α forest, b Γ , as the linear variation ofthe mean transmission fluctuation δ F in response to a fractional variation δ Γ in the radiationintensity, analogously to the density and peculiar velocity gradient bias factors. Hence, neara quasar the mean transmission fluctuation will vary by δ F = b Γ δ Γ owing to the fractionalradiation perturbation δ Γ , where b Γ = − F Z dF P ( F ) F ln( F ) . (5.4)If we use as the transmission distribution a log-normal function in the optical depth,with mean ¯ F = 0 . σ F = 0 . z = 2 . 3, we find b Γ = 0 . g is f ν = 10 − . . g ) , expressedin cgs units ([40]). The mean value of f ν we obtain for our quasar sample is f ν = 2 . × − erg s − cm − Hz − at the center of the g -band (at 4800 ˚A; this corresponds to a g magni-tude of 20.5) At the mean redshift ¯ z q = 2 . 38 of our quasar sample, the implied mean luminos-ity at the shifted g -band center λ = 1420 ˚A is L ν = 4 πD L f ν / (1+ ¯ z q ) = 3 . × erg s − Hz − .We assume a mean spectral slope from this wavelength to the Lyman limit wavelength L ν ∝ ν − ([56], [57]) , which results in L ν HI = 2 . × erg s − Hz − . The correspond-ing quasar flux at a characteristic comoving distance of interest of d (1 + ¯ z q ) = 20 h − Mpc,or proper distance d = 8 . 33 Mpc, is then f ν HI = 2 . × − erg s − cm − Hz − . Finally, usingequation (5.2) with the spectral index α = 1 . ν > ν HI , the derived photoionization rateis Γ q = 5 . × − s − . – 14 –f we assume a mean photoionization rate from the average of all sources Γ = 10 − s − ,then at this comoving distance of 20 h − Mpc, we have δ Γ ≃ . 05, and the mean perturbationon the transmission should be δ F ≃ b Γ δ Γ ≃ . d − .Comparing to figure 3, we see that this radiation effect should displace the value of thecross-correlation by one contour at r = ( σ + π ) / = 20 h − Mpc. In the third panel of figure2 (for the range 7 h − Mpc < σ < h − Mpc), the increase of the cross-correlation wouldbe ∼ . 03 at small π , which is larger than the difference between the data points and ourfiducial model fit, and in the fifth panel (for the range 15 h − Mpc < σ < h − Mpc), theincrease would be ∼ . b q in order to compensate for the radiation effect to a value b q ≃ α forest depends on the time delay, ct d = r + π : the Ly α forest is illuminated by the luminosity of the quasar at a time t d before the epoch when weare observing. The selection effect again causes the average quasar in a flux-limited sampleto have lower luminosity in the past compared to the present, and this should introduce anasymmetry depending on the sign of π . A hint of this signature of the radiation effect is seenin figure 2, in the region π ≃ − h − Mpc and σ < | π | , which is consistent with the increaseof δ F ≃ . 007 we have estimated at r = 20 h − Mpc from the radiation effect. If this hint iscorrect, quasars might shine at close to the expected luminosities within ∼ ◦ of the lineof sight and for time delays t d < years, and have effective lower luminosities at largerangles from the line of sight and for longer time delays.In any case, obtaining solid conclusions on the contribution of the radiation effect tothe cross-correlation and the statistical significance of any detection requires a more detailedmodeling with more fitting parameters, additional data and a more careful inclusion of allthe important effects. We plan to perform this study in the future when the BOSS survey iscompleted. We mention here, however, that the radiation effects we have discussed may bealtering the value of the linear model parameters we have fitted. In particular, the redshiftoffset ∆ z may in part be the result of the attempt to adjust the radiation and time-delayeffect which introduces the asymmetry depending on the sign of π .Non-linear effects in the clustering of both quasars and Ly α absorption, as well as thenon-linearity in the relation between optical depth and transmitted flux fraction, may alsohave an impact on the cross-correlation at small scales. However, we compared the lineartheory predictions to the results of numerical simulations of [30] for the cross-correlation, andwe found that non-linearities become important only at scales smaller than a few h − Mpc,whereas the radiation effects are a more important correction at intermediate scales of ∼ h − Mpc. – 15 –inally, we note that HeII reionization may alter the IGM temperature near quasarsbecause of the additional heating involved ([58], [59]), which may result in an additional effecton the cross-correlation. The effect would likely also vary as the inverse squared distancefrom the quasar. However, the duration of the HeII reionization is probably much longerthan the typical lifetime of a quasar, implying that most of the temperature fluctuationsoriginated from the HeII reionization would be caused by quasars that have long been dead. Our fitted linear theory model for the mean overdensity around a quasar can be used to makea prediction on the impact of this overdensity on any measurements of the proximity effect ofquasars on the line of sight due to their ionizing radiation. In figure 7, we show the predictionfor the line of sight cross-correlation, as πξ ( σ = 0 , π ), for our two first models in table 1:1) the fiducial model fit using pixels at r > h − Mpc (solid, black line), and 2) the samemodel using all pixels at r > h − Mpc (dashed, blue line; in this subsection we choose theconvention that π is positive even though for the three-dimensional discussion π is negativein front of the quasar). Also shown as the red, dotted line is the expected radiation effectin our sample of quasars, which we have plotted according to our simple estimate aboveas ξ = 0 . h − Mpc /π ) . Note that this curve is proportional to the mean quasarluminosity, so the proximity effect due to the radiation dominates over the overdensity effectfor quasars of much greater luminosity than the ones in the BOSS sample. -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 10 100 π ξ ( π , σ = ) π (Mpc/h)clustering (r>15)clustering (r>7)ionization Figure 7 . Cross-correlation function along the line of sight, i.e., using pixels and their backgroundquasar. The black solid (blue dashed) line shows our best fit theory for r > h − Mpc ( r > h − Mpc). The dotted red line shows the expected radiation effect. As shown in this figure, the overdensity effect should roughly cancel the radiation effectat π ∼ h − Mpc for the luminosities of typical BOSS quasars. Most studies of the proximityeffect have used brighter quasars to obtain higher signal-to-noise spectra, but figure 7 providesa correction that should be applied to any future measurements of the proximity effect dueto radiation under the assumption that the quasar bias factor does not depend on quasarluminosity. For π > h − Mpc, the overdensity effect causes an increased δ F (decreasedabsorption) that is therefore added to any radiation effect, because the peculiar velocity– 16 –radient due to the infall of matter toward the quasar halo is a more important effect thanthe overdensity. We have measured the cross-correlation of quasars and the Ly α forest transmission in redshiftspace using more than 60000 quasars from BOSS. This unprecedentedly large number ofquasar spectra for such a study has allowed a statistically significant detection of this cross-correlation out to separations of ∼ h − Mpc, and a detailed determination of its radial andangular dependence. The cross-correlation is consistent with the linear theory prediction ofthe standard ΛCDM model, with the expected redshift distortions, on scales r > h − Mpc.Fitting these large-scale measurements to a linear model with four parameters, we find thatthe BOSS quasars at ¯ z q = 2 . 38 have a mean bias factor b q = 3 . +0 . − . . This result isconsistent with the quasar bias measured from the auto-correlation of a sub-sample of thesame quasars, b q = 3 . ± . 3, as presented in [12]. The halo mass having this bias factor is M h ≃ × h − M ⊙ . This measurement can be compared to the halo mass correspondingto the mean bias factor inferred for DLAs, the other population of objects for which the biasfactor was measured from the same approach measuring the cross-correlation with the Ly α forest in BOSS, which was M h ≃ × h − M ⊙ , although a large scatter in halo masses isexpected for both populations of objects and only the mean bias factor is measured. We donot detect any dependence of quasar bias on luminosity or redshift, also consistent with theresults from [12].The large-scale fit to the cross-correlation also provides a measurement of the Ly α forest redshift distortion parameter, β F = 1 . +0 . − . . This value is ∼ . σ higher than theone measured in [4], reducing the tension with previous numerical simulations ([60]). Thevalue of β F measured from this cross-correlation has the advantage of being less sensitiveto systematic errors in the spectrophotometric calibration, although it should be equallyaffected by the presence of DLAs, Lyman limit systems and metal lines in the spectra whichtend to decrease β F ([61]).The cross-correlation at scales r < h − Mpc is not well fitted by the simple lineartheory model we have used, because its amplitude is lower than expected in the model fittedto large scales, as seen in figure 5. We have argued that a likely explanation is the effects ofthe ionizing radiation of the quasars, which are of the right order of magnitude to explain thisdiscrepancy for the luminosity of the BOSS quasars. If all the BOSS quasars emitted theirlight isotropically and with constant luminosities, the enhanced ionization of the surroundingmedium would also affect the cross-correlation we have fitted at r > h − Mpc, implyinga higher quasar bias factor to compensate for this ionization effect. However, the quasarradiation effect is likely to be decreased owing to anisotropic emission and finite quasarlifetimes.The impact of the quasar ionization can also be studied by means of the line of sightproximity effect, and using the cross-correlation measured from especially targeted quasarpairs at smaller angular separations than in the BOSS sample. In the future, these studieswill need to model the superposed effects of the mean overdensity around the quasar hosthalos and the ionization effects. Here, we have presented predictions from our fitted modelsfor the correction that needs to be applied to the line of sight proximity effect for the mean– 17 –verdensity around quasars, before attempting to infer anything about the ionization effectof the quasar. This correction has been neglected in the past ([17], [18],[19], [20]), and is infact small for the most luminous quasars that have been used to study the proximity effect,but becomes important for quasars of luminosities typical of the BOSS sample.The expected redshift distortions also imply that the effect of the overdensity is smalleron the line of sight compared to the perpendicular direction, and changes sign at separations | π | & h − Mpc owing to the induced peculiar velocity gradient. We note that this predictedcorrection assumes linear theory, so it can be altered at small scales.The quasar-Ly α cross-correlation can also be useful to constrain the redshift errors ofthe quasars. Our results for the mean redshift offset ∆ z indicate that the quasar estimatorsthat have been used are systematically too low, except for the one based on MgII which isclosest to zero. We have warned, however, that the result for this redshift offset may beaffected by the quasar radiation effect with finite quasar lifetimes, which can introduce anasymmetry depending on the sign of π .Future studies of the quasar-Ly α cross-correlation have a promising potential for prob-ing both the large-scale distribution of matter around quasars and the characteristics of theionizing emission. The Baryon Acoustic Oscillation peak, detected recently in the Ly α au-tocorrelation ([54],[62]), can also be detected in the quasar-Ly α cross-correlation. The ratioof the BAO peak amplitudes in the monopole, ξ BAOqα /ξ BAOαα , should be equal to ([35]) ξ BAOqα ξ BAOαα = b q b F β F + β q ) / β F β q / 51 + 2 β F / β F / . 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