Baryon Acoustic Oscillations in the Ly forest of BOSS quasars
Key Words.:cosmology, Ly forest, large scale structure, dark energy
We report a detection of the baryon acoustic oscillation (BAO) feature in the three-dimensional correlation function of the transmitted flux fraction in the Ly forest of high-redshift quasars. The study uses 48,640 quasars in the redshift range from the Baryon Oscillation Spectroscopic Survey (BOSS) of the third generation of the Sloan Digital Sky Survey (SDSS-III). At a mean redshift , we measure the monopole and quadrupole components of the correlation function for separations in the range . A peak in the correlation function is seen at a separation equal to times the distance expected for the BAO peak within a concordance CDM cosmology. This first detection of the BAO peak at high redshift, when the universe was strongly matter dominated, results in constraints on the angular diameter distance and the expansion rate at that, combined with priors on and the baryon density, require the existence of dark energy. Combined with constraints derived from Cosmic Microwave Background (CMB) observations, this result implies , indicating that the time derivative of the cosmological scale parameter is significantly greater than that measured with BAO at . This demonstrates that the expansion was decelerating in the range , as expected from the matter domination during this epoch. Combined with measurements of , one sees the pattern of deceleration followed by acceleration characteristic of a dark-energy dominated universe.
Baryon acoustic oscillations (BAO) in the pre-recombination universe have striking effects on the anisotropies of the Cosmic Microwave Background (CMB) and on the large scale structure (LSS) of matter (Weinberg et al. (2012) and references therein). The BAO effects were first seen in the series of peaks in the CMB angular power spectrum (de Bernardis et al., 2000). Subsequently, the BAO relic at redshift was seen (Eisenstein et al., 2005; Cole et al., 2005) as a peak in the galaxy-galaxy correlation function at a co-moving distance corresponding to the sound horizon at recombination. For the WMAP7 cosmological parameters (Komatsu et al., 2011), the expected comoving scale of the BAO peak is , with an uncertainty of .
The BAO peak in the correlation function at a redshift appears at an angular separation and at a redshift separation , where and are the angular distances and expansion rates. Measurement of the peak position at any redshift thus constrains the combinations of cosmological parameters that determine and . While the possibility of measuring both combinations is beginning to be exploited (Chuang & Wang, 2012; Xu et al., 2012), most present measurements have concentrated on the combination , which determines the peak position for an isotropic distribution of galaxy pairs and an isotropic clustering strength. The “BAO Hubble diagram”, , now includes the Sloan Digital Sky Survey (SDSS) measurement (Eisenstein et al., 2005) updated to the DR7 (Abazajian et al., 2009) sample and combined with 2dF data (Percival et al., 2010), the 6dF point at (Beutler et al., 2011), the WiggleZ points at (Blake et al., 2011a), and a reanalysis of the SDSS DR7 sample that uses reconstruction (Eisenstein et al., 2007; Padmanabhan et al., 2009) to sharpen the precision of the BAO measurement (Padmanabhan et al., 2012; Mehta et al., 2012). Recently, the Baryon Oscillation Spectroscopic Survey (BOSS; Dawson et al. 2013) of SDSS-III (Eisenstein et al., 2011) has added a precise measurement at (Anderson et al., 2012). BOSS has also reported a measurement of based on the BAO structure in the angular power spectrum of galaxies (Seo et al., 2012).
In this paper, we present an observation of the BAO peak at found in the flux correlation function of the Ly forest of BOSS quasars. This is the first such observation at a redshift where the expansion dynamics is matter-dominated, . The possibility of such a measurement was suggested by McDonald (2003) and White (2003) and first studied in detail by McDonald & Eisenstein (2007). While the galaxy BAO measurements are most sensitive to , the Ly flux transmission is more sensitive to peculiar velocity gradient effects, which enhance redshift distortions and shift our sensitivity to the expansion rate. As we shall show below, the most accurately measured combination from the Ly forest BAO peak is , and the present BOSS data set allows us to determine its value to a precision of 3.5%. Combining this result with constraints from CMB observations allows us to deduce the value of accurate to 4%. Comparing our results with measurements of and of reveals the expected sequence of deceleration and acceleration in models with dark energy.
The last decade has seen increasing use of Ly absorption to investigate large scale structure. The number of quasars in early studies (Croft et al., 1999; McDonald et al., 2000; Croft et al., 2002; Viel et al., 2004; McDonald et al., 2006) was enough only to determine the Ly absorption correlation along individual lines of sight. With the BOSS project the surface density of quasars is sufficient to probe the full three-dimensional distribution of neutral hydrogen. A study using the first 10,000 BOSS quasars was presented by Slosar et al. (2011). This sample provided clear evidence for the expected long-range correlations, including the redshift-space distortions due to the gravitational growth of structure. With the SDSS data release DR9 (Ahn et al., 2012), we now have quasars at (Pâris et al., 2012), with a high enough surface density to observe the BAO peak.
The use of Ly absorption to trace matter has certain interesting differences from the use of galaxies. Galaxy surveys provide a catalog of positions in redshift space that correspond to points of high over-densities. On the other hand, the forest region of a quasar spectrum provides a complete mapping of the absorption over a (comoving) range starting about in front of the quasar (so as to avoid the necessity of modeling the quasar’s Ly emission line). To the extent that quasar lines of sight are random, a large collection of quasars can provide a nearly unbiased sample of points where the absorption is measured. Cosmological simulations (Cen et al., 1994; Petitjean et al., 1995; Zhang et al., 1995; Hernquist et al., 1996; Miralda-Escudé et al., 1996; Theuns et al., 1998) indicate that most of the Ly absorption is due to cosmic filamentary structures with overdensities of order one to ten, much lower than the overdensities of virialized halos sampled by galaxies. These simulations have also indicated that, on large scales, the mean Ly absorption is a linear tracer of the mass overdensity (Croft et al., 1997, 1998; Weinberg et al., 1998; McDonald et al., 2000; McDonald, 2003), implying a relation of the power spectrum of the measured absorption to that of the underlying mass fluctuations. Finally, the forest is observable in a redshift range inaccessible to current large galaxy surveys and where theoretical modeling is less dependent on non-linear effects in cosmological structure formation. These factors combine to make Ly absorption a promising tracer of mass that is complementary to galaxy tracers.
With Ly forest measurements along multiple sightlines, one can attempt to reconstruct the underlying 3-dimensional mass density field (Nusser & Haehnelt, 1999; Pichon et al., 2001; Gallerani et al., 2011), from which one can investigate topological characteristics (Caucci et al., 2008) or the power spectrum (Kitaura et al., 2012). However, the BOSS sample is fairly sparse, with a typical transverse sightline separation (comoving), and the signal-to-noise ratio in individual spectra is low (see Fig. 2 below), which makes it poorly suited to such reconstruction techniques. In this paper we take the more direct approach of measuring the BAO feature in the correlation function of transmitted flux fraction (the ratio of the observed flux to the flux expected in the absence of absorption).
The use of the Ly forest is handicapped by the fact that not all fluctuations of the transmitted flux fraction are due to fluctuations of the density of hydrogen. Because the neutral hydrogen density is believed to be determined by photo-ionization equilibrium with the flux of UV photons from stars and quasars, variations of the UV flux may contribute to fluctuations of the transmitted flux (Worseck & Wisotzki, 2006). In addition, hydrogen can be self-shielded to ionizing photons, leading to much higher neutral fractions and damping wings in strong absorption systems (Font-Ribera et al., 2012b). Metals present in the intergalactic medium provide additional absorption superimposed on the Ly absorption (Pieri et al., 2010). A further complication results from the fact that the flux correlation function uses the fluctuations of the transmitted flux about its mean value; this requires an estimate of the product of the mean absorption (as a function of absorber redshift) and the unabsorbed flux for individual quasars (Le Goff et al., 2011).
All of these complications are important for a complete understanding of the statistics of the Ly forest. Fortunately, they are not expected to produce a sharp peak-like feature in the correlation function, so the interpretation of a peak as due to BAO should give robust constraints on the cosmological parameters.
The BOSS collaboration has performed three independent analyses to search for BAO in Ly forest. This paper presents two of them, both of which aim to analyze the forest using simple procedures at the expense of some loss of sensitivity. The third analysis, with the goal of an optimal measurement of the flux correlation function with a more complex method, is described in a separate publication (Slosar et al., 2013).
Our methods are tested extensively on a set of detailed mock catalogs of the BOSS Ly forest data set. These mock catalogs, which use the method presented by Font-Ribera et al. (2012a), will be described in detail in a forthcoming public release paper (S. Bailey et al., in preparation). In addition, the BOSS collaboration have also released a fiducial version of the DR9 Ly forest spectra (Lee et al., 2012b), with various per-object products including masks, continua, and noise correction vectors designed to aid in Ly forest analysis. While our analyses implement their own sample selection criteria and continuum determination procedures, we have also applied our measurement to this Lee et al. (2012b) sample.
This paper is organized as follows. Section 2 presents the BOSS quasar sample used in this analysis and the procedure used to produce the quasar spectra. Section 3 describes the analysis to measure the correlation function. Section 4 derives the monopole and quadrupole components of the correlation function and determines the significance of the peak observed in these functions at the BAO scale. The cosmological implications of our detection of a BAO peak are discussed in Section 5. Finally, Appendix A provides a brief description of the mock spectra used to test our methodology and Appendix B shows the result of our BAO measurement applied to the BOSS Ly sample of Lee et al. (2012b).
2 The BOSS quasar sample and data reduction
The BOSS project (Dawson et al., 2013) of SDSS-III (Eisenstein et al., 2011) is obtaining the spectra of luminous galaxies and quasars. The project uses upgraded versions of the SDSS spectrographs (Smee et al., 2012) mounted on the Sloan 2.5-meter telescope (Gunn et al., 2006) at Apache Point, New Mexico. BOSS galaxy and quasar spectroscopic targets are selected using algorithms based primarily on photometry from the SDSS camera (Gunn et al., 1998; York et al., 2000) in the bands (Fukugita et al., 1996; Smith et al., 2002) reduced and calibrated as described by Stoughton et al. (2002), Pier et al. (2003), and Padmanabhan et al. (2008). Targets are assigned to fibers appropriately positioned in the diameter focal plane according to a specially designed tiling algorithm (Blanton et al., 2003). Fibers are fixed in place by a pierced metal plate drilled for each observed field and fed to one of two spectrographs. Each exposed plate generates 1,000 spectra covering wavelengths of 360 to 1000 nm with a resolving power ranging from 1500 to 3000 (Smee et al., 2012). A median of 631 of these fibers are assigned to galactic targets and 204 to quasar targets. The BOSS spectroscopic targets are observed in dark and gray time, while the bright-time is used by other SDSS-III surveys (see Eisenstein et al. 2011).
The quasar spectroscopy targets are selected from photometric data with a combination of algorithms (Richards et al. 2009; Yeche et al. 2009; Kirkpatrick et al. 2011; Bovy et al. 2011; Palanque-Delabrouille et al. 2011; for a summary, see Ross et al. (2012)). The algorithms use SDSS fluxes and, for SDSS Stripe 82, photometric variability. When available, we also use data from non-optical surveys (Bovy et al., 2012): the GALEX survey (Martin et al., 2005) in the UV; the UKIDSS survey (Lawrence et al., 2007) in the NIR, and the FIRST survey (Becker et al., 1995) in the radio.
The quasar spectroscopy targets are divided into two samples “CORE” and “BONUS”. The CORE sample consists of 20 quasar targets per square degree selected from SDSS photometry with a uniform algorithm, for which the selection efficiency for quasars is . The selection algorithm for the CORE sample (Bovy et al., 2011) was fixed at the end of the first year of the survey, thus making it useful for studies that require a uniform target selection across the sky. The BONUS sample was chosen from a combination of algorithms with the purpose of increasing the density on the sky of observed quasars beyond that of the CORE sample. The combined samples yield a mean density of identified quasars of with a maximum of , mostly in zones where photometric variability, UV, and/or NIR data are available. The combined BONUS plus CORE sample can be used for Ly BAO studies, which require the highest possible quasar density in a broad sky area but are insensitive to the uniformity of the quasar selection criteria because the structure being mapped is in the foreground of these quasar back-lights.
The data were reduced with the SDSS-III pipeline as described in Bolton et al. (2012). Typically four exposures of 15 minutes were co-added in pixels of wavelength width . Besides providing flux calibrated spectra, the pipeline provided preliminary object classifications (galaxy, quasar, star) and redshift estimates.
The spectra of all quasar targets were visually inspected, as described in Pâris et al. (2012), to correct for misidentifications or inaccurate redshift determinations and to flag broad absorption lines (BAL). Damped Ly troughs are visually flagged, but also identified and characterized automatically (Noterdaeme et al., 2012). The visual inspection of DR9 confirms 60,369 quasars with 2.1 3.5. In order to simplify the analysis of the Ly forest, we discarded quasars with visually identified BALs and DLAs, leaving 48,640 quasars.
For the measurement of the flux transmission, we use the rest-frame wavelength interval
The range is bracketed by the Ly and Ly emission lines at 121.6 and 102.5 nm. The limits are chosen conservatively to avoid problems of modeling the shapes of the two emission lines and to avoid quasar proximate absorbers. The absorber redshift, , is in the range . The lower limit is set by the requirement that the observed wavelength be greater than 360 nm below which the system throughput is less than 10% its peak value. The upper limit comes from the maximum quasar redshift of 3.5, beyond which the BOSS surface density of quasars is not sufficient to be useful. The distribution of absorber redshift is shown in figure 2 (top panel). When given the weights used for the calculation of the correlation function (section 3.3), the absorbers have a mean redshift of .
For the determination of the correlation function, we use “analysis pixels” that are the flux average over three adjacent pipeline pixels. Throughout the rest of this paper, “pixel” refers to analysis pixels unless otherwise stated. The effective width of these pixels is , i.e. an observed-wavelength width . The total sample of 48,640 quasars thus provides measurements of Ly absorption over a total volume of .
Figure 2 (bottom panel) shows the distribution of the signal-to-noise ratio for pixels averaged over the forest region. The relatively modest mean value of reflects the exposure times necessary to acquire such a large number of spectra.
In addition to the BOSS spectra, we analyzed 15 sets of mock spectra that were produced by the methods described in appendix A. These spectra do not yet reproduce all of the characteristics of the BOSS sample, but they are nevertheless useful for a qualitative understanding of the shape of the measured correlation function. More importantly, they are useful for understanding the detectability of a BAO-like peak and the precision of the measurement of its position.
3 Measurement of the correlation function
The flux correlation function can be determined through a simple two-step process. In the first step, for each pixel in the forest region (equation 1) of quasar , the measured flux at observed wavelength is compared with the mean expected flux, , thus defining the “delta field”:
Here, is the unabsorbed flux (the so-called “continuum”) and is the mean transmitted fraction at the HI absorber redshift. The quantities and in equation 2 are not independent but related via .
In the second step, the correlation function is calculated as a weighted sum of products of the deltas:
where the are weights and each or indexes a measurement on a quasar at wavelength . The sum over is understood to run over all pairs of pixels of all pairs of quasars within defining a region in space of pixel separations, . The region is generally defined by a range and with:
where is the component along the line of sight. Separations in observational pixel coordinates (ra,dec,) are transformed to in units of by using a CDM fiducial cosmology with matter and vacuum densities of
In the sum (3), we exclude pairs of pixels from only one quasar to avoid the correlated errors in and coming from the estimate of . Note that the weights in eq. 3 are set to zero for pixels flagged by the pipeline as having problems due, e.g., to sky emission lines or cosmic rays.
A procedure for determining is defined by its method for estimating the expected flux and by its choice of weights, . The two methods described here use the same technique to calculate weights but have different approaches to estimate . We will see that the two methods produce correlation functions that have no significant differences. However, the two independent codes were invaluable for consistency checks throughout the analysis.
The two methods were “blind” to the extent that many of the procedures were defined during tests either with mock data or with the real data in which we masked the region of the peak in the correlation function. Among those aspects fixed in this way were the quasar sample, the continuum determination, the weighting, the extraction of the monopole and quadrupole correlation function and the determination of the peak significance (section 4). This early freezing of procedures resulted in some that are suboptimal but which will be improved in future analyses. We note, however, that the procedures used to extract cosmological information (section 5) were decided on only after de-masking the data.
3.1 Continuum fits, method 1
Both methods for estimating the product assume that is, to first approximation, proportional to a universal quasar spectrum that is a function of rest-frame wavelength, (for quasar redshift ), multiplied by a mean transmission fraction that slowly varies with absorber redshift. Following this assumption, the universal spectrum is found by stacking the appropriately normalized spectra of quasars in our sample, thus averaging out the fluctuating Ly absorption. The product for individual quasars is then derived from the universal spectrum by normalizing it to account for the quasar’s mean forest flux and then modifying its slope to account for spectral-index diversity and/or photo-spectroscopic miscalibration.
where is a normalization, a “deformation parameter”, and is the mean wavelength in the forest for the quasar and is the mean normalized flux obtained by stacking spectra in bins of width :
Here is the redshift of the absorption line at observed wavelength (), is the observed flux of quasar at wavelength and is the average of the flux of quasar for nm. The weight is given by . The quantity ivar is the pipeline estimate of the inverse flux variance in the pixel corresponding to wavelength . The quantity is the contribution to the variance in the flux due to the LSS. We approximate it by its value at the typical redshift of the survey, : (section 3.3).
Figure 4 shows the resulting mean as a function of observed wavelength. The mean fluctuates about zero with up to 2% deviations with correlated features that include the H and K lines of singly ionized calcium (presumably originating from some combination of solar neighborhood, interstellar medium and the Milky Way halo absorption) and features related to Balmer lines. These Balmer features are a by-product of imperfect masking of Balmer absorption lines in F-star spectroscopic standards, which are used to produce calibration vectors (in the conversion of CCD counts to flux) for DR9 quasars. Therefore such Balmer artifacts are constant for all fibers in a plate fed to one of the two spectrographs and so they are approximately constant for every ’half-plate’.
If unsubtracted, the artifacts in figure 4 would lead to spurious correlations, especially between pairs of pixels with separations that are purely transverse to the line of sight. We have made a global correction by subtracting the quantity in figure 4 (un-smoothed) from individual measurements of . This is justified if the variance of the artifacts from half-plate-to-half-plate is sufficiently small, as half-plate-wide deviations from our global correction could, in principle add spurious correlations.
We have investigated this variance both by measuring the Balmer artifacts in the calibration vectors themselves and by studying continuum regions of all available quasars in the DR9 sample. Both studies yield no detection of excess variance arising from these artifacts, but do provide upper limits. The study of the calibration vectors indicate that the square-root of the variance is less than 20% of the mean Balmer artifact deviations and the study of quasar spectra indicate that the square-root of the variance is less than 100% of the mean Balmer artifacts (and less than 50% of the mean calcium line deviations).
We then performed Monte Carlo simulations by adding a random sampling of our measured artifacts to our data to confirm that our global correction is adequate. We found that there is no significant effect on the determination of the BAO peak position, even if the variations are as large as that allowed in our tests.
3.2 Continuum fits, method 2
Method 1 would be especially appropriate if the fluxes had a Gaussian distribution about the mean absorbed flux, . Since this is not the case, we have developed method 2 which explicitly uses the probability distribution function for the transmitted flux fraction , , where . We use the that results from the log-normal model used to generate mock data (see appendix A).
Using , we can construct for each BOSS quasar the PDF of the flux in pixel , , by assuming a continuum and convolving with the pixel noise, :
The continuum is assumed to be of the form
where is the mean flux as determined by stacking spectra as follows:
as in equation 7 except that here there is no redshift binning. The parameters and are then determined for each quasar by maximizing a likelihood given by
Figure 3 shows the estimated for a typical quasar (blue line).
The last element necessary to use equation (2) is the mean transmitted flux fraction . If derived from the mocks were the true distribution of the transmitted flux fraction, then could simply be computed from the average of this distribution. Since this is not precisely true, we determine from the data by requiring that the mean of the delta field vanish for all redshifts. The we obtain is shown in figure 5. The unphysical wiggles in the derived are associated with the aforementioned residuals in for method 1 (figure 4).
There is one inevitable effect of our two continuum estimating procedures. The use of the forest data in fitting the continuum effectively forces each quasar to have a mean absorption near that of the mean for the entire quasar sample. This approach introduces a spurious negative correlation between pixels on a given quasar even when well separated in wavelength. This negative correlation has no direct effect on our measurement of the flux correlation function because we do not use pixel pairs from the same quasar. However, the physical correlation between absorption on neighboring quasars causes the unphysical negative correlation for individual quasars to generate a negative contribution to the correlation measured with quasar pairs. Fortunately, this distortion is a smooth function of scale so it can be expected to have little effect on the observability or position of the BAO peak. This expectation is confirmed by analysis of the mock spectra (section 5).
A discussion on the optimal use of weights for the Ly correlation function is found in McQuinn & White (2011). Here we simply choose the weights so as to approximately minimize the relative error on estimated with equation (3). In the approximation of uncorrelated pixels, the variance of is
where the pixel variance, , includes contributions from both observational noise and LSS. The signal-to-noise ratio is:
Because of LSS growth and redshift evolution of the mean absorption, the depend on redshift and we use the measured dependence of the 1d correlation function (McDonald et al., 2006)
Maximizing the signal-to-noise ratio with respect to this gives:
For this expression to be used, we require a way of estimating the . We assume that it can be decomposed into a noise term and a LSS term ():
where is the pipeline estimate of the noise-variance of pixel and is a factor that corrects for a possible misestimate of the variance by the pipeline.
We then organize the data in bins of and redshift. In each such bin, we measure the variance of , which serves as an estimator of for the bin in question. The two functions and can then be determined by fitting equation (16).
These fits are shown in figure 6. The top panel shows that the measured inverse variance follows the inverse pipeline variance until saturating at the redshift-dependent LSS variance (shown on the bottom left panel). For , there are not enough pixel pairs to determine and . In this high redshift range, we assumed and extrapolated with a second-degree polynomial fit to the data.
The procedure described above was used to determine through equation 3 in -bins of width (centered at 2,6,…, ) and in -bins of width 0.02, (centered at 0.01, 0.03, … 0.99). The bins have an average of terms in the sum (3) with an average nominal variance of for individual bins of as given by (eqn.12).
Figure 7 shows an example of for the bin centered on . The blue dots are the data and the red dots are the mean of the 15 mocks. The function falls from positive to negative values with increasing , as expected from redshift distortions. The effect is enhanced by the deformation due to the continuum subtraction.
Figure 8 presents averaged over three bins in . A clear peak at the expected BAO position, , is present in the bin corresponding to separation vectors within of the line-of-sight. The curves show the best fits for a CDM correlation function, as described in section 5.
The data were divided into various subsamples to search for systematic errors in . For example, searches were made for differences between the northern and southern Galactic cap regions and between higher and lower signal-to-noise ratio quasars. No significant differences were found in the overall shape and amplitude of the correlation function. We also verified that the BAO peak position does not change significantly when wavelength slices of Ly forest data are eliminated, in particular slices centered on the Balmer features in figure 4. The peak position also does not change significantly if the subtraction of the mean (figure 4) is suppressed.
4 The Monopole and quadrupole
The analysis of the correlation function was performed in the framework of the standard multipole decomposition (Hamilton, 1992). For each bin in we fit a monopole () and quadrupole () to the angular dependence:
where is the -Legendre polynomial. We ignore the small and poorly determined term. This fit is performed using a simple minimization with the nominal variance (equation 12) and ignoring the correlations between bins. This approach makes the fit slightly sub-optimal. (Later, we will correctly take into account correlations between -bins of the monopole and quadrupole.) We also exclude from this fit the portion to avoid residual biases due to correlated sky subtraction across quasars; this has a negligible impact on the fits and, at any rate, there is little BAO signal at low .
Figure 9 displays the monopole and quadrupole signals found by the two methods. The two methods are slightly offset from one another, but the peak structure is very similar. Figure 9 also shows the combination which, because of the small monopole-quadrupole anti-correlation (section 4.1), is a better-determined quantity. The peak structure seen in figure 8 is also present in these figures.
Because of the continuum estimation procedure (sections 3.1 and 3.2), we can expect that the monopole and quadrupole shown in figure 9 are deformed with respect to the true monopole and quadrupole. The most important difference is that the measured monopole is negative for while the true CDM monopole remains positive for all . The origin of the deformation in the continuum estimate is demonstrated in appendix A where both the true and estimated continuum can be used to derive the correlation function (figure 22). As expected, the deformation is a slowly varying function of so neither the position of the BAO peak nor its amplitude above the slowly varying part of the correlation function are significantly affected.
4.1 Covariance of the monopole and quadrupole
In order to determine the significance of the peak we must estimate the covariance matrix of the monopole and quadrupole. If the fluctuations in equation (3) in different pixels were uncorrelated, the variance of would simply be the weighted products of the fluctuation variances. This yields a result that is smaller than the true correlation variance that we compute below. The reason is, of course, that the -pairs are correlated, either from LSS or from correlations induced by instrumental effects or continuum subtraction; this effect reduces the effective number of pairs and introduces correlations between bins.
Rather than determine the full covariance matrix for , we determined directly the covariance matrix for and by standard techniques of dividing the full quasar sample into subsamples according to position on the sky. In particular we used the sub-sampling technique described below. We also tried a bootstrap technique (e.g. Efron & Gong, 1983) consisting of substituting the entire set of subdivisions of the data by of these subdivisions chosen at random (with replacement) to obtain a “bootstrap” sample. The covariances are then measured from the ensemble of bootstrap samples. Both techniques give consistent results.
The adopted covariance matrix for the monopole and quadrupole uses the sub-sampling technique. We divide the data into angular sectors and calculate a correlation function in each sector. Pairs of pixels belonging to different sectors contribute only to the sector of the pixel with lower right ascension. We investigated two different divisions of the sky data: defining 800 (contiguous but disjoint) sectors of similar solid angle, and taking the plates as defining the sectors (this latter version does not lead to disjoint sectors). The two ways of dividing the data lead to similar covariance matrices.
Each sector in each division of the data provides a measurement of that can be used to derive a monopole and quadrupole, , . The covariance of the whole BOSS sample can then be estimated from the weighted and rescaled covariances for each sector:
The average denoted by is the simple average over sectors, while denotes the correlation function measured for the whole BOSS sample. The are the summed pixel-pair weights for the radial bin for the sector and is the same sum for the whole BOSS sample.
The most important terms in the covariance matrix are the terms, i.e. the monopole and quadrupole variances. They are shown in figure 10 as a function of . In the figure, they are multiplied by the number of pixel pairs in the -bin. The product is nearly independent of , as expected for a variance nearly equal to the pixel variance divided by . For the monopole, the variances are only about 30% higher than what one would calculate naively assuming uncorrelated pixels and equation (12). Figure 10 also displays the monopole-quadrupole covariance times number of pairs, which also is nearly independent of .
We used the 15 sets of mock spectra to test our method for calculating the covariance matrix. From the 15 measurements of one can calculate the average values of and compare them with those expected from the covariance matrix. Figures 12 shows this comparison for the monopole and quadrupole variance, the monopole and quadrupole covariances between neighboring r-bins and the monopole-quadrupole covariance. The agreement is satisfactory.
4.2 Detection significance of the BAO peak
In this section, we estimate the significance of our detection of a BAO peak at . At the statistical power of the present data, it is clear that the peak significance will depend to some extent on how we treat the so-called “broadband” correlation function on which the peak is superimposed. In particular, the significance will depend strongly on the -range over which the correlation functions are fitted. To the extent that the BAO peak is known to be present in the matter correlation function and that the Ly absorption is known to trace matter, the actual significance is of limited interest for cosmology. Of greater interest is the uncertainty in the derived cosmological parameter constraints (section 5) which will be non-linear reflections of the peak significance derived here.
A detection of the BAO peak requires comparing the quality of a fit with no peak (the null hypothesis) to that of a fit with a peak. Typically, this exercise would be performed by choosing a test statistic, such as the , computing the distribution of this quality indicator from a large number of peak-less simulations and looking at the consistency of the data with this distribution. Since our mock data sets are quite computationally expensive and only a handful are available, we chose a different approach.
Our detection approach uses the following expression to fit the observed monopole and quadrupole.
where is a broadband term to describe the LSS correlation function in the absence of a peak, is a peak term, and is a “distortion” term used to model the effects of continuum subtraction. The broadband term is derived from the fiducial CDM cosmology defined by the parameters in equation (31). It is obtained by fitting the shape of the fiducial correlation function with an 8-node spline function masking the region of the peak . The peak term is the difference between the theoretical correlation function and the broadband term. Finally, the distortion term is calculated from simulations, as the difference in the monopole or quadrupole measured using the true continuum and that measured from fitting the continuum as described in appendix A. The three components are shown in figure 13.
Expression (19) contains three parameters each for the monopole and quadrupole (so six in total). We have performed fits leaving all six parameters free and fits where we fix the ratio to be equal to its nominal value used to generate our mock spectra (the value given by assuming a “redshift distortion parameter” , see appendix A). We define the test statistic as the difference between fitting equation (19) simultaneously to monopole and quadrupole by fixing to zero (a “peak-less” four or five-parameter fit) and fitting for (a five or six-parameter fit). In our detection fits we do not fit for the BAO position but fix it to the theoretical prediction. The distribution for this test statistic (“”) under the null hypothesis is a distribution with one degree of freedom. The significance is then given by .
Figure 14 shows the fits to monopole (top panel) and quadrupole (bottom panel) and the corresponding fits with and without peaks and fixing . For method 2, we obtain (111.8/86) with (without) a peak, giving for a detection significance of . For method 1, we obtain (102.2/86) for a significance of . Allowing to be a free parameter gives essentially the same detection significances.
The detection significance of is typical of that which we found in the 15 sets of mock spectra. For the mocks, the significances ranged from 0 to with a mean of .
Our significance depends strongly on the fitting range. For a lower boundary of the range of we obtain a significance of , 3.2 and 2.3, respectively (method 2). The reason for this result is illustrated in figure 15, where the results of the fits with and without peaks are compared to data for different values of . Reducing the fitting range poses less stringent constraints on the distortion and broadband terms, thus allowing some of the peak to be attributed to the broadband. In particular, the statistically insignificant bump in the quadrupole at causes the fitted broadband to increase as is increased to , decreasing the amplitude of the BAO peak. For the monopole, the fit predicts a positive slope for that decreases the amplitude of the peak but predicts a to be much less than what is measured.
5 Cosmology with the BAO peak
The observed position of the BAO peak in is determined by two sets of cosmological parameters: the “true” parameters and the “fiducial” parameters. Nature uses the true cosmology to create correlations at the true sound horizon, . The true cosmology transforms physical separations between Ly absorbers into angles on the sky and redshift differences: and . We, on the other hand, use a “fiducial” cosmology (defined by equation 31) to transform angular and redshift differences to local distances at the redshift in question to reconstruct . If the fiducial cosmology is the true cosmology, the reconstructed peaks will be at the calculated fiducial sound horizon, . Limits on the difference between the fiducial and reconstructed peak position can be used to constrain the differences between the fiducial and true cosmological models.
5.1 The peak position
The use of incorrect fiducial , and leads to shifts in the BAO peak position in the transverse and radial directions by the multiplicative factors and :
where the subscript refers to the fiducial model. Following Xu et al. (2012), we will use a fitting function, , for the monopole and quadrupole that follows the expected peak position as a function of :
Here, the two functions , derived from the power spectrum given in appendix A, describe the underlying mass correlation function, the linear bias and redshift distortion parameter , and the movement of the BAO peak for . The functions take into account distortions, as described below.
For , there is a simple isotropic scaling of the coordinates by and is given by , where are the fiducial monopole and quadrupole and the normalizations are the functions of the bias and redshift-distortion parameter given by Hamilton (1992). For , Xu et al. (2012) found an approximate formula for that was good in the limit . We take the more direct route of numerically expanding in Legendre polynomials, , to directly calculate the .
The functions describe broadband distortions due to continuum subtraction and the fact that the broadband correlation function is not expected to change in the same way as the BAO peak position when one deviates from the fiducial model. They correspond to the term in equation (19). We have used two forms to represent :
The observed monopole and quadrupole can then be fit to (22) with free parameters , , bias, , and the nuisance parameters ( and ).
We first fixed to determine if we find reasonable values of . These two parameters are highly degenerate since both the quadrupole and monopole have amplitudes that are proportional to times polynomials in . A well-determined combination is , for which we find a value ; this is in agreement with found at by Slosar et al. (2011). The larger error of our fit reflects the substantial freedom we have introduced with our distortion function.
We next freed all parameters to constrain . The contours for the two methods and two broadbands are shown in figure 16 and the for the fiducial and best-fit models are given in table 1. The broadband term in equation (24) fits the data better than that in equation (23) both for the fiducial parameters and for the best fit. For broadband in equation (24), the for the fiducial model is acceptable for both methods: for method 1 and for method 2.
The contours in the figure are elongated along the direction for which the BAO peak position stays approximately fixed at large (near the radial direction, where the observations are most sensitive). The best constrained combination of and of the form turns out to have . This low value of reflects the fact that we are mostly sensitive to the BAO peak in the radial direction. At the one standard-deviation level, the precision on this combination is about 4%. However, even this combination is sensitive to the tails in the contours. A more robust indicator of the statistical accuracy of the peak-position determination comes from fits imposing , as has generally been done in previous BAO studies with the exception of Chuang & Wang (2012) and Xu et al. (2012). This constraint does not correspond to any particular class of cosmological models. It does however eliminate the tails in the contours in a way that is similar to the imposition of outside data sets. The two methods and broadbands give consistent results, as seen in table 1.
|Method 1 (24)||85.0/80||84.6/78|
|Method 2 (24)||71.5/80||71.4/78|
|Method 1 (23)||104.3/82||99.9/80|
|Method 2 (23)||88.4/82||87.7/80|
We used the sets of mock spectra to search for biases in our measurement of . The mean value reconstructed for this quantity on individual mocks is , suggesting that there are no significant biases in the determination of the BAO scale. Figure 17 shows the values and errors for the individual mocks along with that for the data. Both the measured value and its uncertainty for the data is typical of that found for individual sets of mock spectra.
5.2 Constraints on cosmological models
Our constraints on can be used to constrain the cosmological parameters. In a CDM cosmology, apart from the pre-factors of that cancel, and evaluated at depend primarily on through and on which, with , determines and . The sound horizon also depends on (required to derive from ), on the effective number of neutrino species (required to derive the radiation density from the photon density), and on the baryon-to-photon number ratio, (required for the speed of sound).
Figure 18 shows the CDM constraints on derived from the contours in figure 16 combined with the most recent measurement of (Riess et al., 2011). We use the contours for method 2 and the broadband of equation 24 which gives better fits to the data than the other method and broadband. The contours also assume and the WMAP7 value of (Komatsu et al., 2011). Also shown are constraints from BAO measurements of (Percival et al., 2010; Anderson et al., 2012; Beutler et al., 2011).
The Ly contours are nicely orthogonal to the lower redshift measurements, reinforcing the requirement of dark energy from BAO data. In fact, our measurement is the only BAO measurement that by itself requires dark energy: . This is because at the universe is strongly matter dominated and the factor in partially cancels the in , enhancing the importance of the dependence of .
Figure 19 shows the constraints on ; where is the dark-energy pressure-density ratio) assuming a flat universe: . Our result is the only BAO measurement that by itself requires negative . Our limit requires matter domination at .
5.3 Constraints on
The contours in figure 16 give the measurements of given in table 1. A measurement of the expansion rate deep in the matter-dominated epoch can be used to demonstrate the deceleration of the expansion at that time. Unfortunately, our data are not yet precise enough to do this. To make a more precise measurement of , we must add further constraints to eliminate the long tails in figure 16. These tails correspond to models where is increased (resp. decreased) with respect to the fiducial value while is decreased (resp. increased). For flat models, this would imply a change in the mean of (averaged up to ) that is opposite to that of the change in , which requires a functional form that strongly differs from the fiducial case. It is possible to construct models with this property by introducing significant non-zero curvature.
Because of the importance of curvature, the tails are eliminated once WMAP7 constraints (Komatsu et al., 2011) are included. This is done in figure 20 within the framework of non-flat models where the dark-energy pressure-density ratio, , is determined by two parameters, and : . As expected, the WMAP7 results in this framework constrain and to migrate in roughly the same direction as one moves away from the fiducial model. Combining WMAP7 constraints with ours gives the values of given in the last column of table 1. For what follows, we adopt the mean of methods 1 and 2 that use the more flexible broadband of equation (24):
The precision on is now sufficient to study the redshift evolution of .
The fiducial model has Mpc and , . These results produce
|0.35||BAO||Chuang & Wang (2012)|
|0.35||BAO||Xu et al. (2012)|
|0.57||BAO + AP||Reid et al. (2012)|
|0.44||BAO + AP||Blake et al. (2012)|
|0.2||AP + SN||Blake et al. (2011b)|
|0||Riess et al. (2011)|
This number can be compared with the measurements of at lower redshift shown in table LABEL:hofztable and figure 21. Other than those of , the measurements that we use can be divided into two classes: those (like ours) that use as the standard of length and those that use as the standard of length.
The comparison with our measurement is simplest with BAO-based measurements that use as the standard of length and therefore measure (as is done here). The first attempt at such a measurement was made by Gaztañaga et al. (2009), a result debated in subsequent papers by Miralda-Escudé (2009), Yoo & Miralda-Escudé (2010), Kazin et al. (2010), and Cabré & Gaztañaga (2011). Here, we use four more recent measurements. Chuang & Wang (2012) and Xu et al. (2012) studied the SDSS DR7 LRG sample and decomposed the BAO peak into radial and angular components, thus extracting directly and . Blake et al. (2012) and Reid et al. (2012) took a more indirect route. They first used the angle-averaged peak position to derive . They then studied the Alcock-Paczynski effect on the broadband galaxy correlation function to determine . Combining the two measurements yielded .
It is evident from comparing our measurement (filled red circle in figure 21) to the other BAO-based measurements (other filled circles) that decreases between and . To demonstrate deceleration quantitatively, we fit the eight BAO-based values of in table 2 to the CDM form . Marginalizing over and we find
clearly indicating deceleration between and . This measurement is in good agreement with the fiducial value of 1.146. We emphasize that this result is independent of , assuming only that the BAO-peak position is redshift-independent in comoving coordinates. The result also does not assume spatial flatness.
To map the expansion rate over the full range , we must adopt the fiducial value of and compare the resulting with and with other BAO-free measurements. Besides the measurement of Riess et al. (2011), we use the WiggleZ analysis combining their Alcock-Paczynski data with distant supernova data from the Union-2 compilation (Amanullah et al., 2010). The supernova analysis does not use the poorly known mean SNIa luminosity, so the SNIa Hubble diagram gives the luminosity distance in units of , . Combining this result with the Alcock-Paczynski measurement of yields . The values are given in table LABEL:hofztable.
We fit all the data in table LABEL:hofztable (filled and open circles in figure 21) to the CDM form of . This yields an estimate of the redshift of minimum
which compares well with the fiducial value: .
In this analysis, we have not used two other sources of information on at high redshift. The first use high-redshift type Ia supernovae to probe the era where the universe transitions from deceleration to acceleration (e.g., Riess et al. (2004, 2007)). The data of Riess et al. (2007) (plotted as the open squares in figure 21)) yielded useful measurements up to . However, this data yields constraints on that are weaker than those of BAO-based methods because of the need to differentiate the distance-redshift relation. Moreover, these inferences of assume spatial flatness. Fitting the SNe data to a model with an evolving deceleration parameter and assuming flatness, Riess et al. (2007) and Riess et al. (2004) were able to demonstrate that , i.e. a negative 3rd-derivative of . However, we point out that in a more general model, the demonstration that at low redshift is not equivalent to a demonstration that becomes negative in the past.
Another approach to determining uses the evolution of stellar populations as a clock to infer (Stern et al., 2010; Moresco et al., 2012). This method yields results that are consistent with expectations, but the uncertainties (statistical and systematic) are larger than those of the determinations in Table 2, so we have not plotted them in figure 21.