On the Observed – Correlation in SDSS QSO Spectra
This paper investigates the effect of differential aperture loss with SDSS fibers and examines whether such selection bias would result in the observed correlation between rest-frame absorption equivalent width of Mg II absorbers, , and mean associated [O II] luminosity, , in SDSS QSO spectra. We demonstrate based on a Monte Carlo simulation that the observed vs. correlation of Mg II absorbers can be well-reproduced, if all galaxies found in deep surveys possess extended Mg II halos and if the extent of Mg II halos scales proportionally with galaxy mass as shown in previous studies. The observed correlation can be explained by a combination of (1) the known vs. anti-correlation in galaxy and Mg II absorber pairs and (2) an increasing aperture loss in the diameter SDSS fiber for galaxies at larger . Galaxies at larger projected distances produce on average weaker Mg II absorbers and weaker (or zero) in SDSS QSO spectra. We show that such correlation diminishes when larger fibers are adopted and is therefore not physical. While under a simple halo model the majority of Mg II absorbers do not directly probe star-forming disks, they trace photo-ionized halo gas associated with galaxies. We show that because of the scaling relation between extended gas cross-section and galaxy mass, the number density evolution of the Mg II absorber population as a whole provides a good measure of the cosmic star formation history.
keywords:galaxies:halos – galaxy: star formation – quasars: absorption lines – survey
Absorption line spectroscopy is a powerful tool for studying the structure of the distant universe. By observing the absorption features imprinted in the spectra of background QSOs, we can study otherwise invisible gaseous structures to redshift as high as background QSOs can be. The Sloan Digital Sky Survey (SDSS; York et al. 2000) has yielded optical spectra of k QSOs at (Schneider et al. 2010). This unprecedentedly large sample of QSO spectra offers a rich resource for studying the distant universe using absorption spectroscopy. For example, Mg II 2796,2803 absorption doublets are commonly seen in QSO spectra obtained using ground-based spectrographs. They provide a uniform probe of intervening gas over a broad redshift range from to . Several groups have conducted systematic searches of Mg II absorption features in SDSS QSO spectra, producing a large sample of these absorbers for constraining their statistical properties (e.g. Bouché et al. 2004; Nestor et al. 2005; Prochter et al. 2006; York et al. 2006; Quider et al. 2011).
Mg II absorbers are routinely found in the vicinities ( kpc) of distant galaxies and provide a convenient probe of galactic halos at high redshifts (e.g. Bergeron 1986; Steidel et al. 1994; Kacprzak et al. 2008; Chen & Tinker 2008; Gauthier & Chen 2011). However, their physical origin, whether the absorbers arise in infalling clouds, outflows from nearby star-forming regions, or a combination thereof, is unclear. The ubiquitous presence of blueshifted Mg II absorption features along the sightlines into the star-forming regions of galaxies indicates that starburst driven outflows are common in distant galaxies and that outflows contribute to some fraction of Mg II absorbers uncovered along random QSO sightlines (e.g. Weiner et al. 2009; Rubin et al. 2010). Although the extent of galactic-scale winds around these galaxies is unknown, the observed self-absorption of Mg II doublets in distant star-forming galaxies has motivated recent works that attribute all strong Mg II absorbers (rest-frame absorption equivalent width Å) along QSO sightlines to outflows (e.g. Ménard & Chelouche 2009; Chelouche & Bowen 2010; Ménard et al. 2011).
Using a sample of 8500 Mg II absorbers at from SDSS DR4 quasar spectra, Ménard et al. (2011; hereafter M11) observed a strong correlation between and their associated median [O II] luminosity per unit area, , in stacked QSO spectra. The observed vs. correlation is characterzed by
where and . This observed correlation applies to Mg II absorbers of Å. A similar trend has also been mentioned in Noterdaeme et al. (2010), but these authors did not find a correlation between and in [O II]-emission selected Mg II absorbers. In addition, Noterdaeme et al. (2010) pointed out that part of their strong Mg II absorbers arise in low galaxies. Because [O II] luminosity provides a measure of current star formation rate (e.g. Kennicutt 1998), M11 interpreted the observed strong correlation as Mg II absorbers tracing on-going star formation. In addition, M11 showed that the frequency distribution function of Mg II absorbers and the [OII] luminosity function share similar shape and amplitude, and that the number density evolution of MgII absorbers follows the cosmic star formation history. Combining these empirical correlations, the authors argue that outflows are the mechanism responsible for the observed Mg II absorption in QSO spectra.
The conclusion of outflows being responsible for the observed Mg II absorbers appears to be discrepant from previous findings that correlates more strongly with galaxy mass and weakly with galaxy colors or recent star formation history (e.g. Steidel et al. 1994; Chen et al. 2010a,b). Such conclusion also makes it difficult to understand the identifications of strong Mg II absorbers in the vicinities of quiescent galaxies (e.g. Gauthier et al. 2010; Bowen & Chelouche 2011; Gauthier & Chen 2011).
While M11 presented a clever approach to estimate the co-moving star formation rate density based on the observations of Mg II absorbers in SDSS QSO spectra, it is important to understand the underlying factors that shape the observed strong vs. correlation in SDSS data. We note that the SDSS fibers have a finite size of diameter on the sky, which corresponds to projected physical separations of kpc at . The observed vs. anti-correlation (e.g. Chen et al. 2010a) implies that galaxies at larger projected distances produce on average weaker Mg II absorbers and lower (or zero) (when the star-forming disks occur at angular distances , or kpc, of the QSO sightlines; see Figure 1). This selection bias strenthens the apparent correlation of Equation (1)111Ménard et al. (2011) applied a fiber selection correction by considering luminosity surface density, , where represents the corresponding physical area of SDSS fiber at redshift . However, as illustrated in Figure 1, the reduction from total observed fluxes to flux surface density does not correct for the differential aperture loss of galaxy fluxes in SDSS QSO spectra.. A similar point on the potential missed galaxy light in the SDSS fibers has also been made in Noterdaeme et al. (2010).
To investigate the selection bias as a result of differential aperture loss, we have carried out a Monte Carlo simulation study. Adopting a empirical model for describing gaseous clouds around galaxies from Chen et al. (2010a) and empirical relations for describing the luminosity and size distributions of the general galaxy population, we demonstrate in this paper that the observed vs. correlation can be well re-produced without any fine-tuning after accounting for the differential aperture loss of galaxy fluxes in the SDSS fiber. While under the simple halo model the majority of Mg II absorbers do not directly probe star-forming disks like high column density damped absorbers (e.g. Wild et al. 2007), they do probe photo-ionized clouds around distant galaxies222Note that the empirical vs. relation of Chen et al. (2010a) was established based on a sample of galaxies at close projected distances to a QSO sightline, including those that may produce a damped absorption feature in the QSO spectrum. We therefore expect that absorber samples generated based on the mean – relation and the observed scatter include contributions from star-forming disks. Based on the observed number densities of damped absorbers (Rao et al. 2006) and Mg II absorbers of Å (Nestor et al. 2005) at , we estimate that no more than 20% of these strong Mg II absorbers have contributions from star-forming disks If more massive galaxies are surrounded by more extended Mg II absorbing gas, then the number density evolution of MgII absorbers naturally follows the cosmic star formation history. Because models which do not require outflows can also reproduce the empirical correlations between absorber abundances and star formation properties, we caution drawing conclusions in favor of an outflow origin for QSO absorbers based on these simple correlations. Throughout this paper, we adopt a CDM cosmology with and and a dimensionless Hubble constant of .
2 Mock Galaxy and Absorber Catalogs
To simulate the SDSS observations, we first generate a mock catalog of 200,000 galaxies distributed uniformly within angular radius of a QSO sightline at redshift between and . The large number is necessary to provide a representative sampling of a wide range of galaxy properties (such as size and luminosity) and to minimize statistical noise in our simulations. The redshift range is selected to match the study of M11. Over the redshift range of , the angular radius of corresponds roughly to kpc.
The working hypothesis of this exercise is that all galaxies are surrounded by extended gaseous halos. This hypothesis is supported by empirical observations that reveal a high gas covering fraction around galaxies of a wide range of luminosity and color (e.g. Chen et al. 2010a). The gaseous halos are expected to produce Mg II absorption features in the spectrum of a background QSO when intercepting the QSO sightline. For every Mg II absorber in the mock catalog, the location and intrinsic properties of the absorbing galaxy are known. It is therefore possible to compute the emission fluxes of the absorbing galaxies recorded in QSO spectra.
2.1 The Mock Galaxy Catalog
We first generate random galaxies following a Schechter probability function of their rest-frame absolute -band magnitude ,
where is the faint-end slope of the galaxy luminosity function. We adopt for the faint-end slope (e.g. Faber et al. 2007) and following Chen et al. (2010a). The mock galaxy sample spans a lumionsity range between and . As discusse below, we also repeat the Monte Carlo simulations with different faint-end slop values, and and our findings remain the same.
Next, we determine the optical size of each galaxy in the mock catalog according to the luminosity-size relation from Cameron & Driver (2007),
where is the half-light radius, the radius within which the galaxy emits half of its total flux. For a galaxy of , the half-light radius is drawn from a Gaussian distribution about the mean luminosity-size relation with a 1- width of 0.4 dex in the space. To include the majority of the galaxy light, the radius that contains 90% of the total flux is more relevant in our study. We therefore convert to , assuming an exponential surface brightness profile.
Next, we determine the [O II] luminosity for each galaxy assuming that the rest-frame -band fluxes trace the mean profile of [O II] emission. The expected is then calculated according to the correlation between and found in deep survey data. Based on the study of Zhu et al. (2007), we find that the vs. correlation can be characterized by
with a 1- scatter of dex. Note that Equation (4) is an empirical relation between observed quantities. It does not include dust extinction corrections. The inferred can therefore be directly compared to what is observed in the stacked SDSS QSO spectra. For a galaxy of , is drawn from a Gaussian distribution about the mean - relation with a 1- width of 0.3 dex in the space.
To examine whether or not our mock galaxy sample is representative of the field galaxy population, we present the [O II] luminosity distribution of our mock galaxy sample in Figure 2 along with the observed [O II] luminosity function of galaxies from different deep surveys (Takahashi et al. 2007; Zhu et al. 2009). All the empirical measurements have been converted to have the same cosmological parameters adopted in our analysis. Figure 2 demonstrates that we reproduce the observed [O II] luminosity function with the mock galaxy sample for the luminous galaxy population. At the faint-end, the observations suffer from survey incompletness (e.g. Zhu et al. 2009) and therefore the observed space density represents a lower limit to the underlying faint galaxy population.
Next, we model each galaxy in the mock catalog as a round disk and randomly place the galaxy within of a QSO sightline and at redshift between and . We compute the corresponding physical projected distance of the galaxy based on the redshift and angular distance to the QSO. For each disk, we also assign a random inclination angle with respect to the observer and a random position angle of the major axis of the inclined disk with respect to the line connecting the galaxy and the QSO.
To determine the fraction of the luminous disk that overlaps the SDSS fiber centered at the QSO, we adopt the size of each disk galaxy and its relative distance and orientation to the QSO. We de-project the inclined disk and determine the galactocentric distance of each point (, ) in the disk according to
where is the projected radius on the plane of the sky, is the position angle of the major axis of the inclined disk, is the azimuthal angle of point (, ) from the major axis, and is the inclination angle of the disk. The expected [O II] emission from the galaxy in the QSO spectral data is then computed by integrating all the light within of the disk that falls in the diameter fiber. We consider two different surface brightness profiles: (1) a flat, top-hat profile and (2) an exponential profile for this calculation. If the luminous disk does not overlap the diameter fiber, then we set . Finally, we divide the computed by the physical area of a diameter fiber at the redshift of the mock galaxy in order to calculate its [O II] luminosity surface density that would be recorded in an SDSS QSO fiber.
2.2 The Mock Mg II Absorber Catalog
The mock Mg II catalog is formed by calculating the expected Mg II absorption strength in the spectrum of the background QSO for every galaxy in the mock galaxy catalog. To determine the associated Mg II absorption strength of a galaxy in the QSO spectrum, we adopt the uniform gaseous halo model of Chen et al. (2010a). Under this model, the Mg II absorber strength for a galaxy at projected distance is characterized by a mean relation of
and dispersion , where is the gaseous radius of the Mg II halo, is the core radius and is and . The gaseous radius is determined based on the galaxy -band luminosity following a power-law model,
for which Chen et al. (2010a) found a best-fit characteristic radius of kpc for an galaxy and a scaling index of . The scaling relation between gaseous radius and galaxy -band luminosity is understood as more massive galaxies are surrounded by more extended halos (e.g. Tinker & Chen 2008; Chen et al. 2010b). In addition to the spatial profile of the extended Mg II gas, Chen et al. (2010a) also measured a high gas covering fraction within . Specifically, they found a mean covering fraction of % for absorbers of Å at and % at . The gas covering fraction increases with decreasing threshold and decreases with (see Figure 10 in Chen et al. 2010a).
For each simulated galaxy, we compute the expected by randomly sampling the mean relation Equations (6) within the observed scatter . For simulated dwarf galaxies with , we set . Including the large scatter () in the computation of allows the possibility of galaxies at small producing Mg II absorbers that are much weaker than the mean. Although every galaxy at in the mock sample is expected to produce a Mg II absorber, the gas covering fraction measured at a given threshold is not 100%.
For a mock sample of 200,000 galaxies, the procedure described above produces a mock sample of Mg II absorbers of Å at . The frequency distribution function of the mock Mg II absorber sample is presented in Figure 3. For comparison, we also include in Figure 3 the model from Prochter et al. (2006) that best describe the incidence of Mg II absorbers identified at in their blind survey. Aside from the offset in the normalization, Figure 3 shows that the frequency distribution function of the mock Mg II absorber sample follows the shape of the best-fit model of Prochter et al. (2006). The difference in the incidence of mock Mg II absorbers (up to 20%) for different adopted faint-end slope of the galaxy luminosity function is understood by the relatively steep dependence of halo gas cross section on galaxy luminosity, from Equation (7). Adopting known galaxy luminosity functions at from Faber et al. (2007) and the scaling relation of Equation (7), we expect to find (following Equation 8 below) a number density of per line of sight for Mg II absorbers of Å. Our model therefore well reproduces the observe absorber statistics at (e.g. Nestor et al. 2005; Prochter et al. 2006).
Using the mock galaxy and Mg II absorber catalogs, we proceed to examine the relation between and of Mg II absorbers. For each Mg II absorber in the mock catalog, we know from our simulation the location and luminosity of the absorbing galaxy. We have also calculated according to the procedures described in Section 2.1 the fraction of galaxy light that would be recorded in the SDSS QSO fibers. We can therefore directly compare our simulation data with the observations of M11.
We present in Figure 4 the distribution of and of the mock Mg II absorber sample. The top panel shows the – distribution of individual Mg II absorbers, including detections (dots) and non-detections (indicated by the arrows). The bottom panel shows the fraction of Mg II absorbers for which the absorbing galaxies are not expected to overlap the fiber and therefore have in the QSO spectra. It is clear that a growing fraction of galaxy light is missed in the QSO spectra for absorbers of decreasing strength due to the vs. anti-correlation (e.g. Chen et al. 2010).
To reproduce the observations in SDSS QSO spectra, we divide the mock catalog of Mg II absorbers into subsamples according to their absorption strengths. Following the proceduce described in M11, who formed a median QSO spectrum at the rest-frame of the Mg II absorbers and measured the associated [O II] emission line flux, we compute the median value of [O II] luminosity surface density for all Mg II absorbers in each bin, including those with . To estimate the scatter, we repeat the Monte Carlo simulation 100 times to generate 100 mock samples of galaxies and Mg II absorbers. We measure the 1- dispersion in among the 100 mock samples. The stars and the associated errorbars in Figure 4 represent the computed and the associated dispersion in each bin.
For comparison, the best-fit power-law model of the observed – correlation (Equation 1) from M11 is included in Figure 4 as the dash-dotted line. The M11 model is found to match well with the simulated data. However, both fall below the locus defined by individual dots because of non-detections quantified in the bottom panel. We also calculate the mean [O II] flux expected in the stacked SDSS QSO spectra, averaged over all Mg II absorbers in each bin including non-detections. The results are shown in solid circles in Figure 4.
4 Discussion and Summary
We have carried out a Monte Carlo study to investigate the effect of differential aperture loss of extended emission from intervening galaxies in SDSS fibers. We generate a mock galaxy sample based on known empirical relations that describe the luminosity and size distribution of the general galaxy population uncovered in deep surveys. The mock galaxy sample is accompanied by a mock Mg II absorber sample that is generated based on a simple assumption that extended gaseous halos are a common and generic feature of distant galaxies and the gaseous extent scales proportionally with galaxy mass. For each absorber in the mock sample, the luminosity and projected distance of the absorbing galaxy are known, allowing us to make predictions for the observed relation between and the associated , in SDSS QSO spectra.
Our study shows that combining the known vs. anti-correlation of Mg II–galaxy pairs (e.g. Chen et al. 2010a) and differential fiber selection of Mg II absorbing galaxies (Figure 1) reproduces the observed vs. correlation in the SDSS QSO spectra without additional fine-tuning or scaling. The results of our study indicate that the observed vs. correlation of Mg II absorbers in SDSS data is likely due to a differential fiber selection bias and does not provide a physical understanding of the origin of the Mg II absorber population.
On the basis of the the observed vs. correlation, M11 further attempted to draw connections between the number density evolution of Mg II absorbers and the cosmic star formation history of the universe as characterized by the comoving [O II] luminosity density . Given a good agreement between and , the authors argue that outflows are the mechanism responsible for Mg II absorption.
The Monte Carlo simulation presented in Section 2 also allows us to address the agreements in these measurements under a simple, generic halo model. It is clear from Figures 3 & 4 show that both the frequency distribution function of Mg II absorbers and the galaxy [O II] luminosity function are well reproduced in our mock galaxy and absorber samples with no preference in starburst systems. Although under the simple halo model Mg II absorbers do not directly probe star-forming disks like high column density damped absorbers (e.g. Wild et al. 2007), they trace the halo gas of distant galaxies. If more massive galaxies are surrounded by more extended Mg II absorbing gas (Equation 7), then is calculated according to
where is the speed of light, is the galaxy luminosity function, is the incidence of extended gas that is the product of halo gas covering fraction and the fraction of galaxies with extended gaseous halos, and is the cross section of the gaseous halo. As mentioned in § 2.2, the mean covering fraction of Mg II absorbing gas is found to be high, roughly 100% for absorbers of Å at (Chen et al. 2010a). Such high covering fraction is consistent with the analysis presented in Kacprzak et al. (2008) for a scaling index of .
Combining Equations (4) and (7) yields
Substituting Equation (9) into Equation (8) leads to
where is defined so that . The same redshift dependent factor is defined in Equation (7) of M11. Equation (10) shows that the number density evolution of absorbers naturally follows the comoving [O II] luminosity density, , if extended gaseous halos are a common and generic feature of field galaxies. One can therefore apply for constraining the cosmic star formation history, or apply known co-moving luminosity density for constraining the fraction of absorbers originating in galactic halos (Chen et al. 2000).
In practice, as illustrated in M11 the SDSS fibers define a survey volume of [O II] emitting galaxies. The comoving [O II] luminosity density, , can be estimated according to
where is the total [O II] luminosity observed in a coadded QSO spectrum, is the total number of Mg II absorbers over the survey redshift pathlength , and is the mean [O II] luminosity averaged over the entire galaxy (absorber) population. It is clear that Equation (11) resembles Equation (10) and that can be estimated based on the product of and the mean [O II] luminosity surface density averaged over all Mg II absorbers.
We present in Figure 4 the mean [O II] luminosity surface density measured in the SDSS fibers and averaged over the number Mg II absorbers in each bin (solid circles). The mean values can be characterized by
where and (solid line in Figure 4). This correlation applies to a survey cylinder at . Note that because of the extreme bimodal distribution of in which a large fraction of absorbers have , the median value is expected to be significantly smaller than the mean .
While Equation (11) provides a useful tool for estimating the co-moving star formation rate density based on the observations of Mg II absorbers in SDSS QSO spectra (as shown cleverly in M11), we note that the observed strong vs. correlation in SDSS data is shaped primarily by a differential fiber loss of the observed [O II] flux and therefore unphysical. To illustrate the fiber selection bias in driving the apparent correlation in Figure 4, we repeat the Monte Carlo analysis described in § 2 for diameter fibers. This is five times the aperture size of SDSS spectra. At , such a large aperture covers an area of roughly kpc projected radius from QSO lines of sight, sufficient to cover the majority of the light from galaxies producing strong Mg II absorbers ( Å). Figure 5 shows the distribution of and measured over diameter fibers for the mock Mg II absorber sample. The mean [O II] flux share a similar slope as the median [O II] flux within the diameter aperture and the slope is consistent with (solid line in Figure 5).
In summary, we have demonstrated that (1) because of the observed vs. anti-correlation (e.g. Chen et al. 2010a), galaxies at larger projected distances produce on average weaker Mg II absorbers and weaker (or zero) in SDSS QSO spectra and that (2) because of the extreme bimodal distribution of in which a large fraction of absorbers have , the median value can be significantly smaller than the mean . Together these effects strenthen the apparent – correlation of M11.
Consequently, empirical correlations between star-forming properties of galaxies and statistical properties of QSO absorbers do not provide the unambiguous evidence necessary to discriminate between infalling clouds and outflows as the mechanism for producing Mg II absorbing clouds at kpc of a galaxy. While outflows are a natural product of starbursts, gas accretion provides the fuels for star formation in galaxies. As discussed by previous authors, mass is likely a more fundamental factor that determines the properties of gaseous halos around galaxies (e.g. Steidel et al. 1994; Ledoux et al. 2006; Chen et al. 2010b) and more massive galaxies can support more extended gaseous halos. Given that models that do not require outflows can also reproduce the empirical correlations between absorber and star formation properties, we caution drawing conclusions in favor of outflows based on simple correlations between absorber and star formation properties.
We thank Jean-René Gauthier, Michael Rauch, Rob Simcoe, Ben Weiner, Vivienne Wild, and Art Wolfe for helpful discussions and comments. G.L. acknowledges support from the Physics REU program at the University of Chicago.
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