Calibrating Milky Way dust extinction using cosmological ... · Fitzpatrick (1999) (FTZ), plotted in Fig. 2 together with the (ar-bitrarily normalised) SDSS ﬁlter transmission curves

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3Astronomy & Astrophysicsmanuscript no. paper c© ESO 2018November 1, 2018

Calibrating Milky Way dust extinction using cosmological sourcesEdvard Mortsell1,2

1 Department of Physics, Stockholm University, AlbaNova University Center, SE 106 91 Stockholm, Sweden2 The Oskar Klein Centre for Cosmoparticle Physics, Stockholm University, SE 106 91 Stockholm, Sweden

e-mail:[email protected]

Received/ Accepted

ABSTRACT

Aims. To constrain the light extinction properties of Milky Way dust.Methods. We investigated the correlations between dust column density as inferred from infrared data and the observed colours ofcelestial objects at cosmological distances with low levels of colour dispersion.Results. Results derived using colours of quasars, brightest central galaxies, and luminous red galaxies are broadly consistent, in-dicating a proportionality constant between the reddeningE(B − V) ≡ AB − AV and the dust column densityDT (given in units ofMJy/sr) of p = E(B−V)/DT ∼ 0.02 and a reddening parameterRV ≡ AV/E(B−V) ∼ 3 with fractional uncertainties of approximately10 %. The data do not provide any evidence for spatial variations in the dust properties, except for a possible hint of scatter in the dustextinction properties at the longest optical wavelengths.

Key words. interstellar medium: dust, extinction – Galaxy: general

1. Introduction

Astronomical data need to be corrected for extinction from dustin the Milky Way. Typical values (albeit with large variations)for the reddeningE(B − V) ≡ AB − AV is 0.02 with AV ∼ 0.06,see Fig. 1. In this paper, we reinvestigate the relationshipbe-tween the Milky Way reddening and the dust maps generated inSchlegel et al. (1998) using colours of quasars (QSOs) observedwithin the Sloan Digital Sky Survey (SDSS). We also comparewith results from SDSS observations of brightest central galax-ies (BCGs) and luminous red galaxies (LRGs). Compared withthe original investigation in Schlegel et al. (1998), the improvedstatistics and multiple band photometry give us the possibilityto decrease the errors on a linear relation while simultaneouslyconstraining the properties of Milky Way dust, in this case thetotal to selective extinction ratioRV ≡ AV/E(B − V).

We also performed a simple check to investigate if there aredeviations from the linear relationship, in other words whetherregions with high dust column density have more or less redden-ing than the linear prediction. No such effect is found, nor dowe find any firm evidence of scatter in the dust properties as afunction of the line-of-sight.

While noting that there is still room for non-linear behaviour,as well as scatter in dust properties at angular positions andscales not probed by the data, we conclude that at the positionsand scales studied in this paper, it is a fair assumption thatthereddening is linear to the dust column densityDT (in units ofMJy/sr) with proportionality constantp = E(B − V)/DT ∼ 0.02andRV ∼ 3. The relative errors are of the order of 10 % and aremainly limited by uncertainties in the filter response functionsand the source spectral templates.

This agrees well with the results of Berry et al. (2011) whereSDSS photometry of 73 million stars is used to obtain the dustextinction towards each star giving a three-dimensional dust mapof the Milky Way in the direction of the observed stripes. The

results from this analysis are consistent withRV = 3.0, with astatistical and systematic error of 0.1 and a scatter of 0.2.

In Peek & Graves (2010), the colours of passively evolvinggalaxies from the SDSS are used to obtain corrections to the re-sults of Schlegel et al. (1998). They find that the reddening inregions of low dust temperature are underpredicted with maxi-mum deviations inE(B − V) of 45 millimagnitudes.

Schlafly et al. (2010) use the blue edge of SDSS main-sequence stars to recalibrate the reddening induced by thedust maps of Schlegel et al. (1998). They find that the redden-ing is best described by a Fitzpatrick extinction curve withRV = 3.1, whereas the extinction curve originally employed inSchlegel et al. (1998) generically overestimates the reddening.Similar results were obtained in Schlafly & Finkbeiner (2011)where very similar methods were used, but the reddening wasmeasured by comparing the observed colours of SDSS stars tothe colours predicted from their spectral features.

In the following, all limits are given as 95.4 % confidencelevels (CL) corresponding toχ2 = χ2

min+ 3.98 for one parameterandχ2 = χ2

min + 6.16 for two parameters.

2. Method

We used the temperature corrected dust maps fromSchlegel et al. (1998) to derive a reddening map by corre-lating the dust maps to the observed colours of celestial objects,preferably with a low intrinsic colour dispersion. The dustmap, given as the dust column densityDT in units of MJy/sr,is derived from Infrared Astronomical Satellite (IRAS) data,in principle allowing for a resolution of approximately fivearcminutes. However, these data are calibrated and temperaturecorrected using degree resolution Cosmic Background Explorer(COBE) Diffuse Infrared Background Experiment (DIRBE)data. We note that in Kohyama et al. (2010), IRAS data are usedto estimate the dust temperature in order to reach a resolutionof five arcminutes also for the temperature corrected data ofthe

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http://arxiv.org/abs/1210.2191v2

Edvard Mortsell: Calibrating Milky Way dust extinction using cosmological sources

Fig. 1. Galactic dust extinction represented asE(B − V) andplotted with degree resolution for Galactic latitudes|b| > 20◦, to-gether with the corresponding probability distribution function.This dust map peaks atE(B−V) ∼ 0.016 mag, with a mean valueof 0.048 mag and a median value of 0.028 mag. Data taken fromSchlegel et al. (1998).

Cygnus region. However, no all sky map data have yet beenproduced using this technique.

As a first attempt, we assumed that the observed colour, e.g.E(B − V), is proportional to the dust column density

E(B − V) = pDT + m, (1)

where we set out to derive the proportionality constantp, andmrepresents an arbitrary offset that is marginalised over (possiblycaused by calibration errors giving the wrong mean colour) andthe dust column density is given in units of MJy/sr, makingp di-mensionless. In Schlegel et al. (1998), a value of the reddeningparameterRV = 3.1 was assumed, yielding a value of the pro-portionality constantp = 0.0184± 0.0014 for Mg2-calibratedBCGs.

Note that this can, in principle, be done for any set of filtersX andY

E(X − Y) = kXY DT + mXY , (2)

and that multiple colour data thus can be used to infer dust prop-erties.

2.1. Fitting for RV and p

Since SDSS data allow for multiple colour combinations, weare also able to fit for the properties of the Milky Way dustand relax any prior assumptions on the reddening parameterRV . The SDSS [u, g, r, i, z] filters have central wavelengths at[3543, 4770, 6231,7625,9134]Å and we thus expectg−r and/oru − r to correspond the most closely toB − V.

For each source and filter, we calculate the transmitted fluxusing a redshifted source template. This is done for a grid ofdust models in order to compute how each dust model will af-fect a given colour. Note that in doing this, we need to calcu-late the transmitted flux for each dust model separately since k-corrections will depend on the dust model. In practice, we havecreated tables that list the necessaryk-corrections over a grid ofredshiftsAV andRV for each filter that we interpolate between.After calculating how a given dust model affects each sourcetemplate in each filter, we can compare with the actual observed

coloursE(X − Y) allowing us to constrain the parametersp andRV .

Normalising such that zero column density corresponds tozero reddening, i.e.m = 0 in Eq. 1, definingf (λ) ≡ A(λ)/AV andusing

AV = pRV DT , (3)

we can write

A1 − A2

DT≡∆mDT= pRV( f1 − f2) , (4)

and

d

(

∆mDT

)

= dpRV ( f1 − f2) + pdRV( f1 − f2) + pRVd( f1 − f2) , (5)

showing how uncertainties inDT , ∆m, and the dust extinc-tion curve f (λ) translates into uncertainties inp and RV . Inthe following, we will make use of the Milky Way extinctioncurvesf (λ) as parametrised in Cardelli et al. (1989) (CCM) andFitzpatrick (1999) (FTZ), plotted in Fig. 2 together with the (ar-bitrarily normalised) SDSS filter transmission curves. Theex-tinction curve of O’Donnell (1994) gives results very closetothose derived using the CCM paramterisation.

We now turn to the expected degeneracy betweenRV andpand how this can be broken using multiple colour data. In theCCM-model, we havef = a + b/RV and f1 − f2 = (a1 − a2) +(b1 − b2)/RV ≡ ∆a + ∆b/RV . Settingd

(

∆m/DT)

= 0 in Eq. 5and puttingd(∆a) = d(∆b) = 0, i.e. no systematic error in theextinction curves, we obtain

dpp=

−dRV

RV + ∆b/∆a. (6)

This can be integrated to give

p = p0RV,0 + ∆b/∆a

RV + ∆b/∆a, (7)

describing the relation between different values ofRV andp giv-ing the same extinction per unit column density∆m/DT . Thedegeneracy lines for the SDSS standard coloursu− g (dark red),g − r (red),r − i (orange) andi − z (yellow) are plotted in Fig. 3,displaying the power of multiple colour data to break the degen-eracy betweenp andRV . We see thatp is anti-correlated withRV , if the difference (AX − AY )/AV is decreasing more slowlythan 1/RV with RV , which is the case for the three reddest SDSSstandard colours. The slopes of these curves compare well withthe confidence contours derived from data in the forthcomingsections.

3. Error analysis

We first note that constant filter calibration errors do not affectour results, since we are marginalising over any constant off-set between the observed and theoretically calculated colours.However, we may get systematic effects if the spectral templateis wrong and/or if the shape of the filter response function isincorrectly modeled since this will change the effective wave-length of the filters. We assume that this will induce fractionaluncertainties in the effective wavelengths of [0.02, 0.01, 0.01,0.01, 0.01] in [u, g, r, i, z] filters (Doi et al. 2010). We note thatthis error will be a true systematic error since the effect will becorrelated between all source redshifts and colours. For compar-ison, for our QSO template, effective wavelengths will vary by

2


Fig. 2. Dust extinction curves for different values ofRV showntogether with the (arbitrary normalised) SDSS filter transmis-sion curves.Top: Extinction curve for the CCM parametrisation.Bottom: Extinction curve for the Fitzpatrick parametrisation.

Fig. 3. Degeneracy between parametersp and RV for SDSScoloursu − g (dark red),g − r (red), r − i (orange) andi − z(yellow). The degeneracy lines correspond to a reddening equiv-alent top = 0.02 andRV = 3.0. It is evident that multiple coloursallow the degeneracy to be broken.

∼ 2 − 3 % when varying the source redshift and dust param-eters. Schlegel et al. (1998) calculate their reddening estimatesto have a dispersion of∼ 16 %. Attributing a major part of thisdispersion to a scatter in the dust column density, we assumeafractional uncertainty of 10 % in the dust map valuesDT . Sincethis uncertainty is treated as a statistical error, and we are sys-tematics dominated, the exact value of this fractional error doesnot affect the end result. A possible problem not taken into ac-count in this paper is that the dust map may be contaminated bythe extragalactic background at low values (Yahata et al. 2007).

We incorporate uncertainties using a Monte Carlo-based ap-proach as follows:

– We pick a random value for the shift in the effective wave-length for each filter and compute the corresponding colouroffset. We also randomise the value forDT at each sky posi-tion within the given uncertainties.

– We derive the likelihood over a grid of fitted parameters. Indoing this, we use the statistical sources of error from obser-vational uncertainties and the intrinsic colour dispersion ofthe sources.

– We repeatN times (we useN ∼ 50) to obtainN likelihoodfunctions.

– We compute the mean of theN likelihood functions.

In this way we take full account of the different correlations be-tween sources of error, and the resulting confidence contourscorrectly incorporate both statistical and systematic effects. Asnoted in Sect. 1, the dominating source of error is the possibilityof shifts in the effective filter wavelengths, originating in uncer-tainties in the filter response functions and the source spectraltemplates.

4. Quasar analysis

It is well known that QSOs form a relatively homogeneous groupof objects in terms of colours and spectral features. They alsohave large flux at short wavelengths which means that we canuse blue colours, e.g.u− r even for high redshift sources, whichallows us to effectively break the degeneracy betweenp andRVas shown in Fig. 3. We use the fifth edition of the Sloan DigitalSky Survey Quasar Catalog (Schneider et al. 2007) consisting of77 429 objects. The area covered by the catalogue is 5 740 deg2,see Fig. 4. QSOs have also been used successfully to constrainthe amount and properties of dust in galaxies at cosmologicaldistances (Ostman et al. 2006, 2008) and for intergalactic dust(Ostman & Mortsell 2005).

We are primarily interested in the colours of QSOs, so itis important to understand how the objects in the SDSS QSOcatalogue are chosen in order to assess whether there are selec-tion effects that need to be considered. The QSO candidate se-lection is based on the position of sources in multidimensionalSDSS colour space. In terms of colours, QSOs are mainly iden-tified as being outside the four-dimensional stellar loci pointsin [u, g, r, i, z] space (Richards et al. 2002). Since the method ispurely differential, any general colour bias will not affect the endresult; however, we need to make sure that the selection doesnotdepend on the dust column. Since the selection is made aftercorrecting for the extinction maps in Schlegel et al. (1998), as-suming that these provide a fair model for the extinction, thereis no reason that the selection should be severely affected byMilky Way dust extinction. By applying an extra colour cut ofincluded objects where the (uncorrected) colour is less than 2σoff the mean value, as evaluated in bins of different dust column

3


Fig. 4. Top: Angular positions of the SDSS QSOs used in thisanalysis with darker colours corresponding to higher observedsurface densities of QSOs, reflecting the given survey strategy.Bottom: Restframe QSO template spectrum shown with arbi-trary normalised SDSS filter transmission curves.

density, we are guaranteed a sample not correlated with the dustcolumn, unless initial dust corrections are grossly incorrect.

Only QSOs in the redshift range 0.5 < z < 3.0 are consideredas QSOs outside this range are less accurately described by thespectral template used, created by combining the Hubble SpaceTelescope radio-quiet composite spectrum (Telfer et al. 2002)with the SDSS median composite spectrum (Vanden Berk et al.2001) as described inOstman et al. (2006, 2008). Generally, allcolours are relatively homogeneous atz <∼ 2. Typically, atz ∼ 1,we haveσ(u − g) = 0.15,σ(g − r) = 0.13,σ(r − i) = 0.093, andσ(i − z) = 0.11. However, when the Lyman-α emission is beingredshifted into the observed filters, the dispersion increases. Wethus cut theu-band data atz > 3000/1210− 1 ∼ 1.5 and theg-band data atz > 4000/1210− 1 ∼ 2.3.

In order to display the close to linear relationship betweenthe observed QSO colour and the dust density, in Fig. 5 weshow the observed QSOg − r colour as a function of galac-tic dust column density at redshiftsz ∼ 0.95, z ∼ 1.72 andz ∼ 2.48. Results for the full fit ofp and RV are shown inFig. 6 for the CCM dust extinction parametrisation (top panel)and the Fitzpatrick parametrisation (bottom panel). Solidcolours

correspond to 95.4 % CL for two parameters. Our constraintsare p = 0.0205± 0.0015 andRV = 3.10 ± 0.43 (CCM) andp = 0.0190±0.0015 andRV = 3.25±0.37 (FTZ) (95.4 % CL forone parameter). Although consistent at the 95.4 % CL, compared

Fig. 6. Constraints onp andRV as derived from the observedcolours of SDSS QSOs and the measured dust column densityat the angular position of each QSO. Solid colours correspond to95.4 % CL for two parameters.Top: Results for CCM dust corre-sponding to 95.4 % CL for one parameter ofp = 0.0205±0.0015andRV = 3.10±0.43.Bottom: Results for Fitzpatrick dust corre-sponding to 95.4 % CL for one parameter ofp = 0.0190±0.0015andRV = 3.25± 0.37.

to ther− i andi−z colours, we have on average less extinction inu − g and more extinction ing − r for the CCM parametrisation.For Fitzpatrick dust, compared tog − r andr − i, u − g has lessextinction andi − z more extinction on average.

Albeit with very limited resolution, multiple colour data al-low us to derive the dust extinction curve directly from thedata, without the need to assume any specific dust extinctionparametrisation. For each colour, we fit the slopekXY in Eq. 2,apply the necessaryk-corrections, and normalise the extinction

4


Fig. 5. Observedg − r colour as a function of Galactic dust column density for SDSSQSOs at three different redshifts. Note thealmost perfect linear relationship between the observed colour and the dust density.

to zero in theV-band. In Fig. 7 we show the dust law as de-rived from QSO colours compared to the extinction curves forsome fiducial dust models. Assuming the errors to be accuratelyrepresented by the uncertainty of the slope in the linear fit,theerror bars are too small to be visible. However, this assumptionis valid only in the case that the relation between the reddeningand the dust column density indeed is linear and constant overthe field. Furthermore, the dominating source of error of possiblesystematic shifts in the effective central filter wavelengths havenot been included. Note that since the normalisation is arbitrary,only the shape of the derived extinction curve is relevant.

Fig. 7. Empirical dust law as derived from QSO colours (di-amonds) compared to the extinction curves of CCM dust withp = 0.01,RV = 5.0 (dotted curve),p = 0.02,RV = 3.0 (dashedcurve), andp = 0.03,RV = 1.0 (dash-dotted curve).

5. Brightest central galaxies analysis

The brightest galaxies at the centre of galaxy clusters are em-pirically known to be relatively homogeneous in colour. Sincehigh-luminosity early-type galaxies are generally redderthanlow-luminosity galaxies (usually attributable to higher metallic-ity), the colour dispersion can be further reduced using colour-magnitude relations. In this paper, however, we defer from em-ploying any such calibration.

We have used data from the MaxBCG catalogue of 13 823galaxy clusters from the Sloan Digital Sky Survey (Koester et al.2007). In order to extract the full photometric data for the bright-est central galaxies (BCGs), we retrieved all objects within 6arcseconds from the BCG positions. We managed to identifyand retrieve [u, g, r, i, z] data for 13 796 BCGs with redshifts0.046 < z < 0.37. The angular positions of the BCGs aredepicted in Fig. 8. Generally, all colours involving theu-bandare not very well-determined. Typically (atz ∼ 0.2), we haveσ(u − g) = 0.32, σ(g − r) = 0.059,σ(r − i) = 0.029, andσ(i − z) = 0.042.

We cut all colours involving the theu-band atz > 0.15 sincethe colour dispersion becomes very large at those redshifts. Forconsistency with the QSO and LRG analysis, we reject objectswith uncorrected colours more than 2σ off the mean value, asevaluated in bins of different dust column density. Results forfitting p andRV to the BCG data are shown in Fig. 9. Our con-straints arep = 0.0205± 0.0030 andRV = 2.89± 0.88 (CCM),andp = 0.0177±0.0020 andRV = 3.42±0.48 (FTZ) (95.4 % CLfor one parameter). Like the QSO constraints, compared tor − iandi−z that givep = 0.02 forRV = 3, we have less extinction inu − g and more extinction ing − r for the CCM parametrisation.For Fitzpatrick dust,g−r andr− i give p = 0.02 atRV = 3 whileu − g has less extinction andi − z more extinction. In Fig. 10 weshow the empirically derived dust law.

6. Luminous red galaxies

Since luminous red galaxies (LRGs) are also known to be homo-geneous in terms of colours, we have extracted data for 114 151LRGs from the SDSS data base. Generally, all colours involv-ing theu-band are not well-determined. Typically, atz ∼ 0.3 wehaveσ(u − g) = 0.84,σ(g − r) = 0.086,σ(r − i) = 0.043, andσ(i − z) = 0.056. We therefore do not use any colours involv-ing theu-band in this analysis. As discussed in Sect. 4, we re-ject objects with uncorrected colours more than 2σ off the meanvalue, as evaluated at different dust column values, in order forthe final sample to be unbiased by dust extinction. Fitting for

5


Fig. 8. Top: Angular positions of the SDSS BCGs used in thisanalysis with darker colours corresponding to higher observedsurface densities of BCGs.Bottom: Rest-frame BCG templatespectrum shown together with arbitrarily normalised SDSS filtertransmission curves. Note that the same template is used fortheLRG analysis.

p and RV gives the results shown in Fig. 11. Our constraintsare p = 0.026± 0.004 andRV = 1.95 ± 0.35 (CCM), andp = 0.019± 0.0025 andRV = 3.16± 0.36 (FTZ) (95.4 % CLfor one parameter). Compared to the QSO and BCG analysis,the absence of theu-band pushes best-fit values towards higherp and lowerRV . AssumingRV = 3, compared tor − i andi − zthat givep = 0.02, we have more extinction ing−r for the CCMparametrisation. For Fitzpatrick dust,r − i givesp = 0.02 whileg − r and i − z have more extinction. In Fig. 12 we show thederived dust law.

7. Discussion

As seen in the previous sections, there are slight differencesin the preferred values ofp and RV , depending on the dustparametrisation and the data set used. Generally, for the CCMparametrisation withRV = 3, for r − i andi − z we getp = 0.02.We have more extinction (higherp) in g− r and less inu−g. ForFitzpatrick dust,r − i givesp = 0.02 whileg − r andi − z havemore extinction andu−g less. How can we understand these dif-

Fig. 9. Constraints onp andRV as derived from the observedcolours of SDSS BCGs and the measured dust column densityat the angular position of each BCG. Solid colours correspond to95.4 % CL for two parameters.Top: Results for CCM dust corre-sponding to 95.4 % CL for one parameter ofp = 0.0205±0.0030andRV = 2.89±0.88.Bottom: Results for Fitzpatrick dust corre-sponding to 95.4 % CL for one parameter ofp = 0.0177±0.0020andRV = 3.42± 0.48.

ferences? First, we need to understand the differences betweenthe CCM and FTZ dust extinction parametrisations. In Fig. 13,we plot

f (λ)FTZ − f (λ)CCM

f (λ)CCM. (8)

The reason that the we need higherp for i− z for the FTZ modelis that the extinction curve is flatter there compared to the CCMmodel. We also see that it is steeper forg − r which means thatwe get a lowerp for that colour. That is, we need a dust lawthat is slightly steeper forg − r and flatter foru − g compared toCCM. As for the differences between the data sets, comparingBCGs and LRGs, we see that the differences can be explainedby having a flatter template spectrum for LRGs in the regioncoveringg−r, compared to the BCG template. Comparing BCGs

6


Fig. 10. Empirical dust law as derived from BCG colours (di-amonds) compared to the extinction curves of CCM dust withp = 0.01,RV = 5.0 (dotted curve),p = 0.02,RV = 3.0 (dashedcurve), andp = 0.03,RV = 1.0 (dash-dotted curve).

and QSOs, the differences can be resolved by having either theBCG template flatter foru − g andg − r or the QSO templatesteeper in the same wavelength region.

8. RV-variations

It is not unreasonable to assume that the distributions of grainsizes, and thus the resulting value ofRV , is different in high- andlow-density environments. Whether grains are generally largeror smaller in high-density environments depends on whethergrain-grain collisions are more likely to cause coalescence orshattering. We therefore check for any evidence of correlationsbetweenRV and the dust column density by dividing our sampleinto subsamples with different column densities. Note, however,that a high column density does not necessarily correspond to ahigh physical density. In Fig. 14 we show constraints from di-viding the QSO sample into two regions with low (DT < 3 MJysr−1) and high (DT > 3 MJy sr−1) dust column density. Our con-straints for CCM arep = 0.020± 0.0016 andRV = 3.06± 0.56(DT < 3 MJy sr−1), andp = 0.019±0.0019 andRV = 3.09±0.65(DT > 3 MJy sr−1). For Fitzpatrick dust,p = 0.019±0.0016 andRV = 3.28± 0.37 (DT < 3 MJy sr−1), andp = 0.018± 0.0018andRV = 3.23± 0.41 (DT > 3 MJy sr−1) (95.4% CL for oneparameter). This simple test does not provide evidence for anycorrelation between the dust column density and dust properties.

8.1. σ(RV ) and σ(AV )

If dust has spatially varying properties, depending on the size ofthe regions we would expect the colour dispersion to be higherbehind regions with high column densityDT . For filtersX andYwe can write the colourE(X−Y) as a combination of the intrinsiccolourEi(X − Y) and the dust induced colourEd(X − Y)

E(X − Y) = Ei(X − Y) + Ed(X − Y) , (9)

Fig. 11. Constraints onp andRV as derived from the observedcolours of SDSS LRGs and the measured dust column density atthe angular position of each LRG. Solid colours correspond to95.4 % CL for two parameters.Top: Results for CCM dust corre-sponding to 95.4 % CL for one parameter ofp = 0.026± 0.004andRV = 1.95±0.35.Bottom: Results for Fitzpatrick dust corre-sponding to 95.4 % CL for one parameter ofp = 0.019± 0.0025andRV = 3.16± 0.36.

and the corresponding colour dispersion

σ[E(X − Y)]2 = σ[Ei(X − Y)]2 + σ[Ed(X − Y)]2 , (10)

where

Ed(X − Y) =

(

AX

AV−

AY

AV

)

AV ≡ fXY (RV )AV , (11)

and

σ[Ed(X − Y)]2 =

(

RV

fXY

∂ fXY

∂RV

)2 (

σRV

RV

)2

+

(

σAV

AV

)2

Ed(X − Y)2(12)

≡ g[X, Y,RV , σ(RV ), σ(AV)])Ed(X − Y)2 .

In Fig. 15, we plot the varianceσ[E(X − Y)]2 as a function ofEd(X − Y)2 for each standard colour of the QSO sample, expect-ing the variance at zeroEd(X − Y)2 to give the intrinsic colour

7


Fig. 12. Empirical dust law as derived from LRG colours (di-amonds) compared to the extinction curves of CCM dust withp = 0.01,RV = 5.0 (dotted curve),p = 0.02,RV = 3.0 (dashedcurve), andp = 0.03,RV = 1.0 (dash-dotted curve).

Fig. 13. The differences between the CCM and FTZ dust extinc-tion curves.

varianceσ[Ei(X−Y)]2 and the slope to be related to the scatter inRV andAV through the functiong[X, Y,RV , σ(RV), σ(AV )] (usingthe CCM dust extinction parametrisation). As can be seen, thereare no obvious correlations betweenσ[E(X−Y)]2 andEd(X−Y)2

(or σ[E(X − Y)]2 and the dust column densityDT ). AssumingσRV /RV andσAV /AV to be constant, although the results de-picted in Fig. 15 indicate that this assumption may be too sim-plistic, we can constrain their values from the fitted straight lineslopes. Results from this analysis are shown in Fig. 16. The bluercolours show no increase in the colour variance with dust columndensity, whereasi − z show a slight increase in the variance in-dicating a possible dispersion ofσAV /AV at the 30 % level. Thissaid, there are a few points that should be remembered: First,this is under the assumption that the relation between the vari-ance and the colour is indeed linear, and that the data pointsare

not correlated. Second, results fori−z are not consistent with thebluer colours that show no signs of a scatter in the dust proper-ties, although it is possible that the scatter only manifests itself atlonger wavelengths. Third, there is a weak binning dependencewhen calculating the dispersion as a function ofEd(X − Y)2. Tosummarize, there is no evidence in the QSO data for any addi-tional scatter in the observed colours in regions with high col-umn density, which indicates that dust properties are relativelyhomogeneous averaged over the size of spatial regions probed inthis paper, except perhaps for a hint of slightly increased scatterat the reddest optical wavelengths probed by the QSO data.

9. Summary

By investigating the correlations between dust column densityas inferred from infrared data and the observed colours of celes-tial objects at cosmological distances with low levels of colourdispersion, we constrain the properties of Milky Way dust. Weexpect analyses of the reddening of extragalactic sources to com-plement analyses of Galactic sources very well since they willhave different systematic uncertainties, and in principle may alsoprobe dust at different distances. Results from QSOs, BCGs,and LRGs colours are broadly consistent, indicating a propor-tionality constant between the reddeningE(B − V) ≡ AB − AVand the dust column densityDT (given in units of MJy/sr) ofp = E(B − V)/DT ∼ 0.02 and a reddening parameterRV ≡

AV/E(B − V) ∼ 3 with relative uncertainties of approximately10 %. In spite of the increased statistics, the fractional error of pis similar to that of Schlegel et al. (1998), the reason beingthatwe simultaneously fit forRV and that the uncertainty is domi-nated by the possibility of systematic shifts in the effective filterwavelengths.

We note that, in accordance with the analyses inSchlafly et al. (2010) and Schlafly & Finkbeiner (2011), the FTZextinction curve may be argued to be preferred before the CCMparametrisation, since it is able to fit QSO, BCG, and LRG datawith similar normalisations and reddening parameters. However,as argued in Sect. 7, such differences may also be traced to un-certainties in the spectral template functions.

The data does not provide evidence for any dependence ofthe dust properties as a function of dust column density, norforany sizable scatter in the dust properties on the scales probedby the data, except for a hint of scatterσAV /AV >∼ 0.3 in theextinction properties of dust at the longest wavelengths probedby the data (corresponding toi − z). We have also checkedthat within the uncertainties there is no evidence of any sys-tematic difference between the dust extinction properties in thenorthern and southern hemisphere. Assuming the dust maps ofSchlegel et al. (1998) to be correct, it is fair to assume thatwecan correct for Milky Way dust by applying a proportionalityconstantp = E(B − V)/DT ∼ 0.02 with reddening parameterRV ∼ 3 to the dust maps.

ReferencesBerry, M., Ivezic,Z., Sesar, B., et al. 2011, ArXiv e-printsCardelli, J. A., Clayton, G. C., & Mathis, J. S. 1989, ApJ, 345, 245Doi, M., Tanaka, M., Fukugita, M., et al. 2010, AJ, 139, 1628Fitzpatrick, E. L. 1999, PASP, 111, 63Koester, B. P., McKay, T. A., Annis, J., et al. 2007, ApJ, 660,239Kohyama, T., Shibai, H., Fukagawa, M., & Hibi, Y. 2010, ApJ, 719, 873O’Donnell, J. E. 1994, ApJ, 422, 158Ostman, L., Goobar, A., & Mortsell, E. 2006, A&A, 450, 971Ostman, L., Goobar, A., & Mortsell, E. 2008, A&A, 485, 403Ostman, L. & Mortsell, E. 2005, J. Cosmology Astropart. Phys., 2, 5

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Peek, J. E. G. & Graves, G. J. 2010, ApJ, 719, 415Richards, G. T., Fan, X., Newberg, H. J., et al. 2002, AJ, 123,2945Schlafly, E. F. & Finkbeiner, D. P. 2011, ApJ, 737, 103Schlafly, E. F., Finkbeiner, D. P., Schlegel, D. J., et al. 2010, ApJ, 725, 1175Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525Schneider, D. P., Hall, P. B., Richards, G. T., et al. 2007, AJ, 134, 102Telfer, R. C., Zheng, W., Kriss, G. A., & Davidsen, A. F. 2002,ApJ, 565, 773Vanden Berk, D. E., Richards, G. T., Bauer, A., et al. 2001, AJ, 122, 549Yahata, K., Yonehara, A., Suto, Y., et al. 2007, PASJ, 59, 205

Acknowledgements. EM acknowledges support for this study by the SwedishResearch Council. EM also acknowledges discussions with Brice Menard andAriel Goobar.

Funding for the SDSS and SDSS-II has been provided by the AlfredP. Sloan Foundation, the Participating Institutions, the National ScienceFoundation, the US Department of Energy, the National Aeronautics and SpaceAdministration, the Japanese Monbukagakusho, the Max Planck Society, andthe Higher Education Funding Council for England. The SDSS is managedby the Astrophysical Research Consortium for the Participating Institutions.The Participating Institutions are the American Museum of Natural History,Astrophysical Institute Potsdam, University of Basel, Cambridge University,Case Western Reserve University, University of Chicago, Drexel University,Fermilab, the Institute for Advanced Study, the Japan Participation Group,Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, theKavli Institute for Particle Astrophysics and Cosmology, the Korean ScientistGroup, the Chinese Academy of Sciences (LAMOST), Los AlamosNationalLaboratory, the Max Planck Institute for Astronomy (MPIA),the Max PlanckInstitute for Astrophysics (MPA), New Mexico State University, The OhioState University, University of Pittsburgh, University ofPortsmouth, PrincetonUniversity, the United States Naval Observatory, and the University ofWashington.

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Fig. 14. Constraints onp andRV when dividing data into regions with low (DT < 3 MJy sr−1) and high (DT > 3 MJy sr−1) dustcolumn density. There is no evidence for any correlation between the dust column density and dust properties.Upper left: Resultsfor CCM dust for all regions withDT < 3 MJy sr−1 corresponding to 95.4 % CL for one parameter ofp = 0.020± 0.0016 andRV = 3.06± 0.56.Upper right: Results for CCM dust atDT > 3 MJy sr−1 giving p = 0.019± 0.0019 andRV = 3.09± 0.65.Lowerleft: Results for Fitzpatrick dust atDT < 3 MJy sr−1 giving p = 0.019± 0.0016 andRV = 3.28± 0.37. Lower right: Results forFitzpatrick dust atDT > 3 MJy sr−1 giving p = 0.018± 0.0018 andRV = 3.23± 0.41.

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Fig. 15. The varianceσ[E(X − Y)]2 as a function ofEd(X − Y)2 for each standard colour of the QSO sample. The total variance atzeroEd(X − Y)2 is the intrinsic colour varianceσ[Ei(X − Y)]2 and the slope is related to the scatter inRV andAV .

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Fig. 16. Constraints on constantσRV /RV andσAV /AV derived from the linear fits depicted in Fig. 15. The colouredregions corre-spond to 95.4 % CL (red) and 68.3 % CL (yellow) for two parameters.

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Calibrating Milky Way dust extinction using cosmological ... · Fitzpatrick (1999) (FTZ), plotted in Fig. 2 together with the (ar-bitrarily normalised) SDSS ﬁlter transmission curves