The local FIR Galaxy Colour-Luminosity distribution: A reference for BLAST, and Herschel/SPIRE sub-mm surveys

0.4 MB·English

by  Edward L. Chapin

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed28January2009 (MNLATEXstylefilev2.2)
The local FIR Galaxy Colour-Luminosity distribution: A
reference for BLAST, and Herschel/SPIRE sub-mm surveys
E.L. Chapin1, D.H. Hughes2, I. Aretxaga2
9 1Dept. of Physics & Astronomy, Univ. of British Columbia, 6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada
0 2Instituto Nacional de Astrof´ısica, O´ptica y Electr´onica (INAOE), Aptdo. Postal 51 y 216, Puebla, Mexico
0
2

n
a 28January2009
J

8
2 ABSTRACT

] We measure the local galaxy far-infrared (FIR) 60-to-100µm colour-luminosity
h
p distribution using anall-skyIRAS survey.This distribution is an importantreference
- forthe nextgenerationofFIR–submillimetre surveysthathaveandwillconductdeep
o extra-galacticsurveysat250–500µm.Withthepeakindust-obscuredstar-formingac-
r
tivityleadingtopresent-daygiantellipticalsnowbelievedtooccurinsub-mmgalaxies
t
s near z ∼ 2.5, these new FIR–submillimetre surveys will directly sample the SEDs of
a
these distant objects at rest-frame FIR wavelengths similar to those at which local
[
galaxies were observed by IRAS. We have taken care to correct for temperature bias

2 and evolution effects in our IRAS 60µm-selected sample. We verify that our colour-
v luminosity distribution is consistent with measurements of the local FIR luminosity
4
function, before applying it to the higher-redshift Universe. We compare our colour-
1
luminositycorrelationwithrecentdust-temperaturemeasurementsofsub-mmgalaxies
2 and find evidence for pure luminosity evolution of the form (1+z)3. This distribu-
2
tionwillbe usefulforthe developmentofevolutionarymodels forBLAST andSPIRE
.
1 surveys as it provides a statistical distribution of rest-frame dust temperatures for
1
galaxies as a function of luminosity.
8
0 Keywords: galaxies:luminosityfunction,infrared:galaxies,submillimetre,galaxies:
: evolution
v
i
X
r
a
1 INTRODUCTION sent an important early stage in the evolutionary sequence
that ultimately produces locally-observed massive elliptical
Deep extra-galactic surveys at sub-mm wavelengths (
galaxies (e.g. Scott et al. 2002; Blain et al. 2004). Thus, to
∼
200–1200µm) over the last 10 years have uncovered a
thisday,thelocalfar-infraredluminosityfunctionmeasured
population of luminous infrared galaxies (L > 1012L⊙) by IRAS continues to be useful for interpreting the results
with star-formation rates inferred to be 1000M⊙/yr−1 of these longer-wavelength surveys.
≫
(e.g. Smail et al. 1997; Hugheset al. 1998; Barger et al.
1998; Eales et al. 1999; Cowie et al. 2002; Scott et al. Additional motivation for studying the IRAS luminos-
2002; Borys et al. 2003; Serjeant et al. 2003; Webb et al. ity function, and its connection to thehigher-redshift SMG
2003; Wang et al. 2004; Greveet al. 2004; Laurent et al. population,comesfromtheshapeandmagnitudeofthecos-
2005; Coppin et al. 2005, 2006; Knudsenet al. 2006; mic infrared background (CIB) measured by COBE which
Bertoldi et al. 2007; Scott et al. 2008; Greve et al. 2008; peaksnear200µm.(e.g.Fixsen et al.1998).Itsbroadshape
Perera et al. 2008). These sub-mm galaxies (SMGs here- resembles the superposition of many thermal SEDs, which
after) are believed to be high-redshift (z > 1) ana- canbeinterpretedasevidenceforapopulationofsourcesat
logues, and in many cases more luminous examples, of lo- redshiftsz <1with a large range ofphysicaltemperatures,
cal Ultra-Luminous Infrared Galaxies (ULIRGs) discovered or alternatively, as a population with a narrower range of
withIRAS20yearsago(Sanders & Mirabel1996).Further- temperatures,butresidingoveragreaterrangeofredshifts,
more, the rest-frame wavelengths sampled by sub-mm sur- including a significant fraction at z > 1 (the SMG popu-
veys of the highest-redshift SMGs approaches those of the lation). This latter possibility is supported by the fact that
far-infrared (FIR) IRAS observations. Appearing in vast thetotalenergydensityoftheCIB(Franceschini et al.2001)
quantities consistent with massive evolution of the local exceeds the contribution of local IRAS galaxies by a factor
ULIRG population, these SMGs are now believed to repre- of 3 (Soifer & Neugebauer 1991).
∼
2 E. L. Chapin, D. H. Hughes, I. Aretxaga
In this paper we examine the colour-luminosity corre- dependence of the fitted distribution on the choice of SED
lation and luminosity function of IRAS galaxies, which to- templates; (iii) a correction for redshift evolution in the
gether are an important reference for constraining models IRAS galaxy population is applied; and (iv) we account
of galaxy evolution with a new generation of sub-mm sur- for a bias against the detection of cooler sources caused
veys at shorter wavelengths. It has been known for some by the 60µm selection criterion for the sample. Our galaxy
time that more luminous IRAS galaxies exhibit warmer sample,SEDfittingprocedure,andmethodsforcalculating
dust temperatures (e.g. Soifer & Neugebauer 1991). This luminosities and volumes are described in Section 2. The
relationship is important for a class of phenomenological luminosity function and colour-luminosity distribution are
models (e.g. Blain & Longair 1993; Guiderdoni et al. 1997; calculated in Section 3. We discuss the choice of luminos-
Blain et al. 1999; Chary & Elbaz 2001; Malkan & Stecker ity variable, and its consequences, in Section 4.1. Finally,
2001; Rowan-Robinson 2001; Lagache et al. 2003, 2004) wecomparethelocal colour-luminositycorrelation withthe
that have been used to predict the source counts observed values for high-redshift sub-mm galaxies in Sec-
and redshift distributions at FIR wavelengths for the tion 4.2 and test a simple evolutionary model. Throughout
Spitzer Space Telescope (Werneret al. 2004) and Akari thispaperastandardcosmologyisadoptedwithΩ =0.23,
M
(Matsuhara et al. 2006), and at sub-mm wavelengths, Ω =0.77 and H =74kms−1Mpc−1.
Λ 0
for instruments such as SCUBA (Holland et al. 1999),
MAMBO (Kreysa et al. 2002), LABOCA (Kreysa et al.
2003), Herschel/SPIRE (Griffin et al. 2006), SCUBA-2 2 SAMPLE PREPARATION
(Holland et al. 2006), AzTEC (Wilson et al. 2008), and
We use the same flux-limited S > 1.2Jy IRAS sample of
BLAST (Pascale et al. 2008). These models often use the 60
Fisher et al. (1995) as in C03. Their catalogue covers most
shape of the CIB as an integral constraint, since the to-
oftheskyandprovides60and100µmflux-densitiesaswell
tal surface brightness of galaxies at each wavelength can-
as spectroscopic redshifts for each galaxy. The cool, high-
notexceedthemeasureddiffusebackground.Theseauthors
luminosity region of the observed colour-luminosity plane
applyevolution tolocal luminosity functionstoobtain esti-
found to contain a large numberof spurious sources in C03
matesoftheredshift-dependentluminosityfunctionΦ(L,z).
hasalsobeenexcised.Weusethissampletofirstcalculatea
In order to compare these models with observations, spec-
non-parametric(binned)FIRluminosityfunction,andthen
tralenergydistribution(SED)templatesareadoptedtoex-
fit it with simple parametric models.
trapolate observed flux densities from the rest-frame lumi-
nosities. In a number of cases, it has been beneficial to fit
dataoverarangeofwavelengthsbydividingthelocallumi-
2.1 SEDs, Luminosities and Colours
nosity function into several discrete populations, each with
different SED templates and separate evolutionary forms To calculate rest-frame luminosities and colours from ob-
(e.g.Blain et al.1999;Rowan-Robinson2001;Lagache et al. served IRAS 60 and 100µm flux densities, we follow the
2003, 2004). Recently Wall et al. (2008) demonstrated di- method of Saunderset al. (1990). A single temperature
rect evidence for the presence of at least two significant modifiedblackbodySEDisassumedforeachsource,S(ν)=
populations in a sample of sub-mm luminous sources in AνβBν(Tobs). The dust emissivity index is fixed at β =1.5
GOODS-N.Theluminosity-colourcorrelation can beuseful which is consistent with typical values measured for local
for such models as a method for assigning dust tempera- ULIRGs with sub-mm follow-up (e.g. Dunneet al. 2000;
tures to SED templates as a function of luminosity. For ex- Klaas et al. 2001; Yang& Phillips 2007). All of the sub-
ample,Lagache et al.(2003)usetheobservedIRAScolour- sequent analysis in this paper has also been repeated us-
luminosity distribution of Soifer & Neugebauer (1991), and ing values β = 1.0 and 2.0, and the variation in the re-
Lewis et al. (2005) use the FIR colour-luminosity distribu- sultsiswellwithinthequoteduncertainties.Theremaining
tionofChapman et al.(2003,henceforthC03)inferredfrom two parameters, the amplitude A, and the observed tem-
a model fit to IRASdata. perature Tobs, are then uniquely determined from the ob-
C03 calculate Φ(L,C), the galaxy volume density as served S60 and S100. For this fit we take into account the
a function of total 3–1100µm luminosity, L , and the 60– broad IRAS passbands (Beichman et al. 1988). Bolometric
T
100µm colour, C log(S /S ). They formulate Φ(L,C) luminositiesarecalculatedbyintegratingthefittedSEDdi-
60 100
astheproductofa≡luminosityfunction,andthedistribution rectly—thebolometricfluxemittedintherest-frame,Sbol
in C as a function of L. The two functions are fit indepen- is simply the integral of the observed SED across the red-
dently, with the latter being constrained directly from the shiftedband,Sbol= cc//λλlu(1(+1+zz))S(ν)dν,whereλl=122.5µm
observed distribution of the ratio of broad-band IRAS 60 and λ =42.5µm foRr FIR fluxes, and λ =1100µm and
u l
and 100µm fluxes. λ =3µm for TIR fluxes. Similarly the colour C is calcu-
u
In this work we provide a more accurate measurement latedfromthelogarithm oftheratioofmonochromaticflux
of the joint colour-luminosity distribution using a single densitiesemittedat 60and100µmintherest-framebythe
maximum-likelihood optimization to solve for all of the model SED.
model parameters simultaneously. Our methodology also Rather than a simple modified blackbody, C03 adopt
differs from that of C03 in several other key respects: (i) the range of model SED templates from Dale et al. (2001).
rather than calculating C with observed IRAS broad-band This difference has a negligible effect on the inferred FIR
fluxesweuserest-framemonochromatic60and100µmflux luminosities since there is very little structure in the
densities derived from fitted SEDs; (ii) we use narrower- Dale et al.(2001)SEDsat42.5–122.5µmthatisnotcharac-
bandwidth 42.5–122.5µm FIR luminosities instead of 3– terizedbythesingletemperaturevariableinourSEDmodel
1100µm Total Infrared (TIR) luminosities to minimize the (despite the correlation between luminosity and β assumed
The local FIR Galaxy Colour-Luminosity Distribution 3
correspond to themean and 68% confidenceintervals using
observed broad-band IRAS fluxes, effectively re-producing
the top panel of Figure 1 in C03. The thick solid line and
dot-dashed lines show the mean and 68% confidence inter-
val of the colour distribution using the ratio of monochro-
matic60and100µmfluxdensitiesemittedintherest-frame.
At luminosities L<1010L⊙ there is little difference in the
shapes of the distr∼ibutions. At L > 1010L⊙, however, the
colour-luminosity correlation is sign∼ificantly steeper.
The dotted lines in Figure 1 show the mean, and 1-σ
envelope of the C03 colour-luminosity correlation. We note
thatalthoughthemeanofthisfittedparametricdistribution
clearlytracksthestarsintheplot,thestandarddeviationof
the distribution, σ = 0.065, appears to have been under-
C
estimated. We find that both the 68% confidence intervals,
and the standard deviations of C, for each luminosity bin,
are typically closer to 0.13.
2.2 Evolution in the Sample
Since the most distant, luminous objects in IRAS sam-
Figure 1. Observed IRAS FIR colour distribution (C ≡
ples exhibit the effects of strong luminosity and/or density
log(S60/S100))asafunctionof3–1100µmTIRluminosity.Stars
evolution (e.g. Saunders et al. 1990; Kim & Sanders 1998;
andtrianglesshowthemeanand68%confidenceintervalswhenC
Lawrence et al. 1999) we must account for its effect in our
iscalculatedfrombroad-bandIRASfluxesfollowingtheprescrip-
tion in C03. The dotted lines show the mean and 1-σ envelope measurement of the local luminosity function. Rather than
ofthefitted C03colour-luminositycorrelationforreference.The fittingforthisevolutionaryformourselves,weinsteadapply
thick solid and dot-dashed lines show the mean and 68% con- an explicit correction based on the luminosity evolutionary
fidence intervals of the distribution when C is calculated with form fit by Saunderset al. (1990): the luminosity of each
monochromatic flux densities emitted at 60 and 100µm in the galaxy is divided by a factor (1+z)3 corresponding to its
rest-frame. redshift.Wehavechosen toapplyaluminosity,ratherthan
a density evolution correction for consistency with the dis-
cussion in Section 4.2.
in Dale et al. 2001). For example, assuming temperatures
rangingfrom30to50K(andβ =1.5),theFIRluminosities
inferred from our modified blackbody templates compared
2.3 Accessible Volumes
withtheDale et al.(2001)SEDswiththesamecorrespond-
ing values of C agree to within 5%. Since the difference The (1/V ) estimator (Schmidt 1968), with accessible
max
∼
is so small, and for the sake of simplicity, we therefore pro- volumPesVmax correspondingtothelargestredshiftatwhich
ceed with the modified blackbody SED model to measure agalaxywouldbedetectedgiventhesurveyfluxlimit,used
theFIRcolour-luminositydistribution.TheTIRluminosity, in C03, is appropriate for the monochromatic 60µm lumi-
however, cannot be estimated from the modified blackbody nosityfunction.Inthiscaseagivenobject’sluminosity,L ,
60
model as there is significant emission in the mid-infrared isafunctiononlyoftheobservedfluxdensityanddistance.
(MIR) spectrum ( 3–60µm) that is missed by the steep Therefore the maximum volume in which the given object
∼
dropontheWienside(e.g.Blain et al.2003,anddiscussion can be detected corresponds to the distance at which the
in Section 4.1). Our modified blackbody SEDs in this tem- observed S , given its L , drops below the flux limit of
60 60
peraturerangeunder-predicttheTIRluminositiesobtained the sample. For the broad-band FIR luminosity function
fromtheDale et al.(2001)SEDsbyabout30%.Weonlyuse described here, however, the relationship between L and
F
their templates to calculate TIR luminosities that are con- S is more complicated and this simple method is invalid.
60
sistentwithC03forthediscussioninthissection(Figure1), There exists a bias against the detection of cooler sources
and Section 3.1 (Figure 3). given the shape of the galaxy SEDs (Saunderset al. 1990).
Anotherfundamentaldifferencebetween thiswork and ThewavelengthofthepeakFIRemission istypicallyinthe
C03isthedefinitionofFIRcolour.Whereaswechoosetode- range 60–200µm. The SEDs of warmer objects peak closer
fineC intermsoftheratioofrest-framemonochromaticflux to 60µm, and colder objects at longer wavelengths, so that
densities,C03useobservedbroad-bandIRASfluxes.Webe- ingeneralforafixed60µmfluxdensityacolderobjectmust
lieveourdefinitionismoreusefulasageneralreferencesince bemore FIR luminous to beincluded in thesample.
no detailed knowledge of the IRAS passbands is required Saunderset al. (1990) derive the FIR luminosity func-
in order to use our colour-luminosity distribution. Further- tion from their 60µm flux-limited survey by selecting a
more,wefindthatthestrengthofthecolour-luminositycor- sub-sample of objects brighter than a FIR flux limit that
relation is relatively diluted when using broad-band fluxes. corresponds to their 60µm flux-limit and the coolest dust
InFigure1weshowthedistributionofIRASgalaxycolours temperature that they observed, 23K (see Section 6.5 in
as a function of TIR luminosity derived from fits of the Saunderset al. 1990). However, this selection reduces the
Dale et al. (2001) SED templates. The stars and triangles size of their sample from 3000 objects, to 1004. There is
∼
4 E. L. Chapin, D. H. Hughes, I. Aretxaga
also an underlying assumption that there is no significant
population of sources with dust temperatures T <23K.
Inthisworkweusetheentiresample,butcalculateac-
cessible volumes using a modified formalism that accounts
for the dependence of L on the FIR colour. Given an ob-
F
served temperature, T , and redshift, z, the rest-frame
obs
temperature,T,andtotalluminosityforanobjectarecalcu-
lated. The accessible volume corresponds to the maximum
redshift at which an object with its rest-frame luminosity
andtemperaturewouldbedetectedinthesample, orcorre-
spondingly the distance at which its observed flux density
in theIRAS60µm passband is 1.2Jy.
3 THE Φ(L,C) DISTRIBUTION
3.1 Non-Parametric (binned) Estimate
Figure 2. The non-parametric 42.5–122.5µm FIR luminosity
With luminosities and accessible volumes for all of the ob-
function(symbolswith68%Poissonerrorbars,andarrowsshow
jects in the sample, we first calculate the non-parametric
95% upper-limits for bins with <2 objects) and two parametric
(binned)colour-luminositydistribution,Φ (L,C).Sincethe
b fits(Equation6asasolidline,andEquation7asadashedline)
accessible volume is now parameterized by both L and C,
derivedfromtheS60>1.2JyIRASsampleofFisheretal.(1995).
themodified (1/Vmax) estimator is simply , TheparametricFIRluminosityfunctionofSaundersetal.(1990)
P 4π 1 isshownforcomparisonasadottedline.
Φ (L,C)dLdC = , (1)
b Ω V
s Xi i
where Φ (L,C)dLdC is the number of sources in the area
b
of the L–C plane, and the sum runs over all of the galax-
ies,i,withluminosity-evolutioncorrectedluminosities(Sec-
tion 2.2) and colours that land within the bin, and with
accessible volumes V . The factor in front of the sum is the
i
fraction of the sky covered by the survey. The binned lu-
minosity function may be derived from this distribution by
marginalizing over C,
Φ (L)dL= Φ (L,C)dLdC, (2)
b b
Xj
wherej runsoverallofthebinsalongtheC axis.Wedefine
a second representation of the luminosity function using a
lower-case φ,
φ(L)=ln(10)LΦ(L), (3)
whichchangestheunitsfromMpc−3L−⊙1tothemoretypical Figure 3.Thenon-parametric3–1000µmTIR luminosityfunc-
Mpc−3dex−1,inordertoassistcomparisonwithotherwork. tion(symbolswith68%Poissonerrorbars,andarrowsshow95%
ThisrepresentationofourFIRluminosityfunctionisshown upper-limitsfor bins with <2 objects) and the parametricfit of
in Figure 2. C03(solidline).Thediscrepanciesat L>109L⊙ areduetothe
At luminosities L > 109L⊙ there is excellent agree- correctionsdescribedinSections 2.2and2.3.
mentbetweenourlumin∼osityfunctionandthemeasurement
of Saunderset al. (1990) (shown in Figure 2 with a dotted
Lawrence et al. 1999). In addition to this effect, Yun et al.
line). However, at fainter luminosities our luminosity func-
(2001)suggest thatsomeoftheflatteningatfaintluminosi-
tionincludesmanymoreobjects.Thereasonforthisdiscrep-
ties in the Saunderset al. (1990) luminosity function may
ancyisprobablyduetoourchoiceof (1/V )estimator,
max bedueto sample incompleteness.
and the fact that over this luminositPy range the sample is We also produce the non-parametric TIR luminosity
dominatedbyanover-densityofgalaxiesintheLocalSuper-
function for comparison with the parametric form of C03
cluster.ItisforthisreasonthatSaunderset al.(1990)used
in Figure 3. For this calculation we have used the same
an alternative estimator that is insensitive to local density
Dale et al.(2001)SEDtemplatesasC03toderiveTIRlumi-
variations. Their method has the potential to more accu- nosities. TheC03 model1 has asignificantly different shape
rately determinetheshape of theluminosity function,how-
ever at the expense of losing the absolute normalization.
At luminosities L > 109L⊙ the galaxies are typically suf- 1 We take the luminosity function to be σC(2π)1/2Φ1(L) from
ficiently distant tha∼t this issue is no longer important, and Section 3.2 in C03. The dimensionless pre-factor is needed since
both estimators give consistent answers (see Section 8 of their colour distribution, Φ2(C), is an un-normalized Gaussian
The local FIR Galaxy Colour-Luminosity Distribution 5
compared with our binned representation at luminosities
L > 109L⊙, the range over which the local over-density of L (1−α) 1 L
gluamlaixnioessitisyirfurenlcetviaonnt.bTyhaeiframctoorde∼lu3n0d%er-aptr5ed×ict1s01t0heLb⊙i,nannedd Φs(L)=ρ∗(cid:16)L∗(cid:17) exph−2σ2 log210(cid:16)1+ L∗(cid:17)1i×
rises to over-predict by a similar factor at 2 1012L⊙. We , (7)
× ln(10)L
havedeterminedthatthisdiscrepancycanbeexplaineden-
tirely by the effects described in Sections 2.2 and 2.3, since Maximumlikelihoodsolutionsforbothformsareshown
the model and the bins are otherwise consistent without in Figure 2. At the faint end (LFIR < L∗) both functions
them.Applyingonlythecorrectionforevolutioninthesam- approach power-laws, and are therefore indistinguishable.
ple we find that the number of objects in the brightest Attheextremebright-endtheydiverge;φ curvesbelowφ
s p
binsdecreases–explainingthefactorof30%atluminosities at L 1012L⊙, although both forms lie mostly within the
>1012L⊙. The reason for this is that these objects are the error∼bars of the non-parametric estimate.
most distant, and therefore exhibit the strongest effects of To characterize the quality of the fits we calculate val-
redshift evolution. This correction has almost no effect by uesofreducedχ2forluminositybinsthatcontainatleast10
1011L⊙. In contrast, applying the correction for accessible objects, approximately luminosities 5 108–1012L⊙ (with
×
volumesincreasesthenumbersofobjectsinbinsatluminosi- thisnumberof objectsthePoisson errordistribution isrea-
tiesprimarily<1011L⊙.Thisincreaseiscausedbythefact sonably approximated by a Gaussian). Over this range the
that less lumin∼ous objects are cooler, with correspondingly power-law form produces a value of reduced χ2 = 2.2, and
smaller volumes in which they could be detected given the the hybrid form 2.4. Given the similarity of these values,
60µm flux limit. Together, these corrections demonstrate and the fact that each form has the same number of pa-
that the luminosity function is in fact significantly steeper rameters, we feel there is no compelling evidence to favour
than the result of C03 at luminosities >1010L⊙, the most one model over the other given the data. While the choice
important range for comparison with re∼sults from newsub- has no impact on the subsequent discussion in this paper,
mm surveysof distant star-forming galaxies. we note that the two forms rapidly diverge at luminosities
>1012L⊙,potentially themost important region of thelu-
minosityfunctionforcomparisonwiththeresultsofsub-mm
3.2 Maximum Likelihood Model Fits surveys. For example, while at 1012L⊙ the power-law only
Next we fit simple parametric models to the data by max- exceeds the Saunderset al. (1990) form by about 10%, at
imizing the likelihood of observing the sample. For the re- 1013L⊙, it is nearly an order-of-magnitude larger. Fitted
mainder of the paper we consider only FIR luminosities to parameters for both models are given in Appendix A, and
avoid dependence on assumptions about the shape of the they should only be considered valid to a maximum lumi-
mid-infrared SED (Section 4.1). At a given position in the nosity of 2 1012L⊙.
∼ ×
L–C plane,theexpectednumberofsourcesfromoursample
to havelanded in that bin, given a model for Φ(L,C), is
3.2.2 Parametric form of p(C L)
Ω |
µ(L,C)=Vmax4πsΦ(L,C)dLdC. (4) In C03 it was shown that the distribution in C is approx-
imately Gaussian at a particular value of L. The precise
These expectations are used to calculate the joint Poisson
functional form we haveadopted is
likelihood of the data.
We express the model, Φ(L,C), as the product of the 1 1 C C 2
p(C L)= exp − 0 , (8)
luminosity function, Φ(L), and the conditional probability | σC√2π (cid:20)−2 ×(cid:16) σC (cid:17) (cid:21)
ofagalaxyhavingacolorC giventheluminosityL,p(C L),
| with themean colour at a given luminosity given by2
Φ(L,C)=Φ(L)p(C L). (5)
| L′ L
C0=C∗−δlog10(cid:18)1+ L(cid:19)+γlog10(cid:16)1+ L′(cid:17). (9)
3.2.1 Parametric forms of Φ(L)
NotethatunlikeC03,the“knee”luminosity,L′,forp(C L)
ForΦ(L),weconsidertwoforms.Thefirstisthedualpower- |
is independent of the knee luminosity for the luminosity
law of C03,
function, L∗. In addition, the width of the colour distri-
L (1−α) L −β bution, σc, is characterized by two different values: σf and
Φp(L)=ρ∗(cid:16)L∗(cid:17) (cid:16)1+ L∗(cid:17) , (6) σb at the faint and bright ends respectively, with a smooth
transition at L′,
where L∗ is the characteristic knee luminosity, ρ∗ is
the number density normalization of the function at L∗ σ =σ (1 2−L′/L)+σ (1 2−L/L′). (10)
(Mpc−3L−1), and α and β characterize the power-laws at c f − b −
⊙
thefaint (L<L∗)andbright (L>L∗) endsrespectively of Figure 4 compares the mean and 1-σ envelope of the para-
theluminosity function. metric p(C L) with a non-parametric estimate created by
|
The other form considered is the hybrid power- factoringthesmoothmodelΦ(L)fromthebinnedΦb(L,C).
law/Gaussian form preferred bySaunders et al. (1990),
2 We note that in Section 3.2 of C03 the expression for C0 is
withstandarddeviationσC.Also,wehaveassumedthattheunits clearly meant to be the logarithm of the third equation in Sec-
forρ∗ areMpc−3dex−1 ratherthanMpc−3L−⊙1 asindicated. tion3.1–theformwehaveadopted here.
6 E. L. Chapin, D. H. Hughes, I. Aretxaga
Figure 4. Comparison between parametric (solid and dotted
lines give the mean and 1-σ envelope of Equation 8 respec-
tively)andnon-parametricestimates ofp(C|L)(greyscaleshows
Φb(L,C)normalizedalong the C axis). The temperature axis is
derived from C assuming a dust emissivity index β = 1.5. At
luminosities L<∼108 and L>∼1012 the sample does not contain
enoughgalaxiestoaccuratelyconstraintheshape,andp(C,L)is
simplyextrapolatedintheparametricmodel.
Figure 5. The distribution of FIR colours, C, about the mean,
C0 (Equation 9), at a range of luminosities. The solid lines
are normalized slices of the measured (non-parametric) p(C|L)
ItisarguedinC03thatthewidthofthedistributionin (Φb(L,C)/Φb(L)–SeeEquations1and5)evaluatedatC−C0,
C is constant as a function of luminosity. The top panel of andnumbersindicatelog10(LFIR).Theparametricmodel(Equa-
tion8)isshownasadottedline.
Figure 2in C03 demonstrates a constant width in S /S
60 100
with a logarithmic axis, in contrast with the bottom panel
in which the width of the distribution is shown to broaden ure 1 and the discussion at the end of Section 2.1 in this
atgreaterluminositieswhenplottedwithalinearaxis.This paper).
behaviourmotivatesthedefinitionC log(S /S ).How-
60 100
≡
ever, this plot appears to be at odds with the top panel
of Figure 4 in C03 which exhibits a systematic broadening 3.2.3 Parameter Uncertainties
at higher luminosities. Such a trend does not appear to be
We characterize the uncertainties in the ten parameters of
present in our measurement of p(C L).For clarity, wecom-
| Φ(L,C), for both parametric forms of Φ(L), using a boot-
pare slices of the parametric estimate of p(C L) at several
| strap Monte Carlo technique. First, 100 realizations of the
fixedluminositieswiththebinnedestimateinFigure5.The
1.2Jy survey are created from the actual survey data by
goodagreementbetweenthesetwoestimates,bothinterms
randomlysamplingsourcesfromthecataloguewithreplace-
of scatter and systematic variations, indicates that Equa-
ment (see Section 6.6 of Wall & Jenkins 2003). We then fit
tion 8 adequately describes the shape of p(C L). We find
| themodeltoeach simulatedsample.From these100fitswe
that the width of the distribution narrows with increasing
calculate the sample variances, and covariances between all
brightness to σ = 0.13 from σ = 0.2 at a transition lu-
b f pairsofparameterstoestimatethefullparametercovariance
minosity of ∼ 3.5×109L⊙ (Appendix A). The broadening matrix. The maximum likelihood values of each parameter,
shown in C03 at greater luminosities does not appear to be
their standard deviations, and the Pearson correlation ma-
causedbytheirchoiceofSEDsorchoiceofbroad-bandover
trices are given in Appendix A. Note that the parameters
monochromatic colours. The most likely explanation is an
for p(C L) are largely independent of the parametric form
artifact oftheC03 gridingscheme.Theyusevariable-width |
chosen for Φ(L).
luminosity bins which contain equal numbers of objects, in
contrast to our method which uses equally-spaced logarith-
micbins.Thewide,sparsely-populatedhigh-luminositybins
maysimplydilutethecolour-luminositycorrelation.Wenote 4 DISCUSSION
thatourfittedvalueforσ appearstobeconsistentwiththe
b 4.1 Choice of Bolometric Luminosity Variable
width of the distribution in the lowest, narrowest, luminos-
ity bins in the top panel of Figure 4 from C03, (although In this work we have chosen to use the 42.5–122.5µm FIR
they claim a smaller standard deviation of 0.065; see Fig- luminosity instead of the wider-bandwidth 3–1100µm total
The local FIR Galaxy Colour-Luminosity Distribution 7
Blain et al. 2002), the observed redshift distribution is a
reasonable proxyfor thetotal dust-obscuredstar-formation
ratehistoryofmassivegalaxies.IfSMGshavethermalSEDs
similar to the ULIRGs that populate the bright end of the
IRAS luminosity function presented here, their SEDs peak
atwavelengths 60–200µmintherest-frame,andtheyare
∼
redshifted into the 200–600µm BLAST and SPIRE band-
passes near the peak of their redshift distribution.
Asan example, Figure 6shows theSED of theULIRG
Arp 220 at a redshift z = 2.5 compared to the BLAST
and SPIRE bandpass region, and the integrated FIR and
TIRfluxes.ClearlytheBLASTandSPIREintegratedfluxes
more closely matches therest-frame FIR than TIR flux.
Thepeak-normalizedSEDofthestarburstgalaxy M82
redshifted to z = 2.5 is shown for comparison as a dashed
line to illustrate the relatively large scatter at mid-infrared
(MIR)wavelengthscomparedtothesmooththermalSEDat
longer wavelengths. The rest-frame MIR SEDs of these ex-
amplegalaxiesexhibitprominentpolycyclicaromatichydro-
carbon(PAH)absorptionandemissionfeatures(the 100–
∼
10µm range of theobserved SED; measurements for actual
Figure 6. The peak-normalized spectral energy distribution of SCUBA-selected SMGs are given in Pope et al. 2008). For
Arp 220 (dotted line), and shifted to z = 2.5 (solid line). The these two examples there is a difference of 30% in the
∼
complete observed BLAST and SPIRE band (200–600µm, in- contribution of the mid-infrared emission to the TIR lumi-
cluding the 30% finite bandpasses for each channel) is indicated nosity.Forthese reasons, evolutionary models based on the
bythesolidgreyshadedrectangle.Therest-frame3–1100µmto- FIRluminosity function are less dependenton assumptions
tal infrared (TIR) flux, and 42.5–122.5µm FIR flux correspond
about the intrinsic near-IR to millimetre-wavelength SEDs
totheintegralsofthehorizontal-linefilled,andcross-hatchfilled
of high-redshift galaxies than the TIR luminosity function,
regions respectively, for the z = 2.5 SED. Clearly for objects at
enablingcleanercomparisonwithnewandfuturedatafrom
this redshift, the BLAST and SPIRE filters more closely match
deep sub-mm cosmological surveys. We emphasize the fact
therest-frameFIRfluxthantheTIRflux.Thepeak-normalized
M82 SED redshifted to z = 2.5 (shown as a dashed line) illus- that one is free to adopt any template library to infer flux
tratestherelativelylargevariationsintherest-framemid-infrared densities at other wavelengths for objects drawn from our
spectrum (∼ 3–60µm observed wavelength). Symbols indicate Φ(L,C) distribution provided that they have roughly ther-
the wavelengths accessible from ground-based sub-mm surveys mal FIR spectra, and span the relevant range of rest-frame
(e.g. SCUBA at 850µm, and BOLOCAM/AzTEC/MAMBO at FIR colours 0.65 < C < 0.25 (see for example the com-
1.1mm), and existing space-based FIR data (e.g. Spitzer at 24, parisoninSec−tion2.∼1bet∼weenourmodifiedblackbodySED
70and160µm),demonstratingtheirinabilitytoaccuratelycon-
model and the library of Dale et al. 2001).
strainthebolometricFIRluminosityforthesegalaxies.
4.2 Evidence for an Evolving Colour-Luminosity
infrared luminosity, L , as in C03. An argument for using
T Correlation
thelatteristhatitincludesasignificantfractionofthetotal
power emitted by a galaxy that is missed at shorter wave- Evolution in the FIR colour-luminosity distribution is dif-
lengths (< 40µm) — an effect which becomes increasingly ficult to probe from IRAS catalogues given their relatively
importantathighluminositiesgiventhepositiveL–C corre- low redshifts. For example, the median galaxy redshift in
lation. By using L , the“clipping” of shorter-wavelength the Fisher et al. (1995) sample is z = 0.019, and the most
FIR
light obscures the physical interpretation of the correlation distantobject isatz=0.326. Despitethis,it hasbeenpos-
between thetwo variables. sible to place weak constraints on the evolutionary form of
Our primary goal, however, is to assist with the de- the FIR luminosity function. Saunderset al. (1990) found
velopment of a model for the luminosity function of SMGs that the most distant galaxies could undergo extremes of
discovered in BLAST (Pascale et al. 2008) and future Her- puredensityevolution oftheform (1+z)7±2,orpurelumi-
schel/SPIRE (Griffin et al. 2006) 250–500µm surveys, as nosity evolution of the form (1+z)3±1 (we use this latter
well as any other surveys at similar wavelengths. It is now formexplicitlytocorrectoursample,seeSection2.1).Since
generallyacceptedthattheredshiftdistributionforthebulk themostdistantIRASgalaxies arealsothebrightestinthe
ofSMGsdiscoveredin850µmSCUBAsurveyspeaksatred- sample (luminosities > L∗) it is not possible to determine
shifts 2.5. For example, the radio-detected spectroscopic which form (or combination) is the more relevant.
∼
sample of Chapman et al. (2003, 2005) finds a median red- At higher redshifts, the best constraints on the evolu-
shift of 2.2 with an interquartile range z = 1.7–2.8, in gen- tion of ultra-luminous infrared galaxies come from SCUBA
eralagreementwithseveralotherstudiesusingradio–FIRor surveys at 850µm. It can be shown, for example, that con-
radioand24µmguidedphotometricredshiftestimates(e.g. tinued luminosity evolution in the bright-end of the local
Aretxaga et al.2003,2007;Pope et al.2006).Sincetheneg- FIRluminosityfunctionofthesameformasSaunders et al.
ativeK-correctionproducesanearlyun-biaseddetectionef- (1990) to redshifts 2–3 can be used to fit mm–submm
∼
ficiencyat850µmforatypicalULIRGSEDatz 1–8(e.g. source counts, assuming ULIRG-like SEDs to extrapolate
∼
8 E. L. Chapin, D. H. Hughes, I. Aretxaga
are compared with our local measurement of p(C L). As
|
noted in Kov´acs et al. (2006) and Coppin et al. (2008), the
rest-frametemperaturesforsuchluminousgalaxies(median
LFIR = 2.3 1012L⊙) are much lower than objects in the
×
localUniverse.Withthecorrelationbetweenluminosityand
FIR colour in-hand, we now ask the question: can pure lu-
minosity evolution of the form (1+z)3 account for the ap-
parentlycoolertemperaturesofSMGsathigh-redshift?Ex-
plicitly,weexpresstheredshift evolutionoftheFIRcolour-
luminosity distribution, Φ(L,C,z), as a simple function of
thelocal distribution,
L
Φ(L,C,z)=Φ ,C . (11)
(cid:18)(1+z)3 (cid:19)
For comparison, we project each observed SMG into the
local p(C L) distribution by dividing their luminosities by
(1+z)3,|shifting the objects to the left in Figure 7 (dia-
monds).Given theuncertainties,the8warmer objects(top
of the plot) are roughly consistent with the local distribu-
tion once we apply this transformation. Ignoring the red-
shift uncertainties and adding the measured colour uncer-
Figure7.Comparisonoflocalp(C|L)(shadedregionisthe68% tainties,σdata,inquadraturewiththeintrinsiccolourwidth
confidence interval, solid line is C0 from Equation 9) with data inthesourcepopulation,σc (Equation10,attheevolution-
fromCoppinetal.(2008). Thetemperature axisisderivedfrom corrected luminosity of the galaxy – the shaded region of
C assuming a dust emissivity index β = 1.5. Stars with dotted the figure), we calculate residuals, R, between the model,
1-σ errorbarsindicatethe10SMGswithspectroscopicredshifts
C (solid line through the centre of the shaded region) and
0
andtemperaturesderivedfromobserved350µm/850µmcolours.
themeasured colours, C :
Squares have been drawn around the symbols for the 6 objects data
at z > 2. High-redshift ultra-luminous galaxies appear system- R= Cdata−C0 . (12)
atically cooler than those in the local universe. Under the as- σ2 +σ2
sumption of pure luminosity evolution of the form (1+z)3, the data c
p
SMGs have been projected into the local colour-luminosity dis- We note that for five of these objects R < 1, and the re-
tribution by shifting them along the luminosity axis (diamonds maining three are in the range R=1–1.5; the approximate
withsoliderrorbars).Themodelisaplausiblefittothedataex- expectation for uncorrelated Gaussian uncertainties. This
ceptforLOCK850.4andLOCK850.41whichappearmuchcooler result is contrary to the conclusion of C03 who found no
(shownaslightersymbols).Theseobjectsmayhaveambiguousor
need to invoke redshift evolution for p(C L) when compar-
incorrectoptical/IRcounterpartidentifications(discussedinSec- |
ing with lower-redshift (z <1) samples.
tion4.2). Future BLAST andSPIRE surveys willconstrainFIR
The two most significant outliers, LOCK 850.4 (C=-
luminosities of ∼1012L⊙ galaxies to ∼20%, with uncertainties
2.2,z=0.526)andLOCK850.41(C=-2.47,z=0.689),also
inC of ∼0.1(several Kelvin).A representative measurement in
theL–C planeisgivenbythethickcross-filledsquare. the two coolest and least luminous objects, deserve further
explanation. These two galaxies are also the only objects
with redshifts z < 1. Since their observed sub-mm colours
observed flux densities from rest-frame luminosities (e.g are otherwise similar to the other galaxies, these low red-
Scott et al.2002;Lagache et al.2003).Amoredirecttestof shifts also imply lower rest-frame dust temperatures (lower
evolution for a small sample of SMGs with known redshifts valuesofC).Thereisapossibilitythatthetrueopticalcoun-
was recently performed by Wall et al. (2008) finding simi- terpart(andhenceredshift)forLOCK850.4isatz=1.482,
larresults.Unfortunately,untilrecently,shorter-wavelength rather than 0.526 as adopted by Coppin et al. (2008).
data that would help to constrain thedust temperatures of Both potential counterparts were proposed in Ivison et al.
objects in these samples is generally unavailable, and it is (2005). Adopting the higher redshift object as the coun-
thereforenotpossibletosearchdirectlyforcolourevolution. terpart, the inferred FIR luminosity increases from 8
Measuringdusttemperaturesforlargesamples( 1000) of 1010L⊙ to 1012L⊙ and the rest-frame dust temperatur×e
≫
SMGs is one of the primary science goals of BLAST and from 13K to 21K, or C=-1, at which point it would ap-
Herschel/SPIRE surveys — with the caveat that redshifts pear to have similar dust properties to the other galax-
mustfirstbedeterminedindependentlyforatleastasubset ies in the sample. There is a similar possibility of a mis-
ofthesenewobjectssincethereisapotentialdegeneracybe- identification for LOCK 850.41. Two counterparts are sug-
tweentheapparent observedtemperatureandredshift (e.g. gested in Ivison et al. (2005), although they were only able
Blain et al. 2003). toobtain thespectroscopic redshift indicated abovefor one
Recent SHARC-II 350µm observations of SMGs (e.g. of them. The other counterpart has an optical photomet-
Kov´acs et al. 2006; Coppin et al. 2008), however, have en- ric redshift estimate of z=2.2 0.2 from Dyeet al. (2008).
abled improved estimates of dust temperatures for smaller A similar value of z=2.1 1.4±based on its FIR colours
±0.6
samples (several tens of galaxies). In Figure 7 ten objects is proposed in Aretxaga et al. (2007). Adopting a redshift
(triangles)withconstraineddusttemperaturesandspectro- of z=2.2, the luminosity for LOCK 850.41 increases from
scopic redshifts (median z =2.1) from Coppin et al. (2008) 4 1010L⊙ to5 1011L⊙ andtherest-framedusttempera-
× ×
The local FIR Galaxy Colour-Luminosity Distribution 9
turefrom 12Kto22K,orC=-0.9,alsoclosertothedistri- evolution. These corrections indicate that thebright-end of
bution for the other objects in the sample. However, if the theluminosity function is significantly steeper than an ear-
lower-redshift candidates for these two galaxies are correct, liercalculation byC03whichneglectedthem.Wehaveveri-
andapopulationofgalaxieswithtemperaturesT <15Kdo fiedthatourdistributionisconsistentwiththeFIRluminos-
existinabundanceatredshiftsz <1,itispossible∼thatthey ity function of Saunderset al. (1990) by marginalizing over
werecompletelymissedinIRASsurveys,andwillappearin colour. We fit a parametric model to the data consisting of
thewide-area SCUBA-2and BLASTand SPIREsurveys. theproductoftheluminosityfunction,Φ(L),withthecondi-
Finally, we note that the higher-redshift objects gen- tional colour probability, p(C L), where C log(S /S ).
60 100
| ≡
erally fall the closest to the colour-luminosity distribution. We fit p(C L) using a normal distribution for C as a func-
|
To emphasize this fact we draw squares around the 6 ob- tionofL,withameancolourgivenbyabrokenlogarithmic
jects in this small sample at redshifts z > 2. Naively this functionalsoofL.Thisischaracterizedbythekneeluminos-
fact is slightly surprising, since one might suppose that ityforthebreak,L′,whichisindependentoftheluminosity
the nearer objects are in fact more similar to the sample functionknee,L∗,andthewidthofthedistribution,bytwo
used to constrain the local distribution. However, an addi- different standard deviations σ and σ , above and below
b f
tionalconsiderationistheselectionfunctionforthisSCUBA thekneeL′.
sample. While the negative K-correction produces approx- We have checked directly for evolution in the colour-
imately the same observed 850µm flux density at redshifts luminosity correlation using observations of high-redshift
z 1–8 for a fixed FIR luminosity and temperature, there SMGs (z > 1) with temperatures constrained by
∼
is also a bias towards the detection of cooler objects at a SCUBA 850µm and SHARC-II 350µm photometry from
fixed redshift and flux density since such objects are less Coppin et al. (2008). These high-z ultra-luminous objects
luminous, and hence more abundant in the rest-frame (e.g. appearmuchcoolerthanlocal galaxies ofcomparablelumi-
Eales et al. 1999; Blain et al. 2002; Chapman et al. 2003). nosities, and there is preliminary evidence that pure lumi-
For the lower-redshift objects, at which point the negative nosity evolution in the local colour-luminosity distribution
K-correction isdiminished, theluminosities arealso fainter oftheform (1+z)3 isconsistent with uncertaintiesin their
foragivenfluxdensity,andthisbiascouldbeincreaseddue measured redshifts and colours. This result is contrary to
to the broadening in the colour-luminosity correlation that C03 who find no evidence for a change in the relationship
we havemeasured at lower luminosities. between luminosity and colour in low-redshift (z <1) sam-
This comparison is by no means an exhaustive study ples.
of evolution in the FIR colour-luminosity distribution. It is
our goal to extend this investigation to much larger sam-
ples of SMGs with FIR colour information in new BLAST
extra-galactic surveys (Devlin et al. in prep.), and fu- 6 ACKNOWLEDGEMENTS
tureHerschel/SPIRE surveys.Understandingthedetails of
We thank Manolis Plionis, Enrique Gaztan˜aga, Min Yun
this evolution is intimately related to the star-formation
and Douglas Scott for valuable discussions. We also thank
rate history of massive galaxies, since the rest-frame FIR
theanonymous referee for theirhelpful comments. EC con-
luminosity is the key observed quantity in SMGs from
ducted a portion of this research with the support of an
whichstar-formationratesarederived(e.g.Kennicutt1998;
NSERC Postgraduate Scholarship. IA and DHH acknowl-
Hugheset al. 1998; Scott et al. 2002; Pope et al. 2006).
edgepartialsupportfrom Conacytgrants50786and60878.
5 CONCLUSIONS
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