Correlations between SNe Ia Rates and Host Galaxy Properties

1.1 MB·English

by  Yan Gao Chris J. Pritchet

Draftversion January29,2013
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
CORRELATIONS BETWEEN SDSS TYPE IA SUPERNOVA RATES AND HOST GALAXY PROPERTIES
Yan Gao1 Chris Pritchet
DepartmentofPhysics&Astronomy,UniversityofVictoria
Draft version January 29, 2013
ABSTRACT
Studying the correlationof type Ia supernova rates(SNR) with hostgalaxyproperties is animpor-
3 tant step in understanding the exact nature of type Ia supernovae. We use SNe Ia from the SDSS-II
1 sample, spectroscopically determined masses and star formation rates, and a new maximum likeli-
0 hood method, to fit the Scannapieco and Bildsten rate model SNR = A×M +B×SFR, where M
2 is galaxy mass and SFR is star formation rate. We find A = 3.5+−00..97×10−14(SNe/yr)(M⊙)−1 and
an B = 1.3+−00..43 ×10−3(SNe/yr)(M⊙yr−1)−1, assuming overall efficiency of 0.5. This is in reasonable
agreement with other determinations. However we find strong evidence that this model is a poor fit
J
to other projections of the data: it fails to correctly predict the distribution of supernovae with host
8 massorSFR.Anadditionalmodelparameterisrequired;mostlikelythisparameterisrelatedtohost
2
galaxy mass. Some implications of this result are discussed.

Subject headings: supernovae: general, galaxies: fundamental parameters
]
O
C 1. INTRODUCTION A+B×(SFR/M), where SNR/M is the rate per unit
. mass,andSFR/M is specificstarformationrate(which
h Supernovae of type Ia (SNe Ia) played a pivotal role
is a function of star formation history, independent of
p in the discovery of the accelerating Universe (Riess et
mass).
- al.1998,Perlmutteretal.1999),andareamongthemost
o Substantial effort has been invested in determining
reliablestandardcandles oncosmologicaldistance scales
r the values of A and B since the work of Scannapieco
t (Sullivan et al. 2011, Barone-Nugent et al. 2012). How-
s and Bildsten (2005). See Fig. 1 for a comparison of
ever, a complete understanding of SN Ia progenitor sys-
a some of these results. Neill et al. (2006) and Dil-
[ temshasyettobeachieved,eventhoughcrediblemodels
for such progenitors exist (e.g. Hillebrandt & Niemeyer day et al. (2008) used volumetric SN Ia rates to infer
1 2000,Nomoto1982,Hachisuetal.1996). Manymethods these parameters. Of particular interest is the work of
v have been developed to discriminate between proposed Sullivan et al. (2006), who obtained stellar masses and
3 models, including the theoretical modelling of progeni- SFRs of each individual host (and field) galaxy by fit-
2 ting PEGASE2 spectral templates to multiband pho-
tor systems (e.g., Han & Podsiadlowski 2006, Wang et
6 tometry (Fioc & Rocca-Volmerange 1997, Le Borgne
al. 2010), and the measurement of delay time distribu-
6 and Rocca-Volmerange 2002). They concluded that
tions (e.g., Mennekens et al. 2010, Totani et al. 2009,
1. Sandetal.2012;Maozetal.2012;seealsoGreggio2005, A=5.3±1.1×10−14(SNe/yr)(M⊙)−1, B =3.9±0.7×
0 2010). 10−4(SNe/yr)(M⊙yr−1)−1,valuesthatwewillreturnto.
3 OneapproachusedinconstrainingthenatureofSNeIa On the other hand, Smith et al. (2012) used photo-
1 progenitors is studying the correlation between SN Ia metricmassesandSFR’s to demonstratethatthe A+B
: rates (SNR) and the properties of the galaxies in which model was not a goodmatch to SDSS-II supernovadata
v
they form (e.g., Maoz & Mannucci 2011). The environ- (Friemanetal.2008). Theyproposedanalternatemodel
i
X mental properties studied so far include redshift (e.g., of the form SNR = A×Mx +B×SFRy, with x ≃ 0.7
r Dilday 2010a and 2010b), host galaxy age (e.g., Gupta and y ≃ 1. This is, on the surface, broadly consistent
a et al. 2011), environment galaxy number density (e.g., withthe massdependence foundbyLietal.(2011)from
Cooper et al. 2009), ejecta velocity (e.g., Foley 2012), theLickObservatorySupernovaSurvey(LOSS)(though
and, most relevant to this work, mass and star forma- see §6).
tion rate of the host (e.g., Sullivan et al. 2006). The primary objective of this work is to obtain fits to
In this paper, we investigate the correlation between SNR models using spectroscopically-determined masses
SN Ia rates and the mass and star formation rate of the and SFRs from the SDSS DR7 MPA/JHU value-added
host galaxies in which they formed. One popular model catalog(Kauffmannet al.2003,Brinchmannet al. 2004,
for this correlation is the “A+B” model (Scannapieco Tremonti et al. 2004), making these values somewhat
& Bildsten 2005), which states that the supernova rate independent of, and possibly more reliable than, those
is proportional to a linear combination of galaxy mass based on multiband photometric data. This is of sig-
and star formation rate: SNR = A×M + B×SFR, nificance since it has been pointed out (e.g., F¨orster et
where SNR is the SN Ia rate, M is galaxy mass in al. 2006) that the uncertainty in measurements of star
stars, and SFR is star formation rate. This relation formationhistoriesis animportantlimiting factorinthe
leads trivially to a more physical relation SNR/M = determination of the values of A and B. In addition, we
use a new fitting algorithm for A and B based on max-
imum likelihood, which is more reliable than previous
[email protected]
1SchoolofAstronomyandSpaceSciences,NanjingUniversity fitting methods.
2 Gao and Pritchet
In §2,we presentthe data samples we use in our stud- 2.2. SDSS Spectroscopic Masses and Star Formation
ies, which include the sample of SNe Ia, the sample of Rates
photometric galaxies, and the subset of spectroscopic
The SDSS DR7 MPA/JHU value-added catalog con-
galaxies(itselfasubsetofthephotometricsample). Also
tains derived masses and SFR values for the spectro-
presentedin§2isthe essentialpreprocessingofthe data,
scopic galaxies, including 19987 galaxies in Stripe 82
most notably the matching of the SNe Ia to their host
(our spectroscopicsample). The mass estimates followa
galaxiesinthephotometricsample,andabriefintroduc-
methodology similar to that of Kauffmann et al. (2003),
tion to the MPA/JHU masses and SFRs, in addition to
namely a grid search over a library of star formation
a comparison with other masses and SFRs. Our results
histories (Bruzual & Charlot 2003) for the most prob-
for A and B are given in §3, and a detailed examination
able mass-to-light ratio. (This is slightly different from
of the fit of the model to the data is given in §4. Finally
the approachofKauffmann et al.(2003),whichmatches
we discuss alternate models in §5, and the implications
D (4000)andHδ ,butithasbeenshownthatthemass
n A
of these alternate models in §6.
estimatesofthetwomethods4 donotdifferbymorethan
∼ 0.1dex.) The methods used for obtaining SFR mea-
2. DATA
surementsweresimilartothoseproposedbyBrinchmann
The Sloan Digital Sky Survey (SDSS, Abazajian et etal.(2004). Galaxiesweredividedintothreeclasses: (1)
al. 2009) Data Release 7 catalog2 (DR7) contains 357 “SF”,(2) “low S/N SF”, and (3) “AGN, Composite and
millionuniqueobjects,ofwhichnearly930,000aregalax- Unclassifiable”. The “SF” class was processed by fitting
ies for which spectra are available. From these spec- 5 emission lines (H , H , O III, N II and S II) to those
α β
tra, masses and star formation rates have been ob- ofsimulatedgalaxiesinaspectrallibraryobtainedusing
tained (Kauffmann et al. 2003, Brinchmann et al. 2004, the code by Charlot & Longhetti (2001). For the “low
Tremonti et al. 2004, hereafter MPA/JHU); see §2.2 for S/N SF” class, Brinchmann et al. used a simple conver-
details. Supernova identifications were taken from the sion factor between attenuation-corrected H luminos-
α
SDSS-II Supernova Survey3, which is a 3-year (2005- ityandSFR.The “AGN,Composite andUnclassifiable”
2007) survey conducted within Stripe 82 of the SDSS. class was processed by means of the empirical relation
Stripe 82 covers an area of nearly 300 square degrees in between SFR and D4000.
a belt along the celestial equator, and contains ∼20,000 One problem with this methodology is that passive
of the spectroscopic galaxies mentioned above. For each galaxies are always assigned a (very small) SFR. This
year the survey was conducted, supernova imaging was can be seen in Fig. 2, where MPA/JHU spectroscopic
conducted for 3 months, resulting in a total observation galaxiesareplottedinthemass-SFRplane. Itisdemon-
period of 9 months. strated in §4 that this does not significantly affect our
results.
2.1. SDSS II Type Ia Supernovae To test the reliability of the MPA/JHU masses and
SFR’s,wecomparetheirresultswiththoseoftheVESPA
From the complete sample of 660 SDSS II supernovae
catalogue (Tojeiro et al. 2009), which estimates the star
(Frieman et al. 2008), we extract 520 spectroscopically
formationhistoryofanumberofgalaxiesinSDSSStripe
confirmedSNeIa,503ofwhichwereobservedduringthe
82 using spectroscopic methods. VESPA masses are ob-
3 observation seasons in 2005, 2006 and 2007. (For the
tained by summing the star formation history; VESPA
rest of this paper, “SNe” refers to SNe Ia unless stated
SFR’s are obtained from a SFH averaged over the red-
otherwise.) OursampleislargerthanthatusedbySmith
shift range 0 to 0.11. A comparison of these quanti-
et al. (2012), since we include SNe within z = 0.05. We
ties with their MPA/JHU counterparts is made in Fig.
omit17SNeobservedin2004,sincetheobservationwin-
3. From this figure, it can be seen that the mass and
dows and completeness of these SNe would be hard to
SFRestimatesfromthetwocataloguesareinreasonable
gauge.
agreement with each other, with offsets of 0.25 and 0.24
Dilday et al. (2008, 2010) have shown that the identi-
for log M and log SFR, and rms scatter of 0.18 dex and
fication efficiency ǫ in the SDSS-II Supernova Survey is
z 0.48 dex, respectively.
& 0.7 out to a redshift of 0.15, and >0.5 out to z=0.25
At this point it is relevant to ask how much bet-
(e.g. Fig. 8 of Dilday et al. 2010). In fact the me-
ter spectroscopically-determinedmasses and star forma-
dian z of SN in our (spectroscopic host) sample is much
tion rates are than their photometric counterparts (as
lower (hzi=0.11) than for the SDSS SN Ia sample as a
used by, for example, Sullivan et al. 2006, and Smith et
whole (hzi = 0.2) because of the severe magnitude limit
al. 2012). The advantage of using spectroscopic data is
r = 17.77 for the spectroscopic sample of hosts. Thus
duetoseveraleffects. 1. Redshiftestimates(requiredfor
incompleteness ǫ = 0.7 appears to be a reasonable as-
z boththeSNandfieldsamples–e.g.,§3)aremuchbetter
sumption. WealsoassumeanobservationwindowT =9
for spectroscopic data. Using the same sample as ours,
months in length, and detection efficiency ǫ = 1. The
t Smithetal. (2012)showedthatphotometricredshiftsre-
size of the SN sample is therefore assumed to be the in-
sultin negligible offsets (0.03 dex) but a scatter of>0.2
trinsicnumberoccuringintheskywithinatimeframeof
dexinlogMandlogSFRrelativetousingspectroscopic
ǫ ǫ T = 0.5 year. This assumption is discussed further
t z redshifts. 2. Photometric methods for obtaining M and
in §6.1, but we note here that different values of ǫ ǫ T
t z SFRfromSDSSdatarelyonfittingevolvingstellarpop-
result in a simple scaling of supernova rates, and hence
ulation models to 5 broad photometric bands. The sim-
of A and B values.
2 http://www.sdss.org/dr7/ 4 http://www.mpa-garching.mpg.de
3http://sdssdp62.fnal.gov/sdsssn/snlist confirmed updated.php /SDSS/DR7/mass comp.html
Correlations between SNe Ia Rates and Host Galaxy Properties 3
plicity of these models, coupled with the wide range of 3. FITTINGAANDB
degeneracies between model parameters, will result in
We adopt a fitting algorithm based on maximum like-
larger random errors,particularly for SFR. By compari-
lihood. Using Poisson probabilities, the likelihoods of a
sonspectroscopicmethodsallowtheuseoffarmoredata,
galaxy hosting/not hosting a SN Ia are
including emission lines, absorption lines, and spectral
breaks that are better tuned to the determination of M P(S |SN)=S e−Si, and P(S |SN)=e−Si, (1)
i i i
andSFR.3. Fromacomparisonoftwodifferentcatalogs
whereS istheexpectedSNIarateforeachgalaxywithin
above, we have already seen that there are significant i
our observing window, calculated for every galaxy indi-
systematicsinthe determinationofspectroscopicMand
vidually by means of the mass and SFR data. (None of
SFR. For all the reasons mentioned in point 2, there are
the hosts in oursample hostedmorethan one SN.) Mul-
likely to be even larger systematic errors in photometric
tiplying these likelihoods for every galaxy, we obtain a
MandSFRmeasurements. These systematicsareofthe
log likelihood of obtaining our dataset of
greatestconcern; none of our conclusions are affectedby
random errors in M and SFR. 19987 53
lnL=− X Sn+ X(lnSm)+constant. (2)
2.3. Host Matching n=1 m=1
We adopt the host matching algorithm of Sullivan et where 53 is the number of galaxies hosting SNe, and
al. (2006). We use SDSS DR7 r band data to calculate 19987 is the total number of galaxies in our Stripe 82
a dimensionless parameter R for every potential host- spectroscopic sample.
25
SN pair. R is the angular separation of a SN from its We perform a grid search to find the maxi-
25
prospective host, measured in units of the size of the 25 mum likelihood A and B values, which are A =
magnitudes/arcsec2 isophote,andallowingforthe shape 3.5+−00..97 × 10−14(SNe/yr)(M⊙)−1 and B = 1.3+−00..43 ×
of this isophote (see Fig. 4). The host candidate is then 10−3(SNe/yr)(M⊙yr−1)−1. Fig. 6 plots the probability
identified to be the galaxy with the lowest R25. SNe Ia contours and error bars for our grid search, while Fig. 7
withredshiftsfallingmorethan3σ fromthephotometric compares these results with the observed SNR at differ-
redshiftofthehostgalaxyarethendiscarded. Usingsuch ent specific SFR (where specific SFR sSFR=SFR/M)
aselectionprocedure,forhostswithspectroscopy,hostz While exceptionally simple, the maximum likelihood
andSNzarealwaysinagreementtoaprecisionof0.001. analytical method above does have the drawback that
How large can R be without introducing significant it neglects statistical uncertainties in the determination
25
contamination? To answer this question, we conducted of M and SFR. The method could be modified to in-
Monte Carlo simulations with randomly distributed ar- clude the effects of these uncertainties, however, given
tificial SN positions, keeping in mind that a majority of our conclusion (§4) that the A+B model is not valid,
contamination is due to SNe for which the host is not we have chosen instead to simply simulate the effects of
seen. We find that the contamination is ∼ 8% when these uncertainties. We have conducted a set of Monte
only SN-host candidate pairs with R < 3.8 are identi- Carlo simulations by seeding random, artificial SNe into
25
fied as genuine matches. Contamination could be larger the spectroscopic sample, and have included the effects
if we were to take clustering effects into account, but of uncertainties in M and SFR. We find that A and B
such effects are hard to quantify. Applying this crite- are biased by +0.05 and -0.05 when the effects of uncer-
rion, we find 351 matches for the SNe within the SDSS tainties in M and SFR are included. (This bias is in the
DR7 database. Of these, 53 are galaxies in the spec- sense “derivedvalue minus true value”.) The mass term
troscopic sample, with MPA/JHU masses and SFRs. It and the SFR term contribute 27 and 26 SNe Ia respec-
maybearguedthatsince∼170SNe wereunmatched, ∼ tively, implying that there are as many prompt SNe Ia
8%×170∼14SNe ofthe 351couldberandommatches. as delayed SNe Ia according to the model.
We consider this number to be relatively small in com- Ourfinalresults(A=3.5+−00..97×10−14(SNe/yr)(M⊙)−1
ptiamriastoenctoorrtehsep3o5n1dsmtaotcthhees.exAplescot,etdhicsocnotnamtaimnaintiaotniofnroems- and B = 1.3+−00..43×10−3(SNe/yr)(M⊙yr−1)−1) are con-
sistentwithin errorbars with mostof the literature. For
usingR aloneasmatchingcriteria. Theadditionalred-
25 a summary/comparisonof these results, see Fig. 1.
shift constraints would help further lower the number of
random matches. The question of contamination is re- 4. INCONSISTENCIESINTHEA+B MODEL
examined in §6.2. The A and B values above are optimal in a maximum
To test the host-matching procedure above, we use 2 likelihoodsense. However,thereisnoguaranteethatthe
separate methods. The first is to visually examine the A+B model provides a good fit to our data; more tests
images of the SN - host pairs, relative to their R25 = are needed.
1 isophotes. The second is to plot the number of SN AsatestoftheA+B model’sconsistency,werankthe
matches versus the R25 criterion adopted (Fig. 5). We spectroscopic galaxy sample of 19987 galaxies by mass,
can see that there is a flat region beyond R25 = 3.8, calculate the cumulative percentage distribution of SNR
whichweassumetobetheregimewhererandommatches for both the A + B model and the observations, and
between SNe and non-host galaxies become dominant. apply a Kolmogorov-Smirnov (K-S) test to the two dis-
We also compare the discrepancies between our host- tributions (Fig. 8). The A+B model with our derived
matching method and the “closest angular distance” valuesforAandBisrejectedatthe99%confidencelevel.
matching method used by many other authors (e.g. This result was also found by Smith et al. (2012).
McGeeandBalogh2010). Wefindthatthetwomatching To investigate the possibility that other A and B val-
methods give different results for ∼10% of the SNe. ues could have passed the test, we plot (Fig. 8) the cu-
4 Gao and Pritchet
mulative distribution functions of our galaxies for both overpredicting the rates in high mass galaxies. The red
an A-only model (SNR=A×M) and a B-only model line is a (somewhat unphysical) modified A+B model
(SNR=B×SFR), keeping in mind that any set of values withaconstantbackgroundsupernovarateadded(§4.1).
of A and B must fall between these two distributions. The two (unmodified) A+B models are in reasonable
(Note that forthe A-onlyandB-only models,the values agreement with each other.
taken for A and B do not affect the result of the K-S To show how large an effect our uncertainties in the
test.) The A-only model is rejected at an even higher determination of the A and B parameters may have on
degree, while the B-only model passes the test (∼ 50% Fig. 12, we plot Fig. 13, which shows the differential
rejection). This shows that our data do not support a distributionsforthe1σ upperandlowerlimitsforAand
larger value of A/B. B. It can be seen that the uncertainties cannot account
Passing the mass-ranked K-S test is a necessary but for the discrepancy at the low mass end.
insufficient condition for a SNR model to be plausible. As mentioned in §2, the MPA/JHU catalog system-
Almost all models pass the sSFR-ranked K-S tests, in- atically overestimates the SFRs of passive (non-star-
cluding the A+B model (Fig. 9), but the same cannot forming)galaxies,assigningthem a smallbut significant
besaidfortheSFR-rankedK-Stests,forwhichboththe SFR. We redo the A+B fits by setting the SFRs of all
generic A+B model, the B-only model, and all mod- galaxieswithansSFRsmallerthan10−11.5(M⊙/yr)/M⊙
els in between are rejected (Fig. 10). This means that to zero; our results are unaffected.
a smaller value of A/B will not pass the test. Since
5. MODIFICATIONSTOTHEA+B MODEL
we already know that a larger A/B value will not work
(see above),we conclude from these tests that no A+B We havedemonstratedthat the A+B modeldoes not
model can match our observed data. The results of these reproduceourdata. Toaccountforthediscrepancy,most
KS tests are shown in Table 2. notably the observed SNR excess in low mass galaxies,
The KS test is usually valid only when the measure- we now try modifications to the A+ B model. Table
ment uncertainties in the observed sample are small. 2 summarizes the results for some of the many models
Therefore the (possibly significant) uncertainties in the tried. Due to our small sample size, and also to intrinsic
mass and SFR measurements brings the validity of our uncertainties in the MPA/JHU measurements of masses
above methods into question. In order to check the va- and star formation rates, there exist strong correlations
lidity of our K-S test under such conditions, we use a betweenmodelparameterswhenmorethan2parameters
MonteCarloapproach. Wegeneratemanyartificialsam- are used.
plesofsupernovaeusingthebestfitAandBvalueswith As previously found by Sullivan et al. (2006), A×M-
the sample of 19987 observed M and SFR values, add and B ×SFR-only models do not match the observed
the effects of uncertainties in M and SFR, and generate SNR distributions. The A×M + B×SFR + C and
cumulative distributions of log M and log SFR. The re- (A×M+B×SFR)(1+C×M−1)modelsareparticularly
sults are completely consistent with the K-S tests: the interesting,sincetheybothpasstheKStests,andmight
A+B model is rejected at >99% probability for log M have a physical explanation (§6).
and >97% probability for log SFR. We also investigate the model proposed by Smith et
The maximum likelihood method for fitting A and B al. (2012): SNR=A×Mx+B×SFRy. We attempt to
returns a probability which can be used as a measure of obtain best-fit values for the parameters of our own us-
goodnessoffit. Thisprobabilitycanbeinterpretedusing ing maximum likelihood, but find that our sample is too
Monte Carlosimulations. Allowing for the fact that this smalltoovercomethecorrelationsbetween4parameters.
probabilitydepends onthe number of supernovaegener- Using Smith et al.’s values of x = 0.72, and y = 1.01,
ated (which varies from run to run), we find that more we refit A and B using maximum likelihood; the result
than 99.9% of all simulations possess a peak probabil- is A = 5.3+1.3 ×10−11,B = 1.0+0.4 ×10−3 (appropri-
−0.9 −0.3
ity greater than that of the best fit to the observations. ate units in the M⊙ - yr system). Using either these
Thus we conclude, independently of the K-S tests, that values, or the A and B values given by Smith et al.
the A+B model is not a good match to the data. (A = 0.41×10−10,B = 0.65×10−3), we find that the
To present an alternate, and somewhat more illustra- model passes the mass-ranked test, but fails the SFR-
tive, view of this discrepancy, we separate the galaxies ranked test at 99% confidence. A very small value of
by mass into two equally-sized groups divided at log x=0.06±0.10 fits the data better, and passes both KS
M=10.7, and plot specific SNR vs. specific SFR (as in tests - this is probably a manifestation of the C term
Fig. 6 of Sullivan et al. 2006). The resulting plot (Fig. mentioned above.
11) shows that low-mass galaxies have a systematically
higher specific SNR than their high-mass counterparts, 6. DISCUSSION
by a factor of 3–5. Applying different mass cuts consis-
tently results in the same trend. In this paper, we have obtained A and B values from
Aplotofthe differentialdistributionfunctionsofSNR spectroscopic data. However, further analysis reveals
predicted by the A + B model vs the actual observed that there is an intrinsic inconsistency between the data
SNR (Fig. 12) further illustrates the issue, where the andthe A+B model: mass-rankedKS tests reject all A
curves have been scaled to show the relative absolute and B values with a higher A/B ratio than our best-fit
SNR obtained by each model. From Fig. 12, it can values, while SFR-ranked KS tests reject those with a
be seen that both the predictions of our best-fit A+B smallerA/B ratio;furthermore,fromFigs. 8,11and12,
model(greenline)andthatofSullivanetal.(2006)(blue we can see that there exists a SNR excess in low mass
line) underpredict the rates of supernovae hosted by low galaxies. §6.1 discusses some possible issues with the
mass, high specific star formation rate galaxies, while data and its interpretation; §6.2 interprets our results.
Correlations between SNe Ia Rates and Host Galaxy Properties 5
6.1. Data Issues ical form SNR/M = A+B(SFR/M)+C/M. In this
case C/M provides some mass modulation of the A+B
We have not applied any completeness corrections to
model, which we have already noted.
thesupernovae. Thecompletenessofthesamplehasbeen
We now turn to this mass dependence of the A+B
discussed in §2; incompleteness will affect the numerical
model, as seen directly in Fig. 11, and also in the fits
valuesofAandB bylessthan0.1dex,butwillnotaffect
to the two M-dependent models in Table 2. An effect
A/B, nor any of the conclusions regarding the A+ B
similar to this has been seen before in the Lick Observa-
model (unless there is strong redshift evolution in the
tory Supernova Survey (LOSS – Li et al. 2011): SNe Ia
SNR – something that is not observed – e.g., Perrett
et al. 2012). To check this, we redo the work with a from LOSS possess a specific rate SNR/M ∝ M−0.5
redshiftcutof0.20insteadof0.25,andfindnosignificant (rather than the expected SNR/M ∝ constant). In the
differences. case of LOSS, however, reliable SFRs were not available
Is it possible that the detection efficiency is different due to a lack of spectroscopic data, rendering it difficult
in galaxies with high star formation, and concomitant to disentangle the mass-dependence of mean galaxy age
higher extinction? We have already noted that the de- or specific star formation rate (e.g. Schiminovich et al.
tection efficiency of our sample is higher than for SDSS 2007), which could produce some or all of this effect. In
SNe as a whole (§2.1). In fact high specific SFR implies fact, a simple model incorporating ages from Gallazzi et
lowerMandluminosity(Brinchmannetal.2004)andas al. (2005) explains the LOSS data rather well (e.g. §3
a result the star forming galaxies(log sSFR >−10.5)in of Kistler et al. 2011), because low mass galaxies tend
oursample haveanevenlowermeanz (about 0.07);this tobeyounger,andhavehigherspecificSFR’s, thanhigh
means that incompleteness is even less of a problem. A mass galaxies.
simulation of an extreme efficiency variation with sSFR Mass-dependence of the simple A+B model has also
shows that, while B will be biased, the main conclusion beeninferredbySmithetal. (2012),whocombineSDSS
regardingtheinapplicabilityoftheA+B modelremains SNIadatawithphotometricmassesandSFR’stoderive
unaltered. SNR/M ∝A×M−0.3+B(SFR/M). As already noted,
The precisionofthe SDSS MPA/JHUmassesandstar however, this model fails the SFR-ranked KS test when
formation rates may affect our conclusions. We have al- their parameters are assumed.
readymentionedtheeffectsofstatisticaluncertaintiesin Inthispaperthemass-dependenceoftheA+B model
the derived M and SFR quantities. We repeat our work is demonstrated in a manner that is different from, but
using VESPA masses and star formation rates. The re- complementary to, the methodology of Smith et al..
sulting values of A and B conformbetter to those found First, we use spectroscopically-determined masses and
by Sullivan et al. (2006) (see Table 1), but we are still star formation rates, which presumably should be more
abletorejectallA+B modelsusingKStests. SeeTable accurate (though resulting in a smaller sample size).
3 for a summary of our results obtained using VESPA Second, we visualize the mass-dependence in a non-
data, and Fig. 1 for our best-fit values for A and B parametricmanner froma plot of SNR/M vs. SFR/M
obtained using VESPA. for two different mass samples (Fig. 11). Note that
Our simplified treatment of the observation windows this is quite different from the SNR/M vs M plot in
(§2) could affect the derived values of A and B (and C, Li et al. (2011), which includes galaxies with a wide
where relevant) by a common factor (which we estimate range of SFR’s, and hence may be affected by the mass-
to be <0.1 dex). Again, this does not affect conclusions dependence of star formation rate (though see below).
regarding the rejection of A+B models. Third, we use the fact that an acceptable model must
successfully reproduce the observed distribution of su-
6.2. Physical Implications pernovae with mass and SFR, a fact that rules out most
models that do not depend on mass.
The (somewhat unphysical) SNR = A×M +
How is this mass dependence of the A+B model to
B×SFR+C modelisagoodfittothe data,butimplies
beinterpreted? Theobviousculpritismetallicity,dueto
a constant rate per galaxy independent of galaxy prop-
the well-established correlation between metallicity and
erties. Could the progenitors of SNe arising from the C
mass for galaxies (Tremonti et al. 2004, Gallazzi et al.
term (∼ 68+−287% of the total number of SNe, 1σ errors 2005 and references therein). Based on the environmen-
quoted) be due to a diffuse intergalactic stellar popula- tal dependence of SDSS SNR, it has been suggested by
tion, not accounted for in MPA/JHU masses? Sand et Cooper et al. (2009) that the SNR of galaxies may in-
al. (2011) find that about 17% of the SNe Ia in rich crease with decreasing gas-phase metallicity. Kistler et
clusters occur in an intergalactic stellar population, a al. (2011) note that for a given initial mass, the end-
number that rises to almost 50% in groups (McGee and point white dwarf is more massive for lower metallicity,
Balogh 2010). As discussed in Sand et al., these num- afactthatmayonitsown(andignoringmanyotherfac-
bers are in general accord with the direct intergalactic tors)leadtoanenhancedSNRinlowmetallicitygalaxies.
light measurements of Gonzalez et al. (2005, 2007). If We note that neither ofthese results constitutes a direct
some supernovae belong to an intergalactic population, proofofaninversemetallicity-SNRrelation. Theoretical
then nearby galaxies (with approximately the same red- models for single degenerate SNe Ia in fact suggest the
shifts as the supernovae) could be incorrectly identified opposite rate-metallicity relation (e.g. Kobayashi et al.
as hosts. A detailed modelling of this effect is beyond 2000,Langer et al. 2000).
the scope of this paper; however,a C term that requires An alternate,and somewhatmore straightforward,in-
>50% of SNe Ia to be intergalactic seems implausible. terpretation for the mass dependence relates to the fact
An alternate interpetation of the constant C model can that the rate of SN Ia from a burst of star formation
beobtainedsimplybywritingtherateinthemorephys-
6 Gao and Pritchet
(the so-called delay time distribution or DTD) declines of A and B can account for this discrepancy be-
withtime asroughly1/t(Totanietal. 2008,Maozetal. tween the model and the observed data. This result
2011). Referring to Fig. 11, the median logmass for our holds regardlessofanyuncertainties inthe observ-
low(high)masssampleis1.7×1010M⊙(1.1×1011M⊙). ing windows and completeness.
Thus, from the mean age-mass relation of Gallazzi et al
• Modifications to improve the A + B model are
(2005), and DTD ∝1/t, we would expect the rate to be
tested. Mass-dependentmodelsperformwellinthe
about∼1.5−2×higherinthelowermasssample(assum-
K-S tests.
ingthatallstarswithinagalaxyformatthesameepoch)
–marginallyconsistentwith whatis observed. However,
A number of explanations of these results exist. Most
we note (Pritchet et al. 2008) that a wide range of evo-
promising is the galaxy mass dependence of mean stel-
lutionarymodels andcompositestellarpopulations(and
lar ages, possibly coupled with variation in the slope of
hence mean stellar ages) converge to a narrow range of
the IMF with galaxy mass. Further work is needed to
loci in the SNR/M vs. SFR/M diagram in Fig. 11.
quantify the effect of these two mechanisms on super-
While the initial mass function (IMF) is generally as-
nova rates.
sumed to be universal, recent evidence – both dynami-
cal (e.g. Cappellari et al. 2012) and spectrophotometric
(e.g. Spiniello et al. 2012, Ferreras et al. 2012) – sug- This work was funded by Natural Sciences and Engi-
gests that massive galaxies possess steeper IMF’s, with neering Research Council of Canada.
a deficit of high mass stars relative to low mass stars Funding for the SDSS and SDSS-II has been pro-
(see also Kroupa et al. 2011). At a deeper level, this vided by the Alfred P. Sloan Foundation, the Partic-
may well be a metallicity effect. Since SNe Ia are ex- ipating Institutions, the National Science Foundation,
plodingWD’s, andsincesuchWD’s originatefrommain the U.S. Department of Energy, the National Aeronau-
sequence stars as massive as 8 M⊙, it follows that low tics and Space Administration, the Japanese Monbuka-
mass galaxies are more efficient (per unit galaxy mass) gakusho,theMaxPlanckSociety,andtheHigherEduca-
inproducingSNe Ia. Adetailedcalculationofthis effect tion Funding Council for England. The SDSS Web Site
is not yet possible because, among other things, it re- is http://www.sdss.org/.
quires a knowledge of the efficiency of SN Ia production The SDSS is managed by the Astrophysical Research
as a function of stellar mass. However, it is interesting Consortium for the Participating Institutions. The Par-
to note that, for fixed total mass and a power-law mass ticipatingInstitutionsaretheAmericanMuseumofNat-
functiondN ∝MαdM,varyingthe powerαby ±0.5(as ural History, Astrophysical Institute Potsdam, Univer-
observed) around the Salpeter value of α = −2.35 pro- sity of Basel, University of Cambridge, Case Western
duces a change in the number of SN Ia progenitors by a Reserve University, University of Chicago, Drexel Uni-
factorof2−4×(dependingonthemassrangeused). An versity, Fermilab, the Institute for Advanced Study, the
IMF effect therefore has some potential as a mechanism Japan Participation Group, Johns Hopkins University,
for the galaxy mass dependence of SN Ia rates. the Joint Institute for Nuclear Astrophysics, the Kavli
Finally, we note the recently discovered host mass ef- Institute for Particle Astrophysics and Cosmology, the
fect on SN Ia peak luminosities (Kelly et al. 2010, Sul- Korean Scientist Group, the Chinese Academy of Sci-
livan et al. 2010). A somewhat simplistic hypothesis is ences(LAMOST), LosAlamos NationalLaboratory,the
thatboththiseffect,andthehostmassmodulationofsu- Max-Planck-Institute for Astronomy (MPIA), the Max-
pernovaspecificratesthatwehavefound,areconnected. Planck-Institute for Astrophysics (MPA), New Mexico
Boththeseeffectscouldbecausedbythepresenceoftwo State University, Ohio State University, University of
distinctclassesofSNIaprogenitors,whosefrequencyde- Pittsburgh, University of Portsmouth, Princeton Uni-
pends on host galaxy mass through metallicity and/or versity, the United States Naval Observatory, and the
progenitor mass. University of Washington.
Special thanks to Sara Ellison,Trevor Mendel and Se-
7. CONCLUSIONS
bastien Fabbro for valuable discussion during the course
In summary, we have studied how Type Ia supernova of this work, and also for helping with the navigation of
rates correlate with host galaxy masses and star forma- the various databases used. We thank the referee, Alex
tion rates. Our supernovae and host galaxieswere taken Conley, for valuable advice concerning the data process-
from the SDSS spectroscopic galaxy sample, which has ing.
spectroscopic measurements of masses and star forma- Thanks also to colleagues at Nanjing University for
tionrates. Thereliabilityofthehostsamplewaschecked hardware support during the final stages of this work.
bycomparingwithVESPAdata. Ourprimaryresultsare
as follows:
• We find that maximum likelihood values A =
3.5+−00..97×10−14(SNe/yr)(M⊙)−1 andB =1.3+−00..43×
10−3(SNe/yr)(M⊙yr−1)−1 provide an optimal fit
to our data for a generic A+B model. This result
is largely consistent with the literature (Fig. 1).
• The A + B model fails to predict the mass-
dependence of the specific SNR - specific SFR re-
lation in our sample. We also show that no set
Correlations between SNe Ia Rates and Host Galaxy Properties 7
Figure 1. ResultsofourfitstotheA+B modelwitherrorbars
(filledcircle)plottedagainstothervalues(squares)intheliterature
(Scannapieco&Bildsten2005,Neilletal.2006,Dildayetal.2008,
Sullivanetal.2006,Smithetal.2012). Theopencircledenotesour
resultsobtainedusingVESPAdata,asmentionedinthediscussion.
AisinunitsofM−1 y−1;B isinunitsofM−1.
⊙ ⊙
222
111
r]
a
e
y
/ 000
M
R [
F---111
S
g
o
l
---222 Figure 3. AcomparisonbetweenMPA/JHUandVESPAentries
formass(top)andSFR(bottom). Notetheconstantoffset(dashed
line), as predicted in Tojeiro et al. 2009, which is the result of a
---333 calibration offset between the VESPA and MPA databases. The
666 777 888 999 111000 111111 111222 111333 scatters(standarddeviation)forthemassesandSFRsare0.18dex
logM [M ] and0.48dexrespectively.
Figure 2. DistributionofspectroscopicgalaxiesintheMPA/JHU
catalog (grey dots), with the 53 hosted SNe (filled red circles) on
alogM -logSFRplane. Alsoshownforcomparisonarethe error
barsonthe massesandstar formationratesof5randomlypicked
galaxies.
8 Gao and Pritchet
Figure 4. Diagram illustrating the concept of R25. The ellipse
is the 25 mag/arcsec2 isophote for the potential host galaxy, O is
the centre, the SN Ia is at B, and A is the point of intersection Figure 7. Specific supernova rate as a function of specific star
betweentheellipseandlineOB.R25=OB/OA. formation rate. The black points are for our observed sample.
Plotted for comparison are the predictions of our best-fit A+B
model (black line), the predictions of Sullivan et al. 2006 (dashed
line),andtheirdata(squares).
Figure 5. R25 criteria used (x-axis) plotted against number of 1111....2222
SNe matched (y-axis). R25 = 3.8, the criterion we use, is shown
by the vertical line. Note that the curve flattens out veryrapidly
beyondthisline. R 1111....0000
N
S
of
n 0000....8888
o
cti
a
Fr 0000....6666
e
v
ati
mul 0000....4444
u
C
0000....2222
0000....0000
8888 9999 11110000 11111111 11112222
log(M/M )
Figure 8. Cumulative distribution functions of mass-ranked K-
S tests for our best-fit A+B model (green), the A×M-term-only
model (red), and the B×SFR-term-only model (blue). The green
andredlinesarerejectedbytheK-Stest,showingthatourbest-fit
A+BmodelandallA+BmodelswithahigherA/Bratiodonot
agreewithourdata.
Figure 6. Probabilitycontours for the values of A and B on an
A-Bplane. Thered,blueandgreencontoursrepresent1σ,2σ,3σ
probabilityrespectively,whilethecircledenotesourfinalmaximum
likelihoodvalue. The errorbars show the 1σ and 2σ limitsof the
marginaliseddistributionsofAandB.
Correlations between SNe Ia Rates and Host Galaxy Properties 9
1111....2222
R 1111....0000
N
S
of
n 0000....8888
o
cti
a
Fr 0000....6666
e
v
ati
mul 0000....4444
u
C
0000....2222
Figure 11. Specific supernova rate as a function of specific star
formationrate. Theopenpointscorrespondtoourlow-masssam-
0000....0000 ple,andthefilledpointsourhigh-masssample. Theverticalerror
----11113333 ----11112222 ----11111111 ----11110000 ----9999 ----8888
bars indicate Poisson uncertainties for our SNe numbers in each
log[sSFR/(M /yr/M )] bin, and the horizontal error bars correspond to bin size. The
trend that the open points tend to lieabove the filled ones exists
Figure 9. Cumulativedistributionfunctions ofthe specific-star- for all mass discrimination criteria we try. Also plotted are the
formation-ranked K-S tests for the various models. The only predictions of our best-fit A+B model. Our results have been
models which were rejected were the mass-only model (red), and scaledtoaccount forobservingwindowissues.
the SFR-only model (blue). The remaining models (A×M +
B×SFR,A×M+B×SFR+C×M−1,A×M+B×SFR+C, and
(A×M +B×SFR)(1+C×M−1)) all fall within the hashed blue
area. As described in the text, this set of K-S tests is largely in-
conclusive.
1111....2222
R 1111....0000
N
S
of
n 0000....8888
o
cti
a
Fr 0000....6666
e
v
ati
mul 0000....4444
u
C
0000....2222
Figure 12. Differentialdistributionfunctionsofsupernovarateas
afunctionofhostmass. Theblacklinecorrespondstoourobserved
0000....0000 data,presentedasnumberofSNeobservedineachrespectivemass
----2222 ----1111 0000 1111 2222 bin. The green and blue lines are the predictions of our best-fit
A+B modelandtheresultsofSullivanetal.(2006) respectively.
log[SFR/(M /yr)] Theredlineisthepredictionofourbest-fitA×M+B×SFR+C
“constantbackground”model,andthecyanlinecorrespondstothe
Figure 10. CumulativedistributionfunctionsofSFR-rankedK-S (A×M+B×SFR)(1+C×M−1)model. Thepredictionshavebeen
testsforourbest-fitA+Bmodel(green),theAM-term-onlymodel
scaled by the number of bins used within the plotted domain, so
(red),andtheBSFR-term-onlymodel(blue). Thegreenandblue
thatthelinesarecomparablewiththoseoftheobserveddata. Our
linesarerejectedbytheK-Stest,showingthatourbest-fitA+B
resultshavebeenscaledtoaccountforobservingwindowissues.
modelandallA+B modelswithalowerA/B ratiodonotagree
withourdata. This,inconjunctionwithFig. 8,rulesoutallA+B
models.
10 Gao and Pritchet
Figure 13. Differentialdistributionfunctionsofsupernovarateas
afunctionofhostmass. Theblacklinecorrespondstoourobserved
data,presentedasnumberofSNeobservedineachrespectivemass
bin. Theredandbluelinescorrespondtotheupperandlower1σ
limitsduetotheuncertaintiesofourresultsforthevaluesofAand
Brespectively,plottedbysettingbothAandBtotheirupperand
lower limits respectively. The green line corresponds to our best-
fitvalues. Thepredictionshavebeenscaledbythenumberofbins
used within the plotted domain, so that the lines are comparable
with those of the observed data. Our results have been scaled to
account forobservingwindowissues.