Ideal Bandpasses for Type Ia Supernova Cosmology

Ideal Bandpasses for Type Ia Supernova CosmologyAuthor(s): Tamara M. Davis, Brian P. Schmidt, and Alex G. KimSource: Publications of the Astronomical Society of the Pacific, Vol. 118, No. 840 (February2006), pp. 205-217Published by: The University of Chicago Press on behalf of the Astronomical Society of the PacificStable URL: http://www.jstor.org/stable/10.1086/499116 .

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Publications of the Astronomical Society of the Pacific, 118:205–217, 2006 February� 2006. The Astronomical Society of the Pacific. All rights reserved. Printed in U.S.A.

Ideal Bandpasses for Type Ia Supernova Cosmology

Tamara M. Davis1 and Brian P. Schmidt

Australian National University, Cotter Road, Weston Creek, ACT 2611, Australia; [email protected], [email protected]

andAlex G. Kim

Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720; [email protected]

Received 2005 August 24; accepted 2005 October 25; published 2006 February 8

ABSTRACT. To use Type Ia supernovae as standard candles for cosmology, we need accurate broadbandmagnitudes. In practice the observed magnitude may differ from the ideal magnitude-redshift relationship eitherthrough intrinsic inhomogeneities in the Type Ia supernova population, or through observational error. Here weinvestigate how we can choose filter bandpasses to reduce the error caused by both these effects. We find thatbandpasses with large integral fluxes and sloping wings are best able to minimize several sources of observationalerror, and are also least sensitive to intrinsic differences in Type Ia supernovae. The most important feature ofa complete filter set for Type Ia supernova cosmology is that each bandpass be a redshifted copy of the first.We design practical sets of redshifted bandpasses that are matched to typical high-resistivity CCD and HgCdTeinfrared detector sensitivities. These are designed to minimize systematic error in well-observed supernovae; finaldesigns for specific missions should also consider signal-to-noise ratio requirements and observing strategy. Inaddition, we calculate how accurately filters need to be calibrated in order to achieve the required photometricaccuracy of future supernova cosmology experiments, such as theSupernova/Acceleration Probe (SNAP), whichis one possible realization of the Joint Dark Energy Mission (JDEM). We consider the effect of possible periodicmiscalibrations that may arise from the construction of an interference filter.

1. INTRODUCTION

Many cosmological studies using Type Ia supernovae (SNeIa) have been restricted to using only those filter sets that areavailable on the telescopes that make the observations.2 How-ever, upcoming dedicated SN Ia cosmology cameras, such asthose being designed for the Joint Dark Energy Mission (R. L.Orbach & E. J. Weiler 2003, unpublished),3 can gain consid-erable advantage by using a set of filters designed specificallyfor measuring SNe Ia. A clever choice of bandpasses can im-prove our ability to constrain systematic errors and thus en-hance our ability to extract scientific information. In this paperwe study the effect of bandpass choice on SN Ia measurementsand design an ideal bandpass set for supernova Ia studies.

For supernova Ia studies, we need bandpasses covering thewhole wavelength range appropriate for the survey depth inorder to probe the same spectral region of highly redshiftedsupernovae that we do for nearby supernovae. Due to the large

1 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA94720.

2 The High-z SN team is one exception, having designed and built a filterset for their SN Ia measurements.

3 See the online article “The NASA-DOE Joint Dark Energy Mission(JDEM),” at www.science.doe.gov/hep/JDEM%20Reports.shtm.

redshift range being sampled, we need to be able to accuratelycorrect for the brightness of supernovae that are redshifted awayfrom the center of a bandpass (K-correction), and we wouldprefer the requiredK-correction to be small in order to minimizeany systematic error we may introduce in our analysis. AccurateK-corrections are most easily achieved through a set of band-passes that are redshifted copies of one another such that therequiredK-corrections are periodic in redshift.

Type Ia supernovae are a very homogeneous set, but theyare not all identical (Filippenko et al. 1992; Phillips et al. 1992;Leibundgut et al. 1993; Benetti et al. 2004, 2005; B. James etal. 2006, in preparation). Therefore, we also benefit from choos-ing bandpasses through which SNe Ia appear most homoge-neous; that is, choosing bandpass shapes that are not overlysensitive to the spectral diversity of the Type Ia supernovapopulation. We would also like bandpasses that give good colorinformation so that we can measure the effect of extinction andcan account for it. The total throughput of the bandpasses isimportant in determining exposure times, and the greater thethroughput, the faster the survey can be completed. However,we are restricted to creating bandpasses that fit in the sensitiveregions of available detector technology and can be createdwith current (or near-future) filter technology.

Type Ia supernova spectra at zero redshift are bright over a



206 DAVIS, SCHMIDT, & KIM

2006 PASP,118:205–217

Fig. 1.—Typical SN Ia spectrum at maximum light (Nugent et al. 2002,with 2005 update) superimposed on the sensitivity of a typical optical detector(high-resistivity, deep-depleted CCD;thick solid line) and the sensitivity of atypical NIR detector (HgCdTe detector with 1.7mm cutoff; thick dashed line).The spectrum that peaks at 4000 A˚ (solid line) is for a SN Ia at zero redshift,while the spectrum that peaks at 11000 A˚ (dotted line) is for a SN Ia at aredshift of 1.8. The wavelength axis is logarithmic.

range of 2500–10000 A˚ . The ultraviolet (UV) cutoff is quitesharp, but the IR falloff is gradual—approximately the Ray-leigh-Jeans tail. The rest-frame peak brightness at the time ofmaximum light occurs at about 4000 A˚ . At earlier epochs thisshifts slightly to lower wavelengths, while at later epochs thespectrum peaks at longer wavelengths. At a redshift of 1.8, thisrange translates to 7000–28000 A˚ , with a peak at 11200 A˚ . Atypical maximum-light SN Ia spectrum is shown superimposedon typical optical and near-infrared (NIR) detector sensitivitiesin Figure 1.

As discussed above, any set of bandpasses for SN Ia cos-mology should consist of logarithmically distributed band-passes—each bandpass being a redshifted copy of the first—in order to minimize uncertainty inK-corrections (see § 2.1).The free variables include:

1. The shape of a bandpass: transmittance as a function ofwavelength. Some shapes are less sensitive to the intrinsic dif-ferences in SNe Ia than others.

2. The width of a bandpass. Wider bandpasses increasethroughput but may decrease color discrimination ability.

3. The spacing of the bandpasses. This is highly constrainedby the detector sensitivities and bandpass width.

4. The number of bandpasses. The survey can be completedmore quickly for fewer bandpasses.

5. The stability and knowledge of the transmission of thebandpass.

Note that throughout this article we concentrate on theband-passes required for supernova studies. The filters that realizethese bandpasses need to be designed while taking into accountthe mirror reflectivity and detector sensitivity of the telescopefor which the filters are made. Also, when designing final filtersfor a specific mission, the signal-to-noise ratio (S/N) advantageof wider bandpasses should be considered in light of theplanned observing strategy.

Many of the examples in this paper can be applied to theSupernova/Acceleration Probe (SNAP; Aldering et al. 2004),a proposal for the JDEM that concentrates on broadband pho-tometry of Type Ia supernovae. This paper is also relevant toother JDEM proposals that use Type Ia supernovae, even ifthey use spectra rather than broadband photometry, since theprocess of converting a spectrum to a measure of luminositygenerally assumes some passband.

This paper has three main parts. In § 2 we discuss generalbandpass shape considerations, examining the benefits of slopingbandpass edges versus top-hat profiles, as well as bandpass widthand spacing. In § 3 we design several bandpass sets, tuned totypical detector sensitivities, that allow measurements of super-novae over a range of redshifts out to at least . Thesez p 1.8first two parts assume that filters designed to realize these band-passes can be manufactured and calibrated perfectly, and thatthe only scatter in our measurements comes from the inhomo-geneities in the SN Ia population. In the final section we in-vestigate the effect of filter calibration errors on SN Ia mag-

nitude calculations. We test both random and periodiccalibration errors (periodic calibration errors may be expectedwhen using interference filters) and identify frequencies thatare particularly damaging for SN Ia measurements.

2. BANDPASS SHAPE CONSIDERATIONS

Here we investigate the role that bandpass shape will playin the systematic and stochastic error of future SN Ia cosmologymissions. Possible bandpass shapes range from the “top hat”profile, with very sharp cutoff, to a very curved bandpass, witha smooth transition from no throughput to maximum through-put. The bandpasses may be symmetrical or asymmetrical.

The bandpass shape can affect the quality of data collected,in a variety of ways. Diversity in the SN Ia population causesa natural scatter in the data (even after the light curve width-luminosity relation has been corrected for). This scatter can beminimized with a clever choice of bandpasses. Bandpasses alsodetermine the accuracy with which we can measure color andthus deduce extinction correction and other color-dependenttraits.

Minimizing the scatter due to SN Ia diversity is importantnot only for minimizing the number of supernovae needed toget good statistics, but also to reduce the impact of many typesof systematic error, such as a bias due to population evolution.For example, if we use a model of the supernova spectrum tocalculate theK-correction for each supernova, and this modelspectrum is not a true representation of the mean, then we mayintroduce a systematic magnitude offset to the supernovae. Anevolution of the SN Ia population with redshift would createa shift in the mean, which would introduce a spurious redshift-dependent trend in average peak magnitude. Any bandpass set



TYPE Ia SUPERNOVA BANDPASSES 207

2006 PASP,118:205–217

Fig. 2.—Trapezoidal bandpass. For the symmetrical bandpasses used in§ 2, we set .c p a

Fig. 3.—K-correction as a function of epoch between�10 and 50 days aboutmaximum for a variety of observed SN Ia spectra, all shifted to , asz p 0.05measured through a trapezoidal bandpass centered on 4500 A˚ , with a slope

A on each side and a flat top A˚ (integral fluxp 1500).a p c p 500 b p 1000We take the rms over epoch to provide the rmsK-correction value for thisKrms

bandpass (providing one point in Fig. 4). We take the dispersion about thepolynomial fit shown here to provide a measure of the scatter in theK-correction,

, due to intrinsic differences between SNe Ia (one point in Fig. 5).DKrms

that minimizes the scatter due to intrinsic differences in theType Ia supernova population should also minimize these sys-tematic effects.

In order to investigate the effect of bandpass shape on SNIa measurements, we consider some idealized bandpass shapes.Each bandpass is a symmetrical trapezoid (Fig. 2) describedby the width of the rising edge of the trapezoid (a) and thewidth of the flat top (b). The integral flux (arbitrary units) isgiven by (this is the area under the trapezoid). We usea � bsymmetrical trapezoids only for the simplicity of reducing thenumber of parameters we test. Asymmetrical bandpasses followsimilar trends and may be more suitable for a final design basedon the detailed matching of bandpasses to detector and mirrorsensitivities.

2.1. K-Correction

The K-correction is one of the major systematic correctionsthat needs to be made on SN Ia observations. It arises becausea given bandpass samples a different spectral region of red-shifted and rest-frame supernovae. TheK-correction is the cor-rection required to convert the observed magnitude to a rest-frame magnitude by accounting for the different spectralregions sampled. We need to perform theK-correction in orderto be able to compare the magnitudes of SNe Ia at differentredshifts.

Because of theK-correction, the most important feature ofany bandpass set for SN Ia cosmology is that each bandpassin the set be a redshifted copy of the first. That is, the band-passes are the same shape, but redshifted, so that we can sampleexactly the same spectral region of high-redshift supernovaethat we do for low-redshift supernovae. This reduces the modeldependence of theK-correction and thus reduces its suscepti-bility to systematic error. Logarithmically spaced bandpasses,where each bandpass differs from the previous one by a mul-tiplicative factor of for some value , are ideal be-1 � z zsep sep

cause they give a pattern ofK-corrections that is periodic inredshift and periodically returns to zero (except for a filter zero-point offset). A systematic error could be identified by theperiodicity in the results and corrected, because dark-energymodels generally do not predict periodic modulation in distancemodulus. The advantage of bandpasses that are self-similar in

redshift is so great that we assume this as a required propertyof any purpose-built filter set for SN Ia cosmology studies.

Using our idealized bandpass shapes, we take a sample of62 Type Ia supernova spectra and calculate theK-correction(Germany et al. 2004; Nugent et al. 2002) required to comparetheir rest-frame magnitude to the magnitude that would beobserved if they were at a redshift of .4 The spectraDz p 0.05can be found in Kirshner & Oke (1975), Leibundgut et al.(1991), Filippenko et al. (1992), Phillips et al. (1992), Kirshneret al. (1993), Wells et al. (1994), Patat et al. (1996), Filippenko(1997), and Riess et al. (1999).

The K-correction varies according to the epoch of the su-pernova (Fig. 3). The spectra we use cover a range of epochsfrom 10 days before maximum to 50 days after maximum. Togive a measure of the magnitude of theK-correction for aparticular bandpass, we take the rms of theK-correction overthis epoch range, . In Figure 4 we plot2 1/2K p [AK(t) S]rms

as a function of bandpass shape, varying the slope of theK rms

bandpass edges (a) and the width of the flat top (b). The integralflux through the bandpass ( ) increases along the diagonal.a � b

4 The exact choice of is arbitrary—K-corrections get larger for higherDz, but this behavior is common to all bandpasses, so taking a snapshot at aDz

particular redshift is sufficient to quantify their relative behavior. Note thatbecause we are using a bandpass set that is self-similar in redshift, thisDzcan be measured from the redshift of any bandpass in the set. For example,if each bandpass is redshifted by from the previous bandpass, thenz p 0.16sep

successive bandpasses are at redshifts , 0.16, 0.35, 0.56, … , andz p 0refers to SNe with , 0.22, 0.41, 0.64, … . The relationshipDz p 0.05 z p 0.05SN

between these quantities is , wheren is an in-n�11 � z p (1 � z ) (1 � Dz)SN sep

teger starting from 1, numbering each bandpass in the set.




2006 PASP,118:205–217

Fig. 4.—K-correction as a function of the slope of the wings of theKrms

trapezoidal shaped bandpass (a) and the width of the flat top (b). The integralflux ( ) increases diagonally from bottom left to top right. We calculateda � bK-correction as a function of bandpass shape, centering the bandpasses on fivedifferent positions between 4500 and 5500 A˚ . The same trend appears irre-spective of the bandpass position, and here we plot the average. In general,the wider the bandpass the smaller theK-correction, because the wider band-passes encompass a larger overlap region of the redshifted and rest-framespectra.

Fig. 5.—Results of fitting a polynomial to theK-correction as a functionof epoch and plotting the rms scatter about this fit, , as a function ofDKrms

bandpass shape. The shape of the curve is similar to that of theK-correction(Fig. 4), showing that a largerK-correction results in a larger scatter in theK-correction, as expected.

For each bandpass shape, we center the bandpass at five dif-ferent wavelength positions between 4500 and 5500 A˚ so thatthey sample different regions of the spectrum. TheK-correctionas a function of shape follows a similar trend, irrespective ofthe central wavelength of the bandpass, and theK-correctionshown is the average over these different positions.

In order to measure the scatter inK-correction, we fit apolynomial to theK-correction as a function of epoch (Fig. 3)and calculate the rms deviation from this fit, (Fig. 5).DK rms

We chose a fourth-order polynomial for the fit, which is thelowest order polynomial that fits the data such that there is noresidual structure visible by eye.

The dominant factor determining both and isK DKrms rms

integral flux, . Wide bandpasses result in lowerK-cor-a � brections and lessK-correction scatter than narrow bandpasses.This is expected, since wider bandpasses naturally have agreater overlap in the spectral region of the redshifted and rest-frame supernovae they sample, so differences in the continuumslope (i.e., the color) of Type Ia supernovae at a particularepoch (Wang et al. 2005) will have a smaller impact. Moreover,a variation in a small feature of the spectrum will have lessimpact on broadband photometry when a wide wavelengthrange is sampled.

Plotting the rmsK-correction versus edge slope (a) for afixed integral flux (Fig. 6) shows there is a slight reduction inthe K-correction when we use bandpasses with sloping edges.This is also expected, since slight velocity shifts, as often seenin SNe Ia (e.g., Branch 1987; Benetti et al. 2005), can movea supernova feature completely out of the passband of a top-

hat bandpass, whereas the same shift would just slightly changethe amplitude of the feature in a bandpass with sloping edges.This is particularly significant for those bandpass and SN red-shift combinations that conspire to place a spectral peak on theedge of the passband. However, integral flux is the dominantfactor; sloping edges should not be added at the expense ofintegral flux.

K-correction uncertainty improves rapidly up to an integralflux of 1000 (equivalent to the flux under a 1000 A˚ wide tophat), and we therefore recommend a minimum preferred in-tegral flux of this amount.

2.2. Extinction Correction

Another important intrinsic feature affecting observed su-pernova magnitudes is dust extinction (Phillips et al. 1999).Dust extinction manifests itself as a wavelength-dependent at-tenuation of the emitted flux, with shorter wavelengths atten-uated more than longer wavelengths. This wavelength depen-dence allows us to measure the amount of extinction presentby measuring, for example, the color of the supernovaB � Vcompared to the color of a supernova with no extinction.B � VThis color difference is known as the color excess,E( ).B � VThe standard extinction law is given in Savage & Mathis(1979), who provide a table of attenuation as a function ofwavelength for 1 mag ofE( ). Giving this attenuation theB � Vsymbol , a supernova spectrum affected by extinction fromA(l)dust in the host galaxy will have a flux

�0.4A[l/(1�z)]E(B�V )F (l) p F [l/(1 � z)] # 10 (1)ext 0

compared to the emitted flux for a galaxy at red-F [l/(1 � z)]0

shift z. The extinction in other galaxies may deviate somewhat




2006 PASP,118:205–217

Fig. 6.—K-correction ( ) vs. slope (a) and vs. integral flux ( ). Edge slope only has a significant effect for narrow filters. The largest improvement comesK a � brms

by increasing integral flux, but as long as the filters are wider than 1000 A˚ , little more improvement is gained. Shaded regions represent 1j uncertainties. Valuesshown are the average over five bandpass positions between 4500 and 5500 A˚ .

Fig. 7.—Color excess as a function of bandpass shape, parameterized bythe width of the flat top (b) and the width of the sloping wings (a) as perFig. 4. Shape is less important than the integral flux. Narrower bandpassesgive larger color excess and thus better color discrimination and extinctionmeasurement. However, this effect is small compared to bandpass separation,and it is also tempered by the increasingK-correction errors as bandpassesnarrow. For this figure, bandpasses are separated by .z p 0.16sep

from the canonical Milky Way extinction law (Draine 2003,§ 2.3), but for our purposes this canonical extinction law issufficient.

The accepted method used to correct for extinction is to warpthe spectrum to match the SN color (the warping is done avariety of ways, but the same general principles apply). In orderto make this correction accurately, we need to be able to mea-sure color accurately. In particular, we need to be able to clearlydetect the color excess caused by extinction, . WeE(B � V )

therefore test how accurately we can measure extinction withdifferent bandpasses.

We take a model SN Ia spectrum at maximum light (Nugentet al. 2002) and add the effect of dust extinction according toequation (1). We perform synthetic photometry on this spec-trum through a variety of bandpasses and measure the colorboth before and after adding extinction. The larger the observedcolor excess for a fixed amount of extinction, the better thelever arm for detecting dust. Therefore, the bandpass set thatgives the biggest color difference after extinction is our pre-ferred set for extinction correction.

Because the measurement of color requires two bandpasses,we have an additional variable to consider, bandpass separation.The bandpass separation in the sets we are considering is bestdescribed by the redshift between bandpasses, , where thezsep

transmission of thenth bandpass is related to the trans-T (l)n

mission of the first bandpass by .n�1T (l) p T [l/(1 � z ) ]n 1 sep

As we see below, the bandpass shape is of secondary impor-tance to the bandpass separation in determining colordiscrimination.

In Figure 7 we plot the color excess as a function of bandpassshape. In this case, each bandpass is separated by a redshift of

. The dominant effect is again due to integral flux,z p 0.16sep

rather than shape, with narrow bandpasses providing bettercolor discrimination ability than wider bandpasses.

In Figure 8 we plot the color excess as a function of integralflux and bandpass separation. Although the previous figureshows that integral flux makes a more important contributionthan shape, its contribution is dwarfed by the effect of sepa-ration. More widely separated bandpasses give a marked im-provement in color discrimination ability over closely spacedbandpasses. This trend has a limit. As bandpasses get furtherapart, the maximumK-correction required becomes higher, and




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Fig. 8.—Color excess as a function of bandpass separation and bandpasswidth. Bandpasses are all top hats of widthb; therefore, width is equivalentto integral flux. Any effect of bandpass width is dwarfed by the effect ofbandpass separation. Widely separated filters are much better at measuringcolor excess than closely spaced filters.

Fig. 9.— plotted as a function of supernova redshift for the first twoKrms

bandpasses in a set. This diagram assumes the bandpasses are separated by, and uses a bandpass 1200 A˚ wide centered on 4200 A˚ (approx-z p 0.16sep

imating the canonical 6-3 bandpass set). increases for supernovae red-Krms

shifted from the center of a bandpass. When the redshift becomes large enough,it is more economical to consider the supernova blueshifted from the nextbandpass in the set. The maximumK-correction required (dotted line) is ap-proximately midway between these two and increases when the separationbetween bandpasses is increased (as demonstrated by the gray curve for

). Values differ from Fig. 6 because the bandpasses are centered atz p 0.2sep

different wavelengths.

the error in theK-correction reduces the quality of colors wemeasure.

An alternative, where possible, is to calculate color basedon nonadjacent bandpasses (e.g., first and third bandpasses ina set), allowing a denser bandpass set forK-corrections thanis used for color calculations. This only fails for the highestredshift SNe being considered, for which supernova opticalwavelengths are observed by only two observer filters.

2.3. Trade-off

We have shown that wider bandpasses are better than narrowbandpasses for measuringK-corrections, with a reduction froma K-correction of about 0.3 mag, with a scatter of 0.2 mag fora 200 Awide bandpass, to aK-correction of about 0.12 mag,with a scatter of about 0.04 mag for a 1000 A˚ wide bandpass(Figs. 4 and 5, for which ). We have also shownz p 0.05SN

that the width of a bandpass does not have a large effect onmeasuring color excess (Fig. 7). Instead, the dominant char-acteristic for the magnitude of the color excess is the separationof the bandpasses, with wide separations giving improved colorexcess measurements (Fig. 8). However, the greater the sepa-ration of the filters, the larger the maximumK-correction re-quired (Fig. 9). In general, when more than two filters areavailable, color excess would be measured using all filters thatcover supernova-frame optical wavelengths, not just adjacentfilters, allowing for both goodK-corrections and good colorexcess measurements. To investigate whether this gives anystrong constraints when only two bandpasses are available (e.g.,for the highest redshift supernovae in a sample), we need tocalculate the effect of separation on theK-correction and cal-culate the trade-off with color.

K-correction increases as a supernova is redshifted out ofthe bandpass. It reaches a maximum about halfway betweentwo bandpasses, beyond which it is better to consider the su-pernova as blueshifted out of the higher bandpass than red-shifted from the lower bandpass. Let the separation of the band-passes be , and the redshift of the supernova be . For az zsep SN

supernova redshifted from the center of a bandpass by ,rzSN

there is an equivalent blueshift from the center of the nextbzSN

bandpass in the set. These are related by

b r1 � z p (1 � z )/(1 � z ). (2)SN SN sep

In Figure 10 we show theK-correction required as a functionof bandpass separation and the redshift of the supernova. Weonly consider supernovae with redshifts between the first andsecond bandpasses. Beyond this the pattern repeats itself, with

measured from the second bandpass and measured fromr bz zSN SN

the third bandpass.The scatter inK-correction for the first and second filters added

in quadrature (Fig. 11) gives the magnitude of the expectedscatter in color due to intrinsic differences in Type Ia supernovae.As our final diagnostic, we calculate the relative error this rep-resents in color excess. This is plotted in Figure 12. This showsthat the error in theK-correction swamps the measurement ofcolor excess for filter separations below about . Asz p 0.13sep

separations increase beyond , the increased error inz p 0.13sep

K-correction balances the increased color discrimination, so there




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Fig. 10.—K-correction as a function of supernova redshift and bandpassseparation. Supernova redshift is measured relative to the closest bandpass inthe set. This graph shows only positive redshifts measured from the lowerbandpass, but at each point there is a corresponding blueshift from the upperbandpass. We are interested in color, so theK-corrections for two adjacentbandpasses are needed (above and below the redshift of the supernova), andhere the two values have been added in quadrature. TheK-correction initiallyincreases as the supernovae are redshifted from the lower bandpass, thendecreases as the supernovae approach the upper bandpass.

Fig. 12.—Relative error in color excess as a function of bandpass separationand supernova redshift. The supernova redshifts are calculated from the nearestbandpass, and this pattern repeats itself up through the bandpass set. Themaximum relative error peaks at low bandpass separations, but is relativelyflat overall, showing that the improvement in color measurements throughmore widely spaced bandpasses is balanced by the degradation ofK-correc-tions. This can be counteracted by calculating colors using nonadjacentbandpasses.

Fig. 11.—Scatter in the color measurement due to the scatter in theK-correction as a function of supernova redshift (relative to the closest bandpass)and bandpass separation. We have added the scatter in the two relevant band-passes (those above and below the redshift of the supernova) in quadrature.The relative values in this diagram are the key to selecting the best bandpasscombination. The absolute values should be interpreted as upper limits, becausewe expect significant correlation between bandpasses, which would reduce thescatter in the color.

is no further advantage in increasing the bandpass separation. Inpractice, for filter sets with smaller separations, the color excesswould be calculated using the first and third (or higher) band-passes in the set. This is only a disadvantage, therefore, for the

highest redshift supernovae in the sample, for which higher band-passes are not available.

When multiple bandpass measurements are available foreach supernova, the best lever arm for extinction is achievedby having many bandpasses over a large wavelength range.The wavelength range is constrained by the redshifts of theSNe we would like to measure and the wavelengths over whichSNe Ia have significant emission, as well as any technologicallimitations. BothK-corrections and extinction corrections ar-gue for many bandpasses in a set. Finite available exposuretimes argue for few bandpasses. Ideally, the full bandpass setfor a Type Ia supernova experiment will have as few band-passes as possible while still keeping the propagatedK-cor-rection error [including ] less than the acceptableE(B � V )maximum dispersion.

Our overall conclusion is that the bandpass pattern shouldbe determined primarily by the need for bandpasses at least1000 Awide, with the widest wavelength range achievable. In§ 3 we design a set of canonical bandpass sets based on theshape and separation considerations we have discussed here,and examine the constraints on bandpass separation imposedby detector sensitivities and mirror reflectivity.

2.4. Deviations from Self-Similarity

Deviations from our assumption of self-similarity of the band-passes in a set are bound to occur in practice. When calculatingthe effect of deviations from the desired template, there are twoaspects to consider. First, if we are unable to accurately measurethe deviation in the bandpass, we have a calibration error. We




2006 PASP,118:205–217

test various types of miscalibration in § 4 by taking an idealmodel spectrum and calculating the error inK-correction themiscalibration produces. The second aspect occurs when weare able to accurately measure the deviated bandpass shape.Error is then introduced because we do not know the exactshape of the underlying SN spectrum, which prevents us fromperfectly correcting for the wavelength-dependent variation inthe bandpass (this was, of course, the reason we chose self-similar bandpasses).

The error introduced by this second aspect is sensitivelydependent on the shape of the deviation: a variable that is verydifficult to quantify, due to the multitude of arbitrary shapechanges that are possible. Nevertheless, some general principlesapply. If a bandpass is warped so that it allows greater fluxthroughput at a wavelength at which Type Ia supernovae areparticularly variable, then the scatter in theK-correction willincrease. Where that wavelength lies is redshift dependent, soa bandpass that causes a large scatter at will not nec-z p 0essarily also cause a large scatter at . Better knowledgez p 0.1of Type Ia diversity would allow special consideration of band-passes with warps in danger regions. If the bandpass is simplyshifted in wavelength from its desired position but does notchange shape, then it remains as easy to calibrate as if it werein the correct position. However, the maximum gap betweenbandpasses becomes larger, and there will be a redshift region(approximately in the middle of this gap) that has a higherscatter than the rest. If necessary, it would be possible to neglectpoints in this region.

In general, deviations from self-similarity of the bandpassesin a set need to be considered on a case-by-case basis afterfilters have been produced.

3. CANONICAL BANDPASS SETS

In reality we do not have complete freedom in choosingbandpass shapes, and many of the bandpass shapes tested inthe previous section are impractical, given the detector sensi-tivities and wavelength coverage we require. Our primary re-strictions are these:

1. A typical high-resistivity, fully depleted CCD is sensitiveto wavelengths between about 3600 and 10000 A˚ . (Althoughsensitive to a little over 10000 A˚ , the detectors may cause focusproblems beyond this wavelength, because the longer wave-length photons penetrate more deeply into their surface.)

2. A typical HgCdTe near-infrared detector has an adjustableband gap from 7000 to 25000 A˚ (Norton 2002). The examplewe show in Figure 1 is the detector being considered for thepossible next-generation dark energy space telescope, theSupernova /Acceleration Probe, which is a HgCdTe device witha 1.7mm cutoff.

Given these restrictions, we can make some canonical band-pass shapes and spacings that best use the detector sensitivities.We design bandpass sets that range from the optical through

the near-infrared to cover the brightest region of a SN Ia froma redshift of zero to a redshift of about . This is thez p 1.8most interesting range of redshifts for cosmological parameterestimation, because it covers the transition from decelerationto acceleration in theLCDM (cold dark matter) concordancemodel. We have designed these bandpass sets for a space-basedmission and so are not constrained by the atmospheric emissionand absorption that limit ground-optimized bandpass sets.

The results from § 2 suggest that we should use bandpassesat least 1000 A˚ wide, ideally with some slope on the edges,and have many filters covering the total detector range in orderto keep the requiredK-corrections low. When the separationof the bandpasses is kept to less than , the scatter inz ∼ 0.22the K-correction remains below 0.02 mag.5 Previous studies(Kim 2003a, 2003b) performed for theSNAP design haveshown that a combination of six optical filters and three infraredfilters can provideK-correction errors of less than 0.02 magand is technically feasible.

Let thenth bandpass range from to . If we call themin maxl ln n

lowest wavelength bandpass , then the rest of the band-n p 1passes are related to the first by

n�1l p (1 � z ) l , (3)n sep 1

where is the redshift spacing between the bandpasses. Ifzsep

the width of the first bandpass is , thenmax minDl l p l �n n

. To fit in the range of the typical detectors we are(1 � z )Dlsep

considering, the optical bandpasses need to fit in the range3600–10000 A˚ , while the NIR bandpasses need to fit in therange 7100–17000 A˚ . Adding the requirement that the band-passes must overlap meansminl z ! Dl.1 sep

These restrictions leave a very limited range of bandpasssizes and spacings that will fit within the detectors. We havefree rein, however, to choose the number of bandpasses to putin each detector. We create a series of bandpass sets with variouspatterns; i.e., various numbers of bandpasses in the optical andNIR. For each pattern, we can choose to optimize the widthand spacing of the bandpasses a number of ways. One way isto maximize the range covered, so we measure the highestredshift supernovae allowed by the detectors. Another way isto maximize the usage of the bandpass, requiring the maximumoverlap of the bandpasses so that the flux is greatest and thesurvey can be done most quickly. In practice, a balance betweenthese two is likely to be most useful. In Table 1 we providethe details of bandpass sets optimized first for maximum rangeand second for maximum usage, with the additional provisothat they must be able to observe supernovae out to at least aredshift of 1.8 (the approximate range of sensitivity expectedfor the next generation of space telescopes) in two bandpasses.

We define “coverage” to be the wavelength range coveredby a bandpass set compared to the range available in the de-

5 See Fig. 11, where the scatter inK-correction for asingle bandpass isapproximately of the scatter values shown.�2




2006 PASP,118:205–217

TABLE 1Bandpass Separations and Widths for Bandpass Sets with Different

Numbers of Optical and NIR Bandpasses

Range Maximized Usage Maximized, z 1 1.8

Pattern(1)

zsep

(2)

Width(A)(3)

Max z(4)

zsep

(5)

Width(A)(6)

Max z(7)

2-2 . . . . . . . 0.469 1741 1.16 … … …3-2 . . . . . . . 0.360 1350 1.52 … … …3-3 . . . . . . . 0.292 1105 1.79 0.294 1093 1.804-2 . . . . . . . 0.288 1068 1.75 … … …4-3 . . . . . . . 0.241 1044 1.94 0.229 1341 1.804-4 . . . . . . . 0.203 1045 2.04 0.187 1513 1.805-3 . . . . . . . 0.203 1045 2.04 0.187 1433 1.805-4 . . . . . . . 0.176 1045 2.11 0.158 1640 1.805-5 . . . . . . . 0.155 1046 2.16 0.146 1406 1.966-3 . . . . . . . 0.166 1023 1.94 0.158 1192 1.806-4 . . . . . . . 0.155 1046 2.16 0.137 1654 1.806-5 . . . . . . . 0.138 1046 2.21 0.121 1814 1.806-6 . . . . . . . 0.125 1047 2.25 0.120 1294 2.107-4 . . . . . . . 0.137 1023 2.17 0.121 1433 1.807-5 . . . . . . . 0.125 1047 2.25 0.108 1791 1.807-6 . . . . . . . 0.114 1047 2.28 0.102 1706 1.917-7 . . . . . . . 0.105 1047 2.30 0.102 1215 2.208-5 . . . . . . . 0.114 1047 2.28 0.098 1593 1.808-6 . . . . . . . 0.105 1047 2.30 0.090 1884 1.808-7 . . . . . . . 0.097 1047 2.33 0.089 1579 2.02

Note.—The two numbers in the “pattern” column represent the number ofoptical and NIR bandpasses, respectively. The “Maxz” column gives themaximum SN redshift we can measure with this bandpass set. (We requirethat supernovae at this maximum redshift be observable in the highesttwofilters in the set, for comparison with low-redshift supernova colors.) The twodifferent major columns represent two different possible choices for each pat-tern. The first maximizes the range, allowing for comparison with the highestredshift supernovae allowed by the detectors. The second maximizes the usage,essentially maximizing the width of each filter (and thus how much theyoverlap), with the proviso that we must be able to measure magnitudes forsupernovae out to redshift of 1.8 in two bandpasses that are comparable tothe spectral regions sampled for zero-redshift supernovae. All these bandpasssets start with the lower end of the first filter at 3600 A˚ .

Fig. 13.—Detector coverage for various bandpass sets, where the sets havebeen chosen so that the maximum wavelength range is covered (Table 1, cols.[2]–[4]). The bandpasses sets are labeled by the number of bandpasses in theoptical and NIR passbands. Coverage is defined as the wavelength range ofthe bandpass sets, divided by the total wavelength range available in thedetectors. There is overlap between the optical and NIR detectors, so the totalcoverage does not equal the sum of the optical and NIR coverage.

Fig. 14.—“Normalized usage” (as defined in text) for a variety of patternswhen usage has been maximized under the proviso that the maximum redshiftmeasurable be 1.8 or more (Table 1, cols. [5]–[7]). This requirement cannotbe satisfied for the 2-2, 3-2, and 4-2 patterns.

tectors. Figure 13 shows the coverage for a variety of bandpasssets, where for each pattern we have chosen the set that max-imizes the wavelength range covered (Table 1, cols. [2]–[4]).Most patterns are able to be designed to completely cover thedetector wavelength region. An exception is the 6-3 pattern,which is too highly constrained by fitting six bandpasses in theoptical region to be able to cover the IR region with just threebandpasses. Similar problems occur in other bandpass sets thathave unbalanced numbers of optical and NIR bandpasses.

We define “usage” to be the sum of the bandpass widths ina set, divided by the total wavelength range of the detectors.To take into account the greater survey time required for alarger bandpass set, we also divide by the number of filters inthe set to get a “normalized usage.” Figure 14 shows the nor-malized usage for various patterns when the bandpass sets havebeen chosen to maximize the usage while still being able toobserve supernovae out to (Table 1, cols. [5]–[7]). Thez p 1.8patterns 2-2, 3-2, and 4-2 cannot satisfy these criteria. Bandpass

sets with slightly fewer NIR than optical bandpasses canachieve the best S/N in the shortest time.

The detectors do not have good performance near the cutoffpoints, so the edges of the bandpasses need to be tuned tomatch the edge of the detector performance if we want to utilizeas much of the detector sensitivity as possible. The numbersgiven here are a rough outline for recommended widths andspacings; the specific shape of the bandpasses is flexible, as




2006 PASP,118:205–217

long as sloping edges do not reduce the flux through the band-pass below the equivalent of a 1000 A˚ wide top hat.

4. CALIBRATION CONSTRAINTS

Errors in filter calibration affect every measurement madewith that filter over the lifetime of the experiment. In particular,because a different filter is used to measure the photometry indifferent redshift bins, a systematic error in filter calibrationmay introduce a spurious redshift-dependent trend. For super-nova observations constraining the luminosity distance, this isamong the worst type of systematic error.

The two main filter types are glass and interference filters.Each has advantages and disadvantages. Interference filtershave a high throughput and can be designed to match arbitrarybandpass shapes, even in the infrared. However, the bandpassesof interference filters are dependent on the angle of incidenceof the incoming light, and this varies with position on the focalplane. This can be calibrated out to some degree by applying,for example, a position-dependentK-correction (Kim 2003a,2003b), or alternatively using the bandpass for the averageincidence angle and allowing the position dependence to con-tribute to the scatter about the mean. This effect contributesan error on the order of 0.01 mag (Kim 2003b). Experienceshows that interference filters are subject to more variabilitywith time than glass filters. In particular, outgassing can changethe layer thickness by a substantial and irreversible amount.They are also more susceptible than glass to radiation damageand are difficult to manufacture to an exact and repeatablespecification.

Glass filters are more stable than interference filters. Theycan be made very uniform over a large area, and very stablewith time (Schott Glaswerke, 2004 product catalog).6 Theirbandpass has negligible angle dependence. However, they can-not be tuned to arbitrary bandpass shapes, and they have lowerthroughput than interference filters (∼60% for glass filters,compared to∼80% for interference filters). Most important,there are no adequate glass filters available in the near-infrared,where a high-redshift SN Ia experiment needs to probe.

It may be that the most stable and accurate filter set for SNIa cosmology could be built from a combination of glass andinterference filters, or possibly hybrid glass/interference filters,where glass provides the low-wavelength cut-on and a smallnumber of interference filter layers provide the cutoff. The novelidea of using high-pass filters with tuned cut-on points and thensubtracting them postobservation to create a narrower filter mayalso be worth considering. However, this procedure increases themeasurement error. Unless glass filters are developed in the in-frared, the NIR filters need to be interference filters.

Experience has shown that in the manufacture of interferencefilters, we can expect a nontrivial difference of∼5% at any

6 See the Schott Glass Filter Catalog, www.schott.com/optics_devices/filter/english.

particular wavelength with respect to transmittance functionspecifications (J. Bland-Hawthorn 2004, private communica-tion). If we can perform calibration perfectly after manufacture,this is not an issue. The difficulties come in several forms:(1) positional dependence due to different incident angles whenthe filters are in a converging or diverging beam, (2) miscal-ibration due to nonuniformity, and (3) dependence of trans-mittance on temperature, humidity, and general decay with age.Transmittance of filters depends on temperature at the level of∼0.2 A K�1 (J. Bland-Hawthorn 2004, private communication),and if filters absorb or lose moisture after the calibration pro-cedure, this will permanently and significantly alter their band-passes. Calibration needs to be done with the filters and de-tectors installed in an operational telescope, and needs to bedone for many positions and many angles.

Miscalibration of filters is less important if the miscalibrationis sufficiently random that the mean transmittance can be ac-curately determined, with the errors caused by the various ef-fects Gaussianly distributed about the measured mean. Theerror would then contribute to the random uncertainty and beaccessible to improvement with better statistics (more SNe).Problems arise, however, if the calibration technique tends toerr in one direction for all filters, or if environmental/age-relatedeffects push all filters in a common miscalibration direction.

It is important, therefore, to consider the accuracy of cali-bration in the choice of a filter set for future SN Ia studies.There are many different filter types and manufacturing tech-niques, and it is beyond the scope of this paper to examineeach in detail. We provide instead the calibration accuracyneeded (noise amplitude) in order to reach a particular pho-tometric accuracy requirement.

4.1. Random Miscalibration

The process of creating a filter involves initially designingan ideal filter that the manufacturers will try to generate. Therewill always be some level of manufacturing error in this pro-cess, so calibration is necessary after manufacture and againonce deployed (e.g., calibration must be done in orbit for aspace mission). We assume that this calibration cannot be doneperfectly and that it may also change with time and wear. Wetherefore assess to what level of accuracy the calibration isneeded in order to reach photometric accuracy requirements.Is it worth using interference filters to create specialized band-passes if glass is more stable and can be calibrated moreaccurately?

As previously discussed, filters are two-dimensional objectsthat cover the focal plane, so their bandpasses may vary withposition. Characterization of the filter must therefore eitherspecify the bandpass as a function of position or assume thefilter is uniform across its surface to a certain precision. In thefollowing, we define a noise amplitude for which we calculatethe corresponding uncertainty in photometry. To permit theassumption of uniformity, the filters must be uniform across




2006 PASP,118:205–217

Fig. 15.—Photometric error as a function of supernova epoch for noiseamplitude levels of 2% applied to three different filters. The first is a BessellB filter. The other two are top-hat filters centered on 4500 A˚ , with widths of1000 and 2000 A˚ . The error induced is independent of the filter shape and isinstead dependent only on the amplitude of noise introduced. Error increasesslightly as the integral flux through the filter increases.

Fig. 16.—Comparison of average error in photometry introduced by un-characterized manufacturing error in a BessellB filter, and two top-hat filtersof different widths (1000 and 2000 A˚ ). For each noise level, we test up to1000 realizations of a noisy filter and calculate the rms error in photometryas a function of epoch of observation. We then calculate the rms over allepochs to determine the final error magnitude vs. noise amplitude plotted here.The fits are quadratic.

their surface to the precision given by the noise amplitude forthe required photometric accuracy.

In order to simulate an unknown calibration error in a filter,we take a filter template and multiply it by noise with a flatpower spectrum. We call the filter function measured duringthe calibration process the “template” filter; the actual filter wehave is called the “noisy” filter. We then take the library ofidealized SN Ia spectra from Nugent et al. (2002) and calculatethe difference in the measured photometry between the tem-plate and noisy filters as a function of epoch (days before andafter maximum light). The resulting photometric difference rep-resents the systematic error introduced by the uncharacterizednoise in the filter.

Each realization of noise produced has a different profileand a different scatter. When calculating tolerances, we generatemany realizations of a noisy filter and take the rms of the error.

White noise is probably not appropriate for allk, since areal filter will not have noise at the smallest scales, but thesesimulations also have a natural lower limit through the finitenumber of array elements we use to describe the filter. Wevaried the density of the array, and it did not affect the results.

We performed the above analysis on three different types offilters: the BessellB filter, a top-hat filter 1000 A˚ wide, and atop-hat filter 2000 A˚ wide. Both of the top-hat filters are cen-tered on 4500 A˚ .

The supernova spectra vary as the supernovae evolve; there-fore, the error in the filters is epoch dependent. In Figure 15we plot photometric error versus epoch for a noise amplitudelevel of 2% in three different filters. We repeat this analysisfor noise amplitude levels from 0.5% to 5%, then average overall epochs to come up with a single error specification as a

function of noise level. This is plotted in Figure 16. The shapeof the filter profile is not significant in this analysis, only thequantity of noise we are adding. The slight differences in thecurves arises because the wider filters sample a wider regionof the supernova spectrum.

It is usually argued that systematic errors should be addeddirectly, not in quadrature. So if each filter has a systematicuncertainty of 0.002 mag, then the color has an uncertainty ofup to 0.004 mag (correlation between bandpasses reduces thisnumber). Extinction is given by , so withA A p 3.1E(B � V )V V

the above parameters the systematic uncertainty in extinctionis up to 0.012 mag. This already represents much of the errorbudget for photometry with proposed space missions (e.g.,SNAP).

To achieve an irreducible systematic photometric accuracyof better than 0.002 mag in a single filter, we need to createfilters that are uniform across their surface (or with a knownposition-transmittance function) and are characterized to betterthan∼1% (noise amplitude less than about 1% of peak trans-mittance). This criterion gets slightly stricter for filters withlarger integrated flux.

Note that although the average error converges on the curvesshown in Figure 16, any one realization of a noisy filter canhave an error that differs significantly from this, and may beup to twice as large. This may more accurately represent thesituation we are faced with, since any filter we build will bejust one realization of a noisy filter.

We tested the photometric error induced when the supernovaspectra are slightly red- or blueshifted with respect to the filter.The profile of photometric error versus epoch appears slightlydifferent, but the average photometric error is unchanged.




2006 PASP,118:205–217

Fig. 17.—Effect of periodic error in filter calibration on photometric errorin SN Ia measurements. We have averaged over phase shifts from 0 to 2p.This indicates that small-scale (short period) variations have less impact thanlarge-scale (large period) variations, and that any filter that has a periodicityof around A(in the rest frame) should be avoided.330� 50

4.2. Periodic Miscalibration

With interference filters, there is a possibility that any mis-calibration may be periodic in wavelength. We therefore ex-amine the effect of a periodic miscalibration to see whetherthere are any frequencies that should be avoided for supernovaobservations.

We take a template filter and multiply it by a single Fouriercomponent (a sine wave) that represents our unknown periodiccalibration error. We have thus created another type of “noisy”filter. We create a series of noisy filters by varying the periodof the noise that we add to the template, while keeping theamplitude of the noise constant in all cases. This allows us tocalculate the error as a function of SN epoch that we wouldintroduce if we used the noisy filter thinking we had the template.Taking the rms of this error gives us a single number quantifyingthe error introduced as a function of the periodicity of the noise.For each period, we also recalculate the rms error for phase shiftsfrom 0 to 2p and take the average of the result. The effect ofphase is significant, due to spectral features moving in and outof peaks and/or troughs of the miscalibrated filter.

Figure 17 shows the effect of period on the rms error betweena template filter and a noisy filter. The noise amplitude has anrms of 5% of the transmittance, and the periods range from 50to 500 A (in the SN rest frame). Over the wavelength rangeshown here, the error increases as the period of the miscalibrationgets higher. This arises because Type Ia supernova spectra arerelatively smooth on small scales (∼50 A) but have significantfeatures on larger scales (several hundred angstroms). When theperiodicity of the miscalibration in the filter matches the dom-inant frequency in the power spectrum of the supernova, theerror due to miscalibration is largest. This occurs around 330 A˚

(multiply this by for the redshifted bandpasses). From this1 � z(Fig. 17) we conclude that any periodicity of around rest-frame

A should be avoided in the manufacture of filters for330� 50SN Ia studies.

5. CONCLUSIONS

K-correction scatter due to intrinsic diversity in the SN Iapopulation can be minimized by using bandpasses wider than1000 A. Adding sloping edges also helps, particularly in thefew cases where a spectral line sits on the edge of a bandpass,but should not be added at the expense of integral flux. Ex-tinction corrections rely on the accurate measurement of color,which is best done using widely spaced filters, irrespective oftheir width or shape. However, there is a compromise, becausemore widely spaced filters mean largerK-corrections, whichin turn increase the error in color calculations. The trade-offsuggests that spacings wider than should be used,z p 0.13sep

but this is not necessarily a strict lower limit, because color

can be calculated using nonadjacent filters. It is a genuine lowerlimit if colors and extinction corrections are required for thehighest redshift supernovae in the detector passband. This paperaddresses the relation between passband set and intrinsic su-pernova dispersion inK-correction. Future work will examineintrinsic dispersion in absolute magnitude.

Bandpass spacing and width are the determining factors foroptimally fitting an evenly spaced bandpass set in the detectorsensitivities. We investigated how well different bandpass pat-terns fill the sensitive regions of the detectors and providedspecifications for several optimized bandpass sets in Table 1.

The photometric accuracy achievable through different kindsof filters is an important aspect of systematic error calculations.Our analysis here shows that characterizing the filters to betterthan∼1%, as defined in § 4.1, keeps the photometric error ina single filter below 0.002 mag. This is independent of filtershape but increases slightly for filters with larger integratedflux. Periodic error in filter calibration is to be expected forinterference filters. Filters that may cause a periodic error ataround A(in the rest frame) should be avoided.330� 50

T. M. D. and B. P. S. acknowledge Australian ResearchCouncil grants LX0454445, DP0209028, and DP0559024.A. G. K. is supported by the Director, Office of Science, ofthe US Department of Energy, under contract number DE-AC02-05CH11231. T. M. D. and A. G. K. would like to thankthe Aspen Center for Physics for their hospitality during earlywork on this project.




2006 PASP,118:205–217

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Ideal Bandpasses for Type Ia Supernova Cosmology