A Precise Distance Indicator: Type Ia Supernova Multicolor Light#Curve Shapes

We present an empirical method that uses multicolor light-curve shapes (MLCSs) to estimate the luminosity, distance, and total line-of-sight extinction of Type Ia supernovae (SNe Ia). The empirical correlation between the MLCSs and the luminosity is derived from a ““ training set ÏÏ of nine SN Ia light curves with independent distance and reddening estimates. We Ðnd that intrinsically dim SN IaÏs are redder and have faster light curves than the bright ones, which are slow and blue. By 35 days after maximum, the intrinsic color variations become negligible. A formal treatment of extinction employing BayesÏs theorem is used to estimate the best value and its uncertainty. Applying the MLCS method to both light curves and to color curves provides enough information to determine which supernovae are dim because they are distant, which are intrinsically dim, and which are dim because of extinction by dust. The precision of the MLCS distances is examined by constructing a Hubble diagram with an independent set of 20 SN IaÏs. The dispersion of 0.12 mag indicates a typical distance accuracy of 5% for a single object, and the intercept yields a Hubble constant on the Sandage et al. Cepheid distance scale of (statistical) km s ~1 Mpc ~1 ( ^ 6 total error). The slope of 0.2010 ^ 0.0035 mag over the H 0 \ 64 ^ 3 distance interval 32.2 \ k \ 38.3 yields the most precise conÐrmation of the linearity of the Hubble law.

Since the Curtis-Shapley debate of 1920 the determination of supernova (SN) luminosities has (Curtis 1921 ;Shapley 1921), been central to the discussion of extragalactic distances.
argued against the "" Island Universe ÏÏ hypothesis Shapley (1921) because it required certain novae (such as S Andromedae of 1885) to reach the astonishing luminosity of M \ [16. He considered this to be "" out of the question.ÏÏ countered, concluding that "" the dispersion of the novae in spirals Curtis (1921) and in our galaxy may reach ten absolute magnitudes . . . a division into two classes is not impossible.ÏÏ This distinction between novae and supernovae, required by the extragalactic nature of the nebulae, was later made explicit by & Baade Zwicky They showed that in addition to their tremendous di †erence in absolute luminosity, the photometric and (1934). spectroscopic behavior of supernovae is distinct from novae.
showed that supernovae were more uniform than Baade (1938) novae, with a dispersion at peak of 1.1 mag, making them suitable as extragalactic distance indicators.
The precision of supernova distance estimates has increased as the SN Ia class has been better understood and more narrowly deÐned. The low dispersion in BaadeÏs sample beneÐted from the fortuitous absence of Type II supernovae, which are signiÐcantly less luminous in the mean. Beginning with SN 1940B, Type II supernovae were classiÐed by the presence of hydrogen in their spectra The growing list of spectroscopically deÐned Type I supernovae had dispersions (Minkowski 1941). at peak of 0.8È0.6 mag et al. & Searle However, this sample (Minkowski 1964 ;Kowal 1968 ;Kirshner 1973 ;Oke 1974). included a number of "" peculiar ÏÏ SN I noted for their lack of silicon, which are now recognized to arise from massive stars that lose their envelope before core collapse (see & Harkness After removing these silicon-deÐcient objects, Wheeler 1990). now classiÐed as IbÏs and IcÏs & Branch & Kirshner & Levreault & (Doggett 1985 ;Uomoto 1985 ;Wheeler 1985 ; Wheeler Harkness the remaining Type Ia supernovae (SN IaÏs) form a more homogeneous set that serves as an even more 1990), precise indicator of astronomical distances. devised a set of standard templates to describe the photometric Leibundgut (1989) behavior of SN IaÏs and to estimate the peak apparent magnitude. Hubble diagrams constructed using the peak of photographic SN Ia light curves had observed dispersions ranging from mag depending on which objects and color p M \ 0. 65È0 SN IaÏs Rood 1994 ;Sandage 1993 ;1992, 1994. through Cepheid variables observed with the Hubble Space T elescope has been undertaken by Sandage et al. (1992Sandage et al. ( , 1994 Assuming SN IaÏs to be homogeneous yields a Hubble constant in the range 50È58 km s~1 Mpc~1 (Sandage et al. 1996). 1992 & Tammann with the most recent measurement giving 57^4 1994, 19961994, 1995a, 1995b, 1996Branch 1992) km s~1 Mpc~1. We show in and that the precision of this measurement is improved and the value of is altered by°°6 7 H 0 including information contained in the light and color curve shapes.
The hypothesis that SN IaÏs are standard candles drew support not only from empirical studies, but also from the earliest theoretical models, which suggested that they arose from ignition of a carbon-oxygen white dwarf at the Chandrasekhar mass & Fowler & McKee In these models, a supersonic shock wave travels through the (Hoyle 1960 ;Arnett 1969 ;Colgate 1969). degenerate star, burning material into 56Ni at a temperature of 5 ] 109 degrees Mu ller, & Ho Ñich (Khokhlov, 1993 ; Mazurek & Wheeler Because the detonation is supersonic, the preshock region cannot expand to decrease the pressure or the 1980). burning temperature. Furthermore, the Fermi pressure of the degenerate material in the postshock region remains insensitive to temperature for longer than the burning timescale. The result is a total incineration and the production of a pure mass of nickel. Such a standard explosion of a uniform mass would lead to a homogeneous light curve and uniform luminosity. Yet, these complete burning models of IaÏs do not reproduce the intermediate-mass elements that are seen in the spectra of SN IaÏs & Harkness A successful model Thielemann, & Yokoi that matched the observational (Wheeler 1990). (Nomoto, 1984) constraints was so persuasive that Branch, & Wheeler and suggested calibration of the Hubble Arnett, (1985) Branch (1992) constant based only on theoretical models of uniform nickel production. However, a variety of models (Livne 1990 ;Khokhlov et al. & Weaver Khokhlov, & Wheeler match the observed features of the spectra and 1993 ; Woosley 1994 ; Ho Ñich, 1995) produce a range of nickel masses and a range of predicted luminosities. These models employ subsonic deÑagration fronts, pulsations, or o †-center explosions to allow the surface layers to preexpand and burn at low temperature. The success of these models in reproducing the observed spectra opens a large range of theoretical possibilities. Unlike the Ðrst monoenergetic models, these models suggest that a wide range of luminosities might result from the ignition of a white dwarf.
Recently, precise observations of SN IaÏs made with CCD detectors show evidence for inhomogeneity in both luminosity and light-curve shape. One of the Ðrst SN IaÏs observed with a distinctly di †erent light curve was 1986G et al. (Phillips 1987 ; which displayed a spectacularly rapid decline in its B and V light curve and unique spectral character-Cristiani 1992), istics including stronger than usual Si features. Although SN 1986G was heavily reddened by dust, reddening cannot alter signiÐcantly the shape of the light curve. SN 1991bg in NGC 4374 is the most extreme SN Ia in an increasingly apparent photometric and spectral sequence. et al.
(also et al. described a number of photometric Leibundgut (1993) Filippenko 1992) abnormalities of 1991bg with respect to his templates. These include the fastest postmaximum decline (2.05 and 1.42 mag decrease drop in B and V in the 15 days after maximum compared to 1.22 and 0.64 mag for the templates in B and V ), a narrow luminosity peak, and an intrinsic red color near maximum. A simple and convincing argument that SN IaÏs have a large spread in luminosity is that SN 1957B, which occurred in the same galaxy, was 2.5 mag brighter in B than SN 1991bg. In addition, SN 1991bg was at least 2 mag fainter than other SN IaÏs in the Virgo Cluster, of which NGC 4374 is a member. This extreme SN Ia seems to have a twin in the subluminous SN 1992K et al. which strongly resembles 1991bg in (Hamuy 1994, photometric and spectral behavior.
At the opposite extreme of the SN Ia class, SN 1991T showed spectral and photometric peculiarities that were di †erent from those seen in the rapid decliners. et al.
found the light curves in B and V to rise and decline more slowly Phillips (1992) than the standard templates near maximum, and a month after peak, this shallower decline resulted in a light curve 0.2È0.3 mag brighter than the templates. Although SN 1991TÏs host galaxy, NGC 4527, appears 1994), 5660 km s~1, but with peak apparent luminosities di †ering by 0.69 mag in B. The large di †erence in apparent magnitude and the small di †erence in recession velocity imply that SN 1992bc was intrinsically brighter by 0.8^0.2 mag than 1992bo. For a di †erence in distance to account for this di †erence in luminosity, the peculiar velocities of the two supernovae would have to di †er by D2500 km s~1, an unlikely alternative. Interestingly, SN 1992bc declined more slowly than the average template, while SN 1992boÏs postmaximum fall was more rapid than the average template, a result in good accord with later analysis by Phillips (1993).
Even before these recent examples of inhomogeneity, less precise photographic measurements by Ciatti, & Rosino Barbon, suggested that there might exist two photometric classes : those which "" fast ÏÏ decline rates after maximum, which were (1973) intrinsically brighter supernovae, and those with "" slow ÏÏ decline rates, which were fainter. With the poor quality of photographic and photoelectric photometry then available, such a real distinction was difficult to demonstrate convincingly. Pskovskii suggested a continuous photometric sequence of SN Ia light curves. He introduced a parameter, b : the slope of (1977,1984) the B-band postmaximum decline in magnitudes per 100 days. Using 54 photographic SN Ia light curves, Pskovskii found a weak correlation between b and the absolute B magnitude at maximum light that was opposite to that of et al. Barbon (1973). Early difficulties in measuring a relation between SN Ia light-curve shape and intrinsic luminosity resulted from noisy photographic data that were poorly sampled and calibrated. These difficulties were compounded by the problem of measuring a decaying light curve on a bright galaxy with a nonlinear photographic detector & Wheeler With the (Boisseau 1991). advantage of better data measured with linear detectors, demonstrated conclusive evidence for a luminosityÈ Phillips (1993) light curve decline relation among SN Ia. Using a set of well-sampled SN Ia light curves with precise optical photometry and accurate relative distances, Phillips found that the absolute luminosity in B, V , and I is correlated with the B-band decline in the 15 days following maximum light. The sense of the correlation is that dimmer SN IaÏs fall more rapidly after maximum than bright SN IaÏs. Application of this relation to his sample results in a signiÐcant reduction of the dispersion in B, V , and I luminosity from 0.8, 0.6, and 0.5 mag to 0.36, 0.28, and 0.38 mag, respectively. A more recent investigation by & Tammann Sandage has examined the luminosityÈlight curve decline relation among "" normal ÏÏ SN IaÏs ; in this case "" normal ÏÏ is ( 1995 ;Ho Ñich 1996). o †-center detonations, surface helium burning, pulsed delayed detonations, and sub-Chandrasekhar progenitors. These models give plausible causes for the observed inhomogeneity of SN IaÏs and for the origin of the empirical relations between light-curve shape and luminosity.  (Riess, 1995a, hereafter have developed a linear estimation algorithm to use the distance-independent light-curve shapes to RPK95a) improve the precision of distance measurements to SN IaÏs. The techniques have much in common, and both yield Hubble diagram dispersions of D0.2 mag for an overlapping set of objects. The light curve shape (LCS) method, described in has the advantage of providing quantitative error estimates for the distance. The present paper extends the LCS RPK95a, technique to use the shapes of B[V , V [R, and V [I color curves, which provide enough information to determine the relation between absolute luminosity and intrinsic color. Knowledge of an SN IaÏs intrinsic color allows us to measure the extinction from the observed reddening. For each well-observed SN Ia we measure the luminosity, the extinction, and the extinction-corrected distance. The multicolor light-curve shape (MLCS) method increases signiÐcantly the precision of distance estimates from SN Ia light curves, as we show in°6.
There are many potential applications for a bright distance indicator with \10% precision. Nearby (0.01 ¹ z ¹ 0.1), it should be possible to measure the Hubble constant to an accuracy that is limited only by the underlying calibration of Cepheids. It is important to compare the Hubble constant derived using the light-curve shape-luminosity relation with determinations that have assumed a homogeneous luminosity for SN IaÏs (Sandage et al. Using the velocity 1994Using the velocity , 1996. residuals from Hubble Ñow, we have measured the motion of the Local Group with respect to the rest frame of galaxies with supernovae Press, & Kirshner At even greater distances (0.3 ¹ z ¹ 0.6), MLCS distance measurements of all (Riess, 1995b). well-observed SN IaÏs could be used to determine the cosmological deceleration parameter, (Perlmutter et al. q o 1995(Perlmutter et al. q o , 1996et al. IAU Circ. 6160 (Branch, 1993 ;1994, 1996. useful in producing smaller dispersion in distance estimates, the difficulty in obtaining good spectra of very distant SN IaÏs makes it hard to identify subtle spectral variations. A sample cut that cannot be applied with equal e †ectiveness for nearby and for distant SN IaÏs could lead to a bias in cosmological parameters determined by them. We prefer to develop a method that can be applied to all SN IaÏs.
Given at least one light curve and one color curve, observed photoelectrically within 10 days after maximum, our MLCS method can distinguish between the e †ects of distance, intrinsic dimness, and dust for all SN IaÏs. In we develop the°°2È5 multicolor light-curve shape method for measuring extinction-corrected distances. In for an independent sample, we°6, compare the extinction-corrected MLCS distances with distances determined by more limited assumptions. In we°7, estimate the Hubble constant and discuss sample membership, selection bias, and the range of progenitors that may be responsible for the empirical range of SN Ia luminosities and colors.

LEARNING CURVES
SN Ia light curves are noisy unevenly sampled time series, which, when observed diligently, are available in four bands, B, V , R, and I. Although some SN IaÏs are found in ellipticals, most are found in spirals and irregulars, where they can be subject to signiÐcant extinction by dust. At present, the number of well-sampled SN Ia light curves available on a modern photometric system is limited (i.e., ¹50) et al. et al. et al. et al. (Hamuy 1995 ;Riess 1996a ;Ford 1993 ;Richmond 1995). We employ a general s2 model to establish the empirical relation between light and color curve shape and luminosity from a nearby subset of SN IaÏs with accurately known relative distances and extinctions. Then we use these results to estimate the extinction-corrected distance for a separate distant sample solely from the observed light and color curves. The observables are a visual band light curve and up to three color curves.
To Ðrst approximation, the observed light and color curves of SN IaÏs are homogeneous, resembling standard template light and color curves with the addition of noise (n) for each color and individual o †sets that result from the apparent visual distance modulus and a color excess due to reddening by dust : Here bold-face type denotes a vector whose entries are measured or determined as a function of time. In standard convention, M is an absolute magnitude, m is an apparent magnitude, and and are unreddened 0 color curves. Each available color curve yields an term which, combined with a standard extinction curve, reduces the E color uncertainty in measuring the total line-or-sight extinction, This description of light and color curves assumes intrinsic A V . homogeneity of SN IaÏs, and is equivalent to the "" template ÏÏ approach of Sandage et al. and Leibundgut (1989), (1994,1996), & Tammann Sandage (1993). Improvements in the quality of the available data set and the success of motivate an approach which, to Ðrst Phillips (1993) order, can account for the observed variations in the light and color curves and intrinsic luminosity. The most economical approach is to adopt a single parameter and correlate it with the variations of the observed curves. The natural parameter to choose is the amount by which the intrinsic luminosity di †ers from an SN Ia of standard brightness and curve shape since, in the end, it is this di †erence we hope to measure. We call this the "" luminosity correction,ÏÏ *, where and * 4 M V [ M V,standard , is the luminosity of the SN Ia chosen to depict the "" standard ÏÏ SN Ia event. This di †erence is measured by M V,standard convention at the date of B maximum. The functions of time which, to Ðrst order in *, correct the observed curves to the template curves are "" correction templates,ÏÏ or Using these deÐnitions, we expect two supernovae whose R V (t) R color (t).
absolute visual luminosities di †er by * magnitudes to have light and color curves that di †er by and The R V (t)* R color (t)*. assumption that luminosity correlates linearly with light and color curve shape need not provide a complete description, but given the limitations of our real data sets, it is a reasonable way to begin. The evidence that this is a useful way to proceed is provided in where we show that this method produces a signiÐcant decrease in the dispersion for an independent set of°6, SN IaÏs in the Hubble Ñow.
Our improved model for the light and color curves is thus where gives the deviation from the template, at each time. Similarly, where and are the unreddened template color curves. It is instructive to compare equations , which allow SN Ia inhomogeneity, to equations which impose SN Ia homogeneity. (1)È(4), In matrix format, we can write equations as one system of equations, where we have combined the terms into one simultaneous measurement of with the aid of the standard reddening 1979). assume the Galactic reddening ratios are valid for the dust in distant galaxies. Elsewhere, we examine the homogeneity of these ratios in distant galaxies, and we Ðnd that the most likely form of the reddening law for dust in distant galaxies is consistent with the Galactic reddening law Press, & Kirshner (Riess, 1996b). The s2 of the Ðt between data and model is where n is the vector of residuals in equations or equivalently in Here C is the correlation matrix whose (5)È(8) equation (9). elements are intended to reÑect the errors in observations and residual, unmodeled correlations in SN Ia light and color curves. The correlation matrix is discussed in detail in°4.

& Press
have derived two analytical minimizations of the s2 given in One gives the best Rybicki (1992) equation (10). estimate of the correction templates, or provided the SN Ia parameters, *], are known, and the other R gives the best estimate of the parameters, provided the correction templates are known. We employ both, using the nearby set with accurate relative distances as a "" training set ÏÏ to estimate and Once trained, we measure the and , parameters for an independent set of SN IaÏs from the shapes of their light and color curves.
Analytical minimization of s2 in with respect to or is independent of C and gives equation (10) where the angle brackets denote an average over the training set weighted by the uncertainty in the given estimates of *.
It is convenient to choose a particular object as the "" standard ÏÏ SN Ia whose light and color curves deÐne the standard template curves, and By deÐnition, the "" standard ÏÏ SN IaÏs have a peak luminosity variation of . utility of the method is insensitive to the choice of the template curves and luminosity, since all quantities are deÐned relative to it. LeibundgutÏs templates, made from SN 1989B, SN 1980N, and SN 1981B, approximate the light curves of the "" normal ÏÏ training set supernovae in B and V . For R and I, we have constructed our own templates from SN 1989B, SN 1980N, and SN 1981B. In solving for the correction templates, we only need consistent relative distances for our training set of SN IaÏs, since we correlate luminosity di †erences with light-curve shape variation. No choice of absolute distance scale is necessary because all luminosity corrections, *, are relative to the standard SN Ia. Surface brightness Ñuctuations (SBFs), planetary nebula luminosity function (PNLF), and Tully and FisherÏs luminosityÈline width relation (T-F) provide accurate and consistent bias-corrected relative distances to the host galaxies of the training set of SN  This approach has the advantage of placing our distance scale (1994,1996). directly onto that of the Cepheid variables. Published SBF distances have been undergoing some modiÐcations (J. Tonry, private communication). We comment on the e †ect of changes in SBF distances in°7.
The training set of supernovae we have used also has precise optical photometry, well-sampled light curves, and estimates of E(B[V ) (see The color excesses in Table 1 are estimated from comparisons with unreddened, photometrically M V (t), Table 2, Cepheid distance scale discussed in°7.
Adding various amounts of the correction templates to the standard templates generates an empirical family of light and color curves (see This family of curves demonstrates some interesting relations between light-curve behavior and Fig. 1). luminosity. The natural history of SN IaÏs is that intrinsically dim SN IaÏs rise and fall rapidly in V as compared to the leisurely rise and fall of intrinsically bright SN IaÏs. These results are similar to those of with the advantage that Phillips (1993) they show the SN Ia behavior from before maximum and more than 15 days after maximum.
An interesting new result is the quantitative relation between SN Ia luminosity and color. Supernova luminosities correlate with their intrinsic colors at early times (see also Before day 35, the dim SN IaÏs are red and the bright ones are Lira 1995).    (Branch 1992 ;Capaccioli 1990 ;Joe ever 1982 ;Tammann 1987). variations at maximum due to reddening for 0.10 mag of visual extinction are expected to be 0.03, 0.03, and 0.05 in B[V , V [R, and V [I. Although the sense of the color changes from absorption is the same as from intrinsic variation, the values are not.
den Bergh noted that the relation between intrinsic B[V color and luminosity predicted by the mean behavior Van (1995) of & KhokhlovÏs theoretical models agrees coincidentally with the standard reddening law. He takes advantage Ho Ñich (1996) of this agreement to deÐne a reddening-free luminosity that employs a single measurement of B[V color to account for both the luminosity variation intrinsic to the supernova and that which results from absorption by dust. Our empirical relation between luminosity and color shows that dust and intrinsic luminosity variation do not cause exactly the same change in color. This di †erence may cause the increase in dispersion around the Hubble line observed after van den BerghÏs prescription is applied et al. (Riess 1996b). More than 35 days past maximum light, all supernovae exhibit nearly uniform colors. Entries in the correction templates for day 35 give di †erences of 0.00, 0.01, and 0.01 mag in B[V , V [R, and V [I color for a 0.10 mag change in visual absolute luminosity. Detailed theoretical modeling of SN IaÏs shows a similar relation between supernova color and luminosity & Khokhlov (Ho Ñich 1996). An alternate view of the photometric di †erences between intrinsically bright and dim SN IaÏs is presented in The Figure 2. family of absolute B, V , R, and I light curves shows recognizable morphological variations. The B light curve family is similar in behavior to the V family ; dim SN IaÏs rise and fall more rapidly in B and V than bright SN IaÏs. In R, the brighter SN IaÏs have a "" shoulder ÏÏ D25 days after B maximum. For dimmer SN IaÏs, this shoulder is less pronounced and disappears completely for the most underluminous objects. In the I band, the brighter SN IaÏs have two maxima. The Ðrst occurs quite early, D5 days before the B-band maximum. The second, broad maximum is at D30 days after B maximum. As the luminosity of the SN IaÏs decreases, a number of changes in the I-band light curve are apparent : the Ðrst maximum is later and broader, while the second maximum is dimmer and occurs earlier. For the most underluminous SN IaÏs the two maxima merge into one maximum that is broad and occurs D5 days after B maximum.
The uncertainty in the empirical relations depicted in is discussed in These uncertainties, shown in Figure 1°4. Figure 3, quantify how useful any SN Ia data are in predicting the distance related parameters.
also shows how well SN Ia data Figure 3 are expected to Ðt the empirical model. This point is worth emphasizing with an example. Our family of V [R color curves predicts that SN IaÏs with di †ering intrinsic luminosity will have the same color after day 60, but shows that this Figure 3 prediction is highly uncertain. Using both our empirical model and our measure of its limitations, we will optimize our measurements of SN Ia distances.
The light and color curve reconstructions in provide a powerful means to measure the extinction-corrected Figure 1 distance at every phase of supernova observation. SN IaÏs appear dim because they are distant, obscured by dust, or intrinsically dim. We can distinguish between these possibilities by using the light and color curve shapes (which determine the intrinsic luminosity and color) and measuring the observed o †sets (which determine the extinction-corrected distance).

EXTINCTION-CORRECTED DISTANCES FROM MULTICOLOR LIGHT-CURVE SHAPES
provides a ready guide for measuring extinction-corrected distances. Given BV RI light-curve photometry, we seek Figure 1 the best set of curves for a Ðxed value of * that minimizes the s2 between model and data. The o †sets between the model and the data provide the best estimates of and Rather than searching the s2 parameter space for a solution, we take k V A V . advantage of the linearity of our model for an immediate solution.
Referring to the matrix form of the model in we use the following deÐnitions ; y is the column of apparent equation (9), magnitude measurements, s is the column of standard templates, L is the three-column matrix with correction templates and o †sets, and q is the three-element column of parameters. With these deÐnitions, we rewrite s2 in as equation (10) s2 The analytical minimization of s2 with respect to the column of free parameters q gives measures simultaneously the distance and the extinction using all the available light-curve observations, Equation (14) reducing the reliance on any particular time of the supernova light curve. According to our deÐnitions in equation or (1) (5), k V is the apparent distance modulus uncorrected for extinction. The extinction-corrected MLCS distance, k, is given by The "" standard candle ÏÏ distance, i.e., without any correction for light-curve shape (luminosity) or reddening, is given by This is the distance derived by correctly Ðtting the shape of the visual light curve to Ðnd the peak, ignoring the color k V ] *. excesses, and assigning the SN Ia a standard luminosity. The standard errors for the parameters in are given by the q best covariance matrix, (LTC~1L)~1. These errors are the Ðtting errors that reÑect the uncertainty in locating a light curveÏs best placement in due to the presence of noise in either the training set light curves or in the independent light curve we Figure 1, are trying to Ðt.
This approach, until now, treats the correction templates derived from our training set as if they were perfect. This is certainly not the case, since the independent distance and estimates for the objects that make up the training set objects are A V themselves not perfect. The same type of uncertainty is seen more easily in the relation, where the light-curve Phillips (1993) decline in the Ðrst 15 days after maximum is correlated against the luminosity derived from independent distance and A V estimates. The uncertainty in the slope of this relation must be considered when it is used. Fortunately, our correction templates are well constrained by the accuracy of the distance and estimates combined with the size of the training set. The A V external source of error on the parameters can be found by varying the training set distances and estimates in a Monte A V Carlo simulation to determine the e †ects on the Ðtted parameters. As expected, the external distance error increases linearly This set is obtained by adding the correction templates, or multiplied by various luminosity corrections, *, to the standard templates to make R V (t) R color (t), the best reconstruction of an SN Ia light and color curves. Intrinsically dim SN IaÏs rise and fall faster in V and have redder colors before day 35 than intrinsically bright SN IaÏs. After day 35, all SN IaÏs have more uniform colors. From the multicolor light-curve shape (MLCS) method, we estimate the luminosity and extinction by dust independently from the distance to measure the extinction-free distance. Data shown as reconstructed.  and shows the di †erences in photometric behavior for bright and dim SN IaÏs. Intrinsically dim SN IaÏs rise and fall faster in B and V than intrinsically Fig. 1 bright SN IaÏs. For the R light curve, a "" shoulder ÏÏ occurs D25 days after B maximum in the bright SN IaÏs. This shoulder is weaker for dimmer SN IaÏs and is absent for the most underluminous ones. In the I band, the bright SN IaÏs have two maxima ; one early (D5 days before B maximum) and one later (D30 days after B maximum). As the luminosity of the SN IaÏs decreases, the Ðrst maximum occurs later and is broader, while the second maximum is dimmer and occurs earlier. For the most underluminous SN IaÏs the two maxima merge into one maximum that is broad and occurs D5 days after B maximum. Data shown as reconstructed.  3.ÈDispersion of noise-free light and color curves around the best-Ðt model. The "" gray snakes ÏÏ comprise 1 p conÐdence regions which, when plotted over a best-Ðt model Ðt (here chosen as the * \ 0 standard templates), are expected to contain 68% of noise-free data points. The square of these functions would be the diagonal entries of the signal correlation matrix (S) which, added to the noise correlation matrix (N), would give the model correlation matrix (C). To compensate for our inability to determine either data covariance (o † diagonal elements) or higher order correlations in luminosity, we rescale each of these functions as described in°4.
with the light curveÏs luminosity correction, *, and it is well described by p \ 0.055* mag. So for a supernova whose luminosity correction * \ 0.30 mag, our external distance error would amount to less than 0.02 mag. This error is negligible for the majority of observed supernovae whose typical o * o ¹ 0.50 mag results in p ¹ 0.03 mag. This external source of error will decrease as the square root of the training set size, which we will be able to expand in the future.
The time of maximum for each supernova is a nonlinear parameter that cannot be solved for analytically in this scheme. It requires an outer iteration to minimize s2 in For our well-observed light curves, the uncertainty in the time of equation (13). maximum, typically well under 1 day, has a negligible e †ect on the parameter errors, with the median error increasing by only 10% if the time of maximum is varied^1 day. For poorly observed light curves whose observations begin D10 days after maximum, the uncertainty in the time of maximum increases substantially, as does its e †ect on the parameter errors. We discard all light curves beginning more than 10 days after maximum to avoid SN IaÏs with large uncertainties while maintaining a useful number of objects. When the set of usable SN IaÏs has grown sufficiently, the precision of this distance indicator might be improved by imposing an even stricter requirement for the time of the Ðrst observation.

CONSTRUCTING THE CORRELATION MATRIX
The correlation matrix, C, used in equations and to determine the best parameters and s2 of the light-curve Ðt, is (13) (14) the sum of two parts, C \ S ] N. The noise correlation matrix, N, is the correlation matrix for the measurement errors. The signal correlation matrix, S, is the correlation matrix which, in the absence of measurement error, estimates the expected deviations of the light curves from our model. It is the S matrix that allows us to use our model despite its known shortcomings. The N matrix is supplied by the conscientious observer.
The S matrix has two parts, diagonal and o †-diagonal entries, which, in principle, are estimated from our training set. The diagonal entries are estimates of the expected deviation of the light curve from the model. We determine the entries along the diagonal of each photometric bandÏs block of S by measuring the dispersion of each training set memberÏs and R V R color around the ensemble average and given in equations and minus each light curveÏs contribution from R V R color (11) (12) measurement error. The result gives, as a function of time, the expected errors of noise-corrected data around the best-Ðt model. We have plotted this dispersion around the best-Ðt model reconstruction in using the standard templates Figure 3 (* \ 0) as an example. The resulting "" gray snakes ÏÏ comprise 1 p conÐdence regions which, when plotted over a best model Ðt, would be expected to contain 68% of the data points. The square root of the diagonal entries of the matrix are S V,B,R,I provided in Table 2.
The o †-diagonal entries of the S matrix provide estimates of the correlation between model residuals. These correlations are likely to be considerable. There are many more elements to consider in determining the point-to-point correlations than we had for the previous autocorrelations. While initially we had only to estimate the expected model residuals for a given band on a given day of the light curve, the o †-diagonal entries require much more information. We would need to estimate the amount of covariance between observations on di †erent days in the same band, in di †erent bands on the same day, and in di †erent bands on di †erent days (In addition, we would have to remove the contribution from measurement covariances, which observers generally do not supply !). Unfortunately, our sparse training set is currently inadequate for quantifying these covariances. A simple two-point correlation function suggests that these covariances are most important in B, R, and I but does not provide enough information to approximate them adequately. This is the same compromise we faced in choosing a simple linear model over one with higher order terms. A detailed description of how our data deviate from a linear model requires the same information as would be required to establish a more detailed model for our data. Either improvement requires a large training set. We echo our minimalist approach to our model with a minimalist approach to the S matrix. We employ a diagonal S matrix (with the diagonals determined as above) with compensation for the possibility that some photometric bands are more adequately modeled with our linear model than others. We increase the diagonal elements of the signal correlation matrix enough to compensate for our inability to estimate its o †-diagonal terms. A simple rescaling of the entire signal correlation matrix would be less e †ective because some photometric bands of data (matrix blocks) have more model covariance (i.e., nonlinear behavior) than others.
We seek to weight each bandÏs data in the correlation matrix by its ability to predict the parameter in common, *. This approach recognizes that in a linear model some bands may be better than others at estimating the parameter * and weights each light curve accordingly. To determine each bandÏs weight, we allow their weights in the correlation matrix to vary and maximize the log-likelihood function for the determination of *, Maximizing L is the desired way to determine parameters of the correlation matrix in which the conventional approach of minimizing s2 would necessarily drive the weights and s2 to zero & Kleyna A simple exercise shows that (Rybicki 1994). maximizing L with respect to the weights drives the residuals to the smallest value they can have while still maintaining a s2 per degree of freedom near unity. By maximizing L, we optimize our dataÏs ability to predict the parameter, *, while requiring simultaneously that the estimated error in * is reasonable. Given more training set data, we could estimate the entire correlation matrix by maximizing L. With our current limited training set, we will use our previous prescription to parameterize the diagonal blocks of the correlation matrix that correspond to each photometric Ðlter and use L to determine the relative weights of those blocks.
Here L takes the form where the subscripts "" ind ÏÏ and "" MLCS ÏÏ denote the value and uncertainty of * as determined by independent methods and MLCS, respectively. Optimal weighting of the B, V , R, and I data minimizes the di †erence between the independently determined * values (in and the MLCS predicted * values that are a function of the weights in the C matrix. Using Table 1) and a downhill simplex method et al.
to Ðnd its maximum, we have determined the relative weights equation (16) (Press 1992) of each band. The diagonal matrix needed for the MLCS Ðt comes from adding the square of the model errors in C V,B,R,I to the square of the observersÏ photometry errors, then multiplying the four diagonal blocks of by 0.37, 11.45, Table 2 C V,B,R,I 8.85, and 5.52. These values, determined by maximizing indicate that the V -band light curves are signiÐcantly equation (16), better predictors of * than B, R, or I data within the framework of our linear model. SpeciÐcally, a V -band observation contributes 5.6, 4.9, or 3.9 times as much as a B-, R-, or I-band observation toward determining the luminosity correction for an SN Ia. This result is not astonishing, since it is well established that light curves in B, R, and I with di †ering luminosity can cross each other A crossing point of such light curves implies the same light or color curve shape for SN IaÏs (Suntze † 1993). with di †erent * values. Such behavior found in B, R, and I diminishes their predictive power in our linear model, but the independence of their measurements provides useful estimates of extinction. A V We have assumed that the correlation matrix for V , B, R, and I data is diagonal, but the correlation matrix for and we readily identify Deriving the desired correlation matrix from equation (18) has the advantage of requiring the much simpler A and diagonal matrices.

C V,B,R,Ĩ 1
Meaningful model parameter errors can only come from models that Ðt the data within statistical expectations. We require that both the V light-curve model and the color curve models give a reduced (per degree of freedom) s2 of 1. To deÐne the conÐdence region for the distance parameters, we use the covariance matrix of the Ðt, (LTC~1L)~1, multiplied by the reduced s2 of the light or color curve Ðts that measures the parameter of interest. This, in e †ect, is renormalization of the C matrix, now done on a supernova-by-supernova basis. This renormalization is not generally a large factor and would presumably become unnecesary with a better nondiagonal model for C.
In the case of the distance modulus error, derived from the visual band data, the error is the (1, 1) entry of the covariance matrix multiplied by the reduced s2 of the V light curve Ðt : Similarly, the extinction error, as derived from the B[V , V [R, V [I, is the (2, 2) entry of the covariance matrix multiplied by the reduced s2 of the color curvesÏ Ðt : The extinction-corrected distance is given by and its variance is the sum of equations and minus twice the (20) covariance of the estimates of k and A V : The extinction-corrected distance error of is the previously mentioned Ðtting error. For a particular SN Ia, its equation (21) size depends on light-curve sampling, measurement errors, and light-curve shape (see These errors provide useful°6). individual estimates of distance uncerainty.

FORMALIZED TRUNCATION OF A V
What is the best way to estimate the absorption by dust given a measurement of excess color ? We have well-founded a priori knowledge that dust scatters or absorbs light but does not amplify it, so the true value of must lie within the range where is a detection limit. Since we measure by dustÏs reddening e †ect, we say, a priori, that dust cannot "" blue-en ÏÏ or m lim A V brighten an SN Ia. We could use this knowledge to improve our estimate of by truncating any measurement of found A V A V to be less than zero. Simple truncation carries the disadvantage of improperly treating our useful estimate of from p AV What is the best way to use our estimate of and its error together with prior knowledge that cannot be equation (20).
A V A V less than zero ? A straightforward Bayesian calculation provides the solution.
Suppose we were to make an estimate of with value and normal error Further, we have some knowledge of the This "" Bayesian Ðlter ÏÏ provides a probability distribution for the true from which we can obtain a best estimate of A V A V and its error. A minimalist approach to would be to assume it is constant over the range in This p(A V ) equation (22). FIG. 4.ÈBayesian estimation of visual extinction by dust. By combining our MLCS measurement of visual extinction, and its error with a priori A V , knowledge that is positive, we can estimate the likelihood and distribution of the true value for T op: Combination of a negative estimate for 25^0.20) with a Gaussian wing (p \ 1) a priori distribution to yield an estimate for (\0.0^0.08). Bottom : Combination of a positive estimate for A V (\0.25^0.20) with the same a priori distribution to give an estimate for (\0.24^0.16).
formalized truncation is too conservative and unrealistic, and we can do better. Since supernovae with very large values of A V are less likely to be part of a sample of detected supernovae, we suggest that a reasonable form for the observed is a p(A V ) one-sided Gaussian that has a maximum at and declines for large (see We have chosen p \ 1 mag of for Fig. 4).
A V our but we show in that our results are insensitive to the particular value used for p. p(A V ),°7 6. COMPARING DISTANCES The above completes our development of an algorithm to measure extinction-corrected distances with multicolor lightcurve shapes. There are, however, a few complications to consider in the practice of measuring distances to supernovae with light-curve shapes.
The "" K-correction ÏÏ & Sandage Mayall, & Sandage corrects for the e †ects of redshift on the (Oke 1968 ; Humason, 1956) measured Ñux through a Ðlter of Ðxed spectral response. These corrections can be approximated by measuring the e †ect of redshifting the spectra of SN Ia supernovae for di †erent redshifts and phases. We can include K-corrections in our templates given the redshift or include them in our measurements given the redshift, using an assumed time of maximum in the outer iteration. For B-and V -band data, we have used the  1996 ;Goldhaber 1996). Using we now measure the distance related parameters for a set of well-observed SN IaÏs assuming that they equation (14), share the same behavior as our training set. By applying the method as developed for the training set to this independent sample, we compute distances for each supernova and construct a Hubble diagram. Analysis of this diagram shows that the MLCS approach gives better precision than the standard candle method. We will also compare the results with a sample selected by the same criteria used by & Sandage to see whether MLCS improves the utility of a "" normal ÏÏ Tammann (1995) set of SNIaÏs. We restrict our attention to light curves obtained on a modern photometric system in which the light curve begins within 10 days of maximum light (as determined by our Ðt). Tests on the training set have shown that in order for a light curve to contain luminosity information in its shape, the Ðrst observation must be within 10 days of maximum. Every one of the SN IaÏs in our samples was recorded on digital images and has accurate subtraction of the host galaxy background to ensure that the light-curve shapes are free from systematic errors & Wheeler (Boisseau 1991). Our independent set of 20 supernovae contains 10 objects from the Calan/Tololo survey (Hamuy et al. 1993a(Hamuy et al. , 1994(Hamuy et al. , 1995 (3) and (4) (5) gives a distance estimate without k V correction for either the luminosityÈlight curve relation or absorption.
In we show the multicolor light-curve shape reconstruction for three SN IaÏs et al. from our Figure 5 (Riess 1996a) independent sample spanning the range of data quality and distance error. For SN 1993ac, we have 23 observations (six epochs) beginning shortly after maximum light, resulting in an extinction-corrected distance error of 0.20 mag. The light and color curves of SN 1994Q contain 42 observations (13 epochs) beginning shortly after maximum and give an extinctioncorrected distance error of 0.13 mag. SN 1995D has one of the best sampled light and color curves with 107 observations (28 epochs) beginning before maximum light and yielding an extinction-corrected distance error of 0.06 mag. In general, the size of our predicted extinction-corrected distance error depends on the number of observations, the noise in the observations, and whether the SN Ia was discovered before or after maximum light. Half of our independent sample of 20 SN IaÏs were observed at or before maximum light.
First we will use the Hubble diagram as an analytical tool, without reference to a distance scale calibration, to determine the precision of our method.
shows, for comparison, two Hubble diagrams for this independent set of supernovae. In Figure 6 Figure 6a we have Ðt the best light-curve shape to each supernova to estimate the distance without any correction for intrinsic luminosity variation or extinction. In we have plotted the MLCS extinction-corrected distances to each SN Ia that Figure 6b accounts for intrinsic luminosity variation and extinction (see The reduction in dispersion is dramatic. The Table 3). improvement in distance precision comes from deriving the correlation between luminosity and light and color curve shape from our training set of SN IaÏs and then applying these relations to the independent sample. Because the training set and the independent set have no overlapping members, the reduction in dispersion is a powerful demonstration of the e †ectiveness of the MLCS method.
compares the distance estimates for di †erent assumptions by measuring the dispersion and s2 on the Hubble Table 4 diagram. In each case we have made a custom reconstruction of the light and color curves. The Ðrst row is the "" standard candle ÏÏ assumption, for which we disregard the light curve shape-luminosity correction, *, by adding it back to the distance, and we make no correction for extinction. Next, we use the MLCS distance modulus, which includes the k V ] *, k V , luminosity information, but we make no allowance for absorption. Following this, we include a correction for only the Galactic component of extinction by  Moving down each successive row represents a reÐnement in our distance measuring technique. The result of Table 4, successive improvements in the method can be seen in the decreasing dispersion on the Hubble diagram. Including the heavily reddened SN 1995E in the sample demonstrates the power of MLCS in dealing correctly with reddened objects but masks the gradual improvement in distance to be made from various aspects of the method. It is important to note that even the distances of the color cut sample can be improved with MLCS. This shows that the distance precision of "" normal ÏÏ or "" Branch-normal ÏÏ SN IaÏs can be enhanced signiÐcantly using light and color curve information. The MLCS method is not just "" reining in ÏÏ the extreme SN IaÏs but rather it is improving the distance measures to all the SN IaÏs.
To demonstrate the statistical signiÐcance of each level of improvement, we have included in the probabilities, Table 4 P r , that the observed improvement in dispersion could occur from a random set of distance corrections. These probabilities give the likelihood for the null hypothesis, that our distance "" corrections ÏÏ have no relation to the true SN Ia distances. In a Monte Carlo simulation, we apply a set of random corrections chosen from the distribution of proposed corrections and see how often the dispersion is as low or lower than the actual improved dispersion. The value of gives the probability that the P r observed (or a greater) decrease in dispersion for each successive distance reÐnement occurred by chance. The results show that MLCS luminosity and extinction corrections are highly signiÐcant regardless of the sample criterion. Both the MLCS luminosity correction and extinction correction reject strongly the null hypothesis that they are unrelated to the true distance of the SN IaÏs. Even the small corrections for galactic extinction do more good than harm. Yet, accounting only for the Milky WayÏs contribution to the total absorption fails to account for host galaxy extinction, which in a few cases can be substantial. Using all the predictive power of MLCS gives remarkably low values for the dispersion with an exceedingly small probability that this improvement in dispersion occurred by chance. With a conservative estimate for the peculiar velocity associated with each Ðeld galaxy of 300 km s~1, our observed dispersion of 0.12 mag implies a typical distance precision of 5%. The improvement in distance precision by including a correction for host galaxy extinction with the conventional reddening law is the Ðrst demonstration that such corrections can be made successfully.
Our Monte Carlo simulation demonstrates that even for a set of SN IaÏs selected by color, such as those used by et Sandage al.
the MLCS method makes a signiÐcant improvement in the precision of the distances. For the color cut sample, the (1996), probability that both our luminosity and extinction corrections would improve the dispersion from 0.33 to 0.13 mag by chance is less than one in a million.
Both SN 1992K and SN 1995E provide instructive examples of how this method leads to improved distance estimates. Both objects appear to be dim, which for a standard candle suggests that they are at a great distance. Assuming a standard candle luminosity places each of them much further away than their redshift implies (see Yet, there are clues in the Fig. 6a). light and color curves of these objects that indicate that they are dim for di †erent reasons. The rapidly declining V light curve of SN 1992K and the color evolution of its B[V curve are nearly identical to the photometric behavior of the subluminous SN 1991bg, a member of the training set et al.
Application of the MLCS method estimates SN 1992K to be (Hamuy 1994). * \ 1.25 mag dimmer than the standard SN Ia, though its is only 0.01 mag. This correction to the luminosity is A V independent of the SNÏs redshift since it depends only on the light-curve shape, but it is reassuring to note that accounting for this objectÏs intrinsic faintness shifts its distance quite precisely onto the Hubble line.
For the case of SN 1995E, the dim appearance that places this SN Ia below and to the right of the Hubble line is not intrinsic to the supernova, but rather is a result of absorption, as shown by MLCS. This supernova was found on the spiral arm of NGC 2441. The shape of its light and color curves suggest a fair resemblance to the standard SN Ia event (* \ [0.17 mag), but all its measured colors are displaced systematically to the red, as would occur from absorption by dust. By Ðtting the shape of the color curves independently from the value of the color, we can measure the color excesses. Assuming a standard reddening law (see et al. we estimate the visual band extinction to be 1.86 mag. As in the case of Riess 1996b), SN 1992K, correcting the luminosity of SN 1995E makes its distance consistent with its redshift measurement. Using a color cut requiring both these objects would be discarded. Yet, we can keep these objects (and others [0.25 ¹ (B[V ) max ¹ 0.25, like them) in the sample and use the MLCS method to distinguish between supernovae that are intrinsically dim and those that are dimmed by dust absorption.
We can make two signiÐcant checks of our extinction-corrected distances by examining the Ðt of the Hubble line to the data. We examine the linearity of the Hubble law by measuring the slope of the relation between the MLCS distance modulus and the logarithm of the redshift. Assuming that space is Euclidean for our modest redshifts, the expectation of this slope is 0.2. & Postman found a slope of 0.1992^0.006 using brightest cluster galaxies, and & Tammann Lauer (1992) Jerjen (1993) found a slope of 0.1988^0.006 using the mean of a number of distance indicators to Ðfteen clusters. Using the extinctioncorrected distances and errors of yields a slope of 0.2010^0.0035, which is consistent with 0.2 and with the two Table 3 previous results. The small error on our slope over the distance interval 32.2 \ k \ 38.3 makes this the most precise check of this classical test of cosmology. Finally, we can examine the goodness of Ðt of our Hubble line to the extinction-correction MLCS distances. The s2 of the Ðt using the independently determined distance errors in Table 3 and km s~1 is 13 p vel \ 300 for 19 degrees of freedom, which is within the expectation of s2. The value of s2 is strongly dependent on the assumed random velocity of the Ðeld galaxies hosting our SN Ia due to our low dispersion and small distance errors. A random velocity error for our Ðeld galaxies in the range 125 km km s~1 gives a s2 within its likely range of 13È25 1994,1996). are 300 km s~1, reÑecting a plausible estimate of random velocities with respect to the Hubble Ñow. (a) Distances estimated with a standard luminosity assumption and no correction for extinction. This method yields and (statistical) km s~1 Mpc~1. (b) Distances from the MLCS p v \ 0.52 H 0 \ 52^8 method, which makes a correction for intrinsic luminosity variation and total extinction as determined from the light and color curve shapes. This method yields and (statistical) km s~1 Mpc~1.
suggest that the MLCS method provides remarkably precise, extinction-corrected distances that are a signiÐcant improvement over SN Ia distances from previous methods.

DISCUSSION
Our intent has been to describe the MLCS method in enough detail so others can take advantage of the precise distance estimates it provides. The strength of MLCS lies in its ability to disentangle the e †ects of absorption and intrinsic luminosity variations while providing meaningful error estimates. These measures are derived from the distance-independent observables of multicolor light-curve shapes. The accuracy of the MLCS relative distance measures has been well established on an independent set of 20 SN IaÏs on the Hubble diagram. To place our MLCS distances on the established absolute distance scale, we use the luminosity calibration for a number of SN IaÏs with an independent distance indicator of high precision. At present, there are three SN IaÏs whose light curves meet our MLCS quality criteria (modern photoelectric photometry with observations less than 10 days after maximum) and whose distances have been measured with Cepheids observed with the Hubble Space T elescope.
The three SN IaÏs, SN 1972E, SN 1981B, and SN 1990N, are listed column (2) gives the extinction-corrected Table 5 : MLCS distances on the distance scale of SBF-PNLF-TF column (3) gives best distances as determined by HST (Table 1), Cepheid measurements (Sandage et al. column (4) gives the di †erences between columns (3) and (2), column (5) 1994, 1996), gives the MLCS luminosity correction, and column (6) Table 3 Hubble constant by Ðtting to the distances and velocities in using the tabulated errors and an assumed random velocity of 300 km s~1. Table 3 p kV~AV The result is a Hubble constant of 64^3 km s~1 Mpc~1 where the uncertainty is internal and incorporates a 0.03 mag uncertainty in the Hubble line and a 0.10 mag uncertainty in the placement onto the Cepheid distance scale. We see also that the MLCS luminosity corrections (col. [5]) are consistent with the luminosity di †erences as determined by the Cepheids (col. [6]).
For comparison, we perform the same calculation for the distance and extinction methods using the three SN Ia samples. We apply each method to the calibration set to determine the o †set onto the Cepheid scale and to the distant set to determine the Hubble line. For the methods that do not provide their own error estimates, we assume a constant distance error of the size necessary to obtain the expected s2 of the Ðt to the Hubble line. In we list the determinations of the Hubble Table 6 constant and internal error for the di †erent methods and samples.
The color cut sample provides us with a set of SN IaÏs that are directly comparable to the "" normal ÏÏ set used by et Sandage al.
Using the assumption of a standard luminosity, the color cut sample gives km s~1 Mpc~1, which is (1996). In and we demonstrate that each of the distance measuring improvements en route to the complete MLCS°6 Table 4, method is highly signiÐcant and should be employed to obtain the best result. Using the MLCS method to measure extinction-corrected distances and errors yields a Hubble constant of km s~1 Mpc~1 or km s~1 H 0 \ 64^3 H 0 \ 63^3 Mpc~1 using constant weighting. The only signiÐcant change in the Hubble constant arises from including the luminosity correction, *. This is because the mean luminosity of the three nearby calibrators is D15% brighter than the mean luminosity of the 20 SN IaÏs in the distant sample. That these nearby SN IaÏs are brighter than the distant SN IaÏs does not necessarily imply an antiselection bias because our sample is neither volume nor magnitude limited. If our sample was complete in volume or magnitude, we would observe a tremendous increase in the number of observed SN IaÏs at large distances. Figure 6 shows that this is clearly not the case. We elaborate on this point below.
A complete error budget for any of the values of in would consist of the stated internal error added in H 0 The MLCS distances to SN IaÏs in the Hubble Ñow combined with redshifts provide the necessary information to estimate the peculiar velocity component of each objectÏs radial motion. Plotted on the sky, the velocity residuals show a dipole pattern indicative of the motion of the Local Group with respect to a frame deÐned by the supernovae. Our preliminary analysis of this motion with a subset of 13 supernovae from the current set and without our MLCS reddening information was consistent with convergence to the cosmic microwave background frame and inconsistent with the & Postman frame at Lauer (1994) 7000 km s~1 et al. The same analysis performed at higher precision using the independent set of 20 SN IaÏs and (Riess 1995b). the MLCS method (which now includes extinction corrections) yields an even stronger detection of the Local Group motion with similar results. In the future, when the sample of SN IaÏs has grown, we will revisit our analysis of the Local Group motion from SN IaÏs.
Sections outline the MLCS technique in sufficient detail to allow for future redetermination of the necessary templates 2È5 and functions as more supernovae become available or are deemed desirable to include in the training set. The training set for this paper uses all supernovae for which accurate and extensive photoelectric photometry is currently available, as well as deÐnitive relative distance estimates and estimates. These include, from the set, SN 1980N, SN 1981B, A V Phillips (1993. We exclude SN 1971I, for which only photographic photometry is available. We also use SN 1994ae et al. for which there are detailed photometric light curves and an indepen-(Riess 1996a), dent distance Ideally we would use only SN IaÏs with no evidence of extinction, but currently we lack a (DellÏAntonio 1995). sufficient number of training set objects to discard any. Instead, we estimate from the color di †erences of unreddened A V SN IaÏs of similar light-curve shape and propagate the resulting uncertainty in to our distance errors. A V It is also likely that in time, the distance estimates used for the training set may improve. Most recently, some of the distance estimates provided by surface brightness Ñuctuation have been reanalyzed (J. Tonry, 1996 private communication (1995) 1994 private communication) and the distance from the shape of the SN 1994D light curve. The di †erence of 0.7 mag for the SBF and supernovae distance moduli to NGC 4526 implies a "" greater than 3 p ÏÏ mutual rejection. The case can be simpliÐed by comparing the results for SN 1992A in NGC 1380 in Fornax to SN 1994D. These supernovae have identically shaped light curves that, according to a light curve shapeÈluminosity relation, implies that they have the same luminosity. The preliminary SBF distance to SN 1994D suggests that it was brighter than SN 1992A by 0.5 mag. Such a di †erence in luminosity between SN IaÏs with similar light-curve shapes, if true, must be unusual.
Inclusion of SN 1994D at the preliminary SBF distance has the e †ect of decreasing the precision of the distance estimates for the independent sample of objects, but the training set is large enough that the e †ect is not substantial. The dispersion of distances on the Hubble diagram for MLCS increased from 0.12 to 0.22 mag, and the s2 increased from 13 to 39. This strong increase in dispersion and s2 reinforces our belief that the preliminary SBF distance to NGC 4526 is inconsistent with the distance derived with SN 1994D. Until either another such object is observed or the SBF distance is Ðnalized, we remain agnostic as to which is "" correct.ÏÏ We Ðnd the MLCS distances to SN 1994D to be 30.83^0.16 and hope that the size and shape of the Virgo Cluster will soon be understood well enough to test this prediction.
In we developed a formal way to combine our measurement of with our a priori understanding of dust. This method°5 A V requires some description of the distribution of values for supernovae discovered in galaxies. The most conservative A V estimate for is that it is constant over the range of This amounts to truncation of extinctions less than p(A V ) equation (22). zero. A Gaussian wing that is a maximum at and is parameterized by its second moment seems more plausible. A V \ 0.0 Either assuming a constant value for of a Gaussian wing with 0.5 ¹ p ¹ O yields a high level of distance precision. The p(A V ) Hubble diagram dispersion is insensitive to the value of p over this range, and the Hubble constant is insensitive to any particular parameterization of including requiring (see Using a with p ¹ 0.5 is too restrictive and Table 6). p(A V ) amounts to discarding any extinction information. Determination of the best value for p by maximizing the log-likelihood of on the Hubble diagram yields p \ 0.9 mag. It is important to consider sample selection e †ects when choosing a Equation (15) form for the observed Clearly the width of could be a function of the search characteristics, since these determine p(A V ). p(A V ) how likely it is to Ðnd an SN Ia obscured by dust. Monte Carlo calculations provide an estimate of the e †ect of misjudging and with enough data, one could solve easily for the distribution of observed The MLCS method estimates the total extinction from the supernova line-of-sight reddening. Therefore, we expect this total extinction measure to be equal to or greater than the galactic extinction as derived from the correlation between hydrogen column density and  (1988a, 1988b, 19994a, 1994b) distances does not apply to the distances in our training set that were derived from the bias-corrected inverse Tully-Fisher relation & Willick (Strauss 1995 ;Schechter 1980 ;Pierce 1996). & Sandage have also commented on the statistical problem raised by a light curveÈluminosity relation Tammann (1995) stating that "" the Malmquist e †ect on the distant SN IaÏs would be overwhelming.ÏÏ SpeciÐcally, & Sandage Tammann (1995) argue that the distant SN IaÏs occupy "" a volume of about 3 ] 105 larger, on average, than the volume of the three nearby local calibrators.ÏÏ Therefore, one might expect a substantial selection bias favoring brighter SN IaÏs in the distant sample as compared to the SN IaÏs nearby. In fact, the mean SN Ia luminosity for the distant sample we use is 0.34 mag dimmer than for the set of three nearby calibrators. Is this surprising ? Not necessarily. The selection bias, as stated by & Sandage Tammann assumes a survey that is complete to a limiting magnitude. This is clearly not case for our sample or for theirs. (1995), Inspection of the Hubble diagrams of (or of Tammann & SandageÏs) does not show the number of SN IaÏs increasing Figure 6 with distance as 100.6k as expected for a complete search of increasing volume. This point can be made quantitatively by performing a simple test which has an expectation value of 0.5 for a uniformly distributed sample. SV /V max T (Schmidt 1968), For the distant set of 20 SN IaÏs which shows that this sample is concentrated nearby. Even limiting the test SV /V max T \ 0.09, to the set of 12 SN IaÏs discovered during the uniform Cala n/Tololo survey yields Real samples of observed SV /V max T \ 0.16. SN IaÏs are not distributed as assumed by & Sandage Tammann (1995). There may be selection parameters other than brightness that inÑuence the discovery of SN IaÏs. It has been suggested (N. Suntze †, 1996 private communication) that at greater distances, photographic surveys, such as the Cala n/Tololo survey, may favor the discovery of SN IaÏs in elliptical galaxies in which supernovae are easier to discern. It has also been suggested et al. that SN IaÏs in ellipticals may be intrinsically dimmer than those in later type galaxies. This (Branch 1996, Hamuy 1996 suggests that at large distances, the sample of SN IaÏs may contain a greater proportion of dim objects. Unfortunately, the Hubble diagrams of do not contain enough information to determine whether dim or bright objects are favored at Figure 6 large distances. If we assume naively that there is no selection bias a †ecting our sample, we Ðnd that the chance of the three local calibrators being 0.34 mag dimmer than the independent sample is 6%. This is about as likely as picking randomly the winner of the next baseball American League Championship Series. An interesting question to consider is what explosion or progenitor parameters could explain our empirically determined variation in light-curve shape, luminosity, and color ? Current models have attempted to explain the inhomogeniety of supernovae in one of two ways. et al. and & Khokhlov have found that a variation in the density Ho Ñich (1995) Ho Ñich (1996) at which the deÑagration burning front transitions to a detonation wave a †ects the amount of 56Ni produced in the explosion. A late transition gives the outer layers time to preexpand, resulting in a small nickel production. The reduced nickel heating diminishes the temperatures in the expanding envelope and photosphere. This results in rapidly dropping opacities. Consequently, the photosphere recedes fast and the stored energy is emitted over a short period of time. Conversely, an early turnover of the deÑagration into a detonation front results in a large amount of nickel, which produces a bright and hot supernova whose opaque layers keep the radiative energy loss comparably low. This mechanism works for deÑagrations, delayed detonations, and pulsating delayed detonations. Qualitatively, the observed correlation between luminosity and the photometric parameters are reproduced & Khokhlov (Ho Ñich 1996). Another theoretical approach to matching the observations involves exploding progenitors of varying and generally sub-Chandrasekhar mass. A layer of helium accumulates at the base of the static hydrogen-burning zone. Sudden burning of this helium at the base of the layer sends a shock that can trigger a carbon detonation at the interface. If this does not occur, a second chance at carbon detonation can come when the shock propagates around the star and converges on the opposite site. The variation in progenitor mass suggests a simpler connection to the amount of nickel produced than the Chandrasekhar models. Again the variation in nickel yield is expected to match the observed correlations in supernova observables. These models have been successful in one and two dimensions, but it remains to be seen if the abundance and the velocity distribution of the intermediate-mass elements produced from nucleosynthesis matches the observations & Arnett (Livne & Weaver & Glasner 1995 ;Woosley 1994 ;Livne 1990 ;Livne 1990). Finally, improvements can be made in the MLCS technique when we acquire a larger training set of supernovae. First, the "" gray snakes ÏÏ of and the relative weights of the B, V , R, and I data determined from suggest that Figure 3 equation (16) greater precision could be attained by including a quadratic term in the light-curve shapes model for B-, R-, and I-band data. An additional order in the Ðt would demand a larger training set of data to avoid overÐtting the details of the training set objects. A larger training set would also support the determination of a more detailed correlation matrix with estimates of model residual covariances.
The MLCS technique provides an exceedingly precise way of measuring extinction-corrected distances with uncertainty estimates. The redshifts to which Type Ia supernova light curves can measure distances have interesting implications for cosmological measurements. Two teams are currently laboring to Ðnd and measure SN IaÏs at 0.3 ¹ z ¹ 0.6 et al. (Perlmutter et al. with the intent of measuring the cosmological deceleration of the universe, At these redshifts, 1996 ; Schmidt 1996) q 0 . SN Ia light curves are difficult to obtain, and informative spectra are even harder, so it is sensible to use all the available SN IaÏs. The uncertainty in is proportional to the distance precision divided by the square root of the number of objects. q 0 The MLCS method should help with both these factors by including all well-observed SN IaÏs in the sample and increasing the precision of SN Ia distance measures.
We are again grateful to Mario Hamuy, Mark Phillips, Nick Suntze †, and the entire Cala n/Tololo collaboration for the opportunity to study their exceptional data before publication. We have beneÐted greatly from discussions with Brian Schmidt, George Rybicki, and Peter Ho Ñich. This work was supported through NSF grants AST 92-18475 and PHY 95-07695.