Standardization of type Ia supernovae
Rodrigo C. V. Coelho, Maurício O. Calvão, Ribamar R. R. Reis, Beatriz B. Siffert
SStandardization of type Ia supernovae
Rodrigo C V Coelho, Maur´ıcio O Calv˜ao, Ribamar R R Reisand Beatriz B Siffert
Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Av. Athos da SilveiraRamos 149, 21941-972, Rio de Janeiro, RJ, Brazil
Abstract.
Type Ia supernovae (SNe Ia) have been intensively investigated due toits great homogeneity and high luminosity, which make it possible to use them asstandardizable candles for the determination of cosmological parameters. In 2011, thephysics Nobel prize was awarded “for the discovery of the accelerating expansion ofthe Universe through observations of distant supernovae.” This a pedagogical article,aimed at those starting their study of that subject, in which we dwell on some topicsrelated to the analysis of SNe Ia and their use in luminosity distance estimators. Herewe investigate their spectral properties and light curve standardization, paying carefulattention to the fundamental quantities directly related to the SNe Ia observables.Finally, we describe our own step-by-step implementation of a classical light curvefitter, the stretch, applying it to real data from the Carnegie Supernova Project.PACS numbers: 95.36.+x, 97.60.Bw, 98.80.Es
Keywords : Cosmology, type Ia supernova, dark energy, light curve standardization
1. Introduction
A supernova (SN or SNe, from the plural supernovae ) is a stellar explosion which mayoccur at the final stage of the evolution of a star or as the result of the interaction betweenstars in a binary system. The current supernova classification follows the historical orderin which these events were observed. Initially, the explosions were divided into types Iand II, according to the presence (type II) or absence (type I) of hydrogen emission linesin their spectra. Later, the observation of SNe with different spectral features resulted inthe introduction of the subtypes which we use nowadays (see figure 1). SNe of types II,Ib and Ic are now believed to occur due to gravitational collapse of massive stars (above ∼ a r X i v : . [ a s t r o - ph . C O ] N ov from a companion, approach the Chandrasekhar limiting mass ( ∼ ‡ .[3]In the single degenerate scenario, the companion is generally considered to be a mainsequence, a red giant or an AGB star, whereas in the double degenerate scenario it isanother white dwarf. The nitty-grity details of the explosion process and the progenitorchannel are still open to debate, both theoretically and observationally.[4, 5, 6] Figure 1.
Schematic classification of supernovae.
As already stated, SNe are classified according to the presence or absence of certainspectral lines in their spectra and, for SNe II subtypes, the shape of their light curves.Type I SNe can be divided into three subtypes: SNe Ia present a silicon absorptionfeature around wavelength λ = 6150 ˚A (their main characteristic); SNe Ib does notpresent silicon lines but present helium absorption lines; and SNe Ic present neithersilicon nor helium features. To learn more about the spectral features of SNe, see [7].The most interesting subtype for cosmological purposes is the Ia, because of their highpower, which allows us to detect them in distant galaxies, and their quite homogeneousemission, which makes possible their use as standard candles.A given class of astrophysical objects (or events) is considered a standard candlewhen their intrinsic luminosity is known or can somehow be estimated. In the case ofSNe Ia, the observation of nearby events showed that all explosions had quite similarluminosities and the relatively small variations (as compared to the typical magnitudesof SNe Ia) can be corrected for (in fact, due to the existence of such fluctuations theseevents should actually be considered standardizable candles). SNe Ia themselves can bedivided into subgroups and a classification scheme much used in the literature is the ‡ The exact value of this limiting mass depends on several properties of the white dwarf: metallicity,Coulomb corrections, temperature, rotation, magnetic fields, etc; in any event, realistically, thesecorrections seem to amount to no more than 10%.[2] one by Branch et al. ,[8] according to which these events can be 1991bg-like § , which aresubluminous, 1991T-like, which are superluminous, and normal ( Branch-normal ). Tohave concrete numbers to express those variations, we calculated the sample standarddeviation in absolute magnitudes M B (cf. section 2) of the SNe Ia in a sample ofVaughan et al. ,[9] comprising 50 SNe Ia, of which 25 are Branch-normal. Consideringonly the Branch-normal SNe Ia, the standard deviation of the distribution of M B is 0.65mag, while its mean is − . (cid:107) and, sincebeing explosions, they are transients (lasting around three months), which makes theirobservation a difficult task. In order to detect a high number of SNe, various projectsare being planned, as this will demand a greater number of researchers in the field. Fora list of these projects and some of the most important past and present experiments,see Table 1. Our goal in this work is to highlight some basic concepts concerning theuse of SNe Ia for cosmology, which we found are not detailed in textbooks. We believethat this work will be of great utility for those who are starting their research in thefield, as well as for researchers who have never worked specifically in this field.In section 2 we present the basic concepts of spectrum, light curve, flux, magnitudes,all of them derivable from the fundamental concept of specific flux. In section 3, wediscuss the influences distance and redshift have on the specific flux of an arbitrarysource. In section 4 we discuss our naive light curve standardization, by taking advantageof a sort of stretch correction that characterizes only the variations in SNe Ia rise-and-decline rates, but not the intrinsic luminosity differences. In section 5 we present our § Supernovae are named for their year of occurrance and an uppercase letter, e.g., “SN 1987A”. Ifthe alphabet is exhausted, double lower case naming is used: [Year] aa .. az, ba .. bz, etc; e.g., “SN1997bs”. (cid:107)
For supernovae relatively close to our galaxy with 0 < z < .
3, the rate of occurrence of SNe Ia pervolume is (3 . ± , × − supernovae/year/Mpc , according to [14]. Table 1.
Past, current and future experiments to detect SNe: Equation of State:SupErNovae trace Cosmic Expansion (ESSENCE) [15], Supernova Legacy Survey(SNLS) [16], Sloan Digital Sky Survey (SDSS) [17], Panoramic Survey Telescope& Rapid Response System (Pan-STARRS) [18], Dark Energy Survey (DES) [19],Javalambre Physics of the Accelerating Universe (J-PAS) [20] and Large SynopticSurvey Telescope (LSST) [21]. The third column gives the number of spectroscopicallyconfirmed SNe Ia for past experiments and the total number of expected detectionsfor current and future ones.Name Running period ∼ ∼ ∼ ∼ conclusions. In Appendix A we describe some usual transformations of an arbitraryfunction, for generic pedagogical reasons.
2. Fundamental quantities
The specific flux ¶ (in the wavelength representation) measured by a detector isgenerically defined as the infinitesimal energy received by the detector per infinitesimaltime interval, per infinitesimal perpendicular area, per infinitesimal wavelengthinterval, + i.e., f λ := dEdt dA ⊥ dλ . (1)The specific flux for a given source will in general depend not only on the wavelength λ , on the distance to the source r (cf. subsection 3.1) and on the source’s specific poweror luminosity L λ , but also, for transient sources, on the time t , and for moving sources,on the redshift z (cf. subsection 3.2); concretely f λ = f λ ( λ, t, r, z, L λ ). For simplicity,in future references to this equation we may suppress one or more dependences in thefunction f λ . More on the discussion present in this section can be found in classicalastrophysics books such as [22].There are basically two techniques used for detecting astronomical objects:spectroscopy and photometry. In spectroscopy one uses a spectrograph to decomposethe incoming light into its different wavelength components and obtain a measure ofthe specific flux at a given time, i.e. the spectrum of the object. Despite the high ¶ The expression specific refers to quantities measured per unit wavelength (or frequency), while bolometric refers to quantities integrated over all wavelengths (or frequencies). + The typical unit of f λ is 1 erg/cm /s/˚A, whereas for the corresponding frequency representation, f ν ( t, ν ) = cf λ ( t, c/ν ) /ν , it is 1 erg/cm /s/Hz = 10 Jy (jansky). spectral resolving power in wavelength ( R := λ/ ∆ λ , where ∆ λ is the resolution of thespectrograph) provided by spectroscopy (a low to intermediate resolution spectrographhas R of the order 1000–10000, whereas state of the art high resolution ones canachieve R (cid:39) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:43) c on s t a n t (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN2007px, z (cid:61) (cid:72) f Λ (cid:43) (cid:76) (cid:230) SN2006fy, z (cid:61) (cid:72) f Λ (cid:43) (cid:76) (cid:244) SN2006py, z (cid:61) (cid:72) f Λ (cid:76) Figure 2.
Observed spectra of some SNe Ia, at four days before B band maximumlight.[23] Here and in all forthcoming figures showing spectra, geometric symbols(circles, squares, and triangles) serve only as a guide for the reader to better identifyto which SN each curve refers. In photometry one uses filters, which let the light pass only for a particularwavelength interval (the filter bandpass), and the resulting observation, called flux,corresponds to specific flux integrated over this interval. Flux measures in a givenfilter at different times (or epochs) constitute a usual (not specific) light curve of theobject. Photometry is a cheaper and faster technique and there are many projects beingdesigned to get a huge amount of data through photometric observations.We now show how to obtain SN Ia light curve templates at a given filter from atheoretical model for f λ ( λ, t ). These templates are necessary for the standardization ofSN Ia light curves, as will be discussed in section 4.First we have to take into account the bandpass of the chosen filter. Thefilters UBVRI (also known as the Johnson-Cousins filter set) are traditionally usedto characterize SNe in the rest frame and will be used in this work. The reader canfind a detailed discussion on photometric systems in Bessell.[24] We show in figure 3the transmissivity curves, i.e. the fraction of energy that passes through the filter as afunction of wavelength, S Xλ , for these filters. It is important to notice that the filtersare not perfect, in the sense that they do not let all photons pass, no matter whatwavelength we consider. We will define the flux in band X , f X , as the energy flux thatis transmitted through filter X , which can be written as f X ( t ) := (cid:90) ∞ f λ ( λ, t ) S Xλ ( λ ) dλ, (2)where we have, for brevity of notation, suppressed the dependence of f λ (and therebyof f X ) on r . λ ( ˚A) S X λ ( % ) U B V R I
Figure 3.
Transmissivity curves for the
UBVRI filters typically used in photometry.
The light curves are generally given in terms of the apparent magnitude in a givenfilter X , which is related to the flux f X by ∗ m X ( t ) := − . (cid:18) f X ( t ) g X (cid:19) , (3)where g X is the reference flux, that can be for instance the flux of a given star to whichall other sources will be compared and defines a magnitude system . A photometricsystem is defined by a set of filters (in our case UBVRI ) and the reference flux definedin all of them. In principle, the reference flux can be different for each filter, howeverthis is not mandatory. In this work we use the AB magnitude system,[25, 26] whichuses as reference a constant specific flux for all frequencies: g ABν = 3631 Jy . Another commonly used magnitude system is the one that uses as reference flux theflux of the Vega star in the chosen filters. Our photometric system will be definedby the filter set
UBVRI and the AB magnitudes. In order to mantain the notationmost commonly used by astronomers, throughout the text we are going to refer to theapparent magnitude in a given filter X by simply the letter X so, for example, theapparent magnitude of an object measured with the B filter will be just denoted B . ∗ Throughout the text log denotes decimal (base 10) logarithm.
The filter reference flux g X is given by g X := (cid:90) ∞ g Xλ ( λ ) S Xλ ( λ ) dλ, (4)where g Xλ ( λ ) is the specfic reference flux for filter X .Since we chose to perform our calculations in wavelength space, we need to rewritethe AB reference specific flux using the relation g ν ( ν ) dν = g λ ( λ ) dλ. Recalling that c = λν , we can obtain the reference specific flux as a function ofwavelength g Xλ ( λ ) = cg ABν λ . Therefore, to build a light curve, we need to evaluate the magnitudes for a given filterusing (3) for spectra at different epochs. In figure 4, we show some light curves fromtypical SNe Ia, whereas in figure 5 we show a SN Ia light curve obtained from the SNIa template generated by Nugent.[28] (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230) (cid:230) (cid:244)(cid:244)(cid:244) (cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244)(cid:244) (cid:244)(cid:244)(cid:244) (cid:244) (cid:244)(cid:244)(cid:244)(cid:244) (cid:244) (cid:244)(cid:244) (cid:244)(cid:244)(cid:244) (cid:244) (cid:244) (cid:244)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) t (cid:45) t (cid:72) days (cid:76) B (cid:224) SN2005M (cid:72) z (cid:61) (cid:76) (cid:230) SN2004ef (cid:72) z (cid:61) (cid:76) (cid:244) SN2006D (cid:72) z (cid:61) (cid:76) Figure 4.
Observed sampling of apparent magnitude B band light curves from twoBranch-normal (SN2004ef and SN2006D) and one 1991T-like (SN2005M) SNe Ia.[27]Notice that a simple visual inspection of the light curves does not allow determiningthe subtypes. With given source and detector, we can display the visual representation of thefunction f λ ( λ, t ) by means of what we will call the spectral surface. In figure 6, forinstance, we show a representation of this surface, constructed from Nugent’s estimatesbased on real SNe Ia data,[28] for fixed z, r and L λ (cf. section 3). The spectral surfacedisplays in one single frame both the time evolution of the spectrum and the wavelengthdependence of the specific light curve. The spectrum of the source, at a given time t ∗ ,is the intersection of the spectral surface with the plane t = t ∗ , and the specific lightcurve, at a given wavelength λ ∗ , is the intersection of the spectral surface with the plane λ = λ ∗ . A spectral surface like the one shown in figure 6 would be the result of ideal (cid:45)
20 0 20 40 6020.19.519.18.518.17.517.16.516.15.5 t (cid:72) days (cid:76) B (cid:43) o ff s e t Figure 5.
Theoretical B band light curve constructed from Nugent’s Branch-normalSN Ia spectral template.[28] observations of a SN, continuous both in wavelength and time. In practice the best wecan do is a discrete sampling of that surface for a given SN; however, even this would beunfeasible for a high number of SNe, because of the time demanded for the observationsand the need for high cost facilities.It is convenient to find a relationship between ideal detected quantities and intrinsic(source rest-frame) ones in a cosmological spacetime. To that end, as a motivatingwarm-up, let us consider an imaginary spherical (2-dimensional) surface, of radius R , concentric with a light source, both at rest in an inertial frame of the Minkowskispacetime. The bolometric (raw or pure) flux is defined as f ( t, R, L λ ) := (cid:90) ∞ f λ ( λ, t, R, L λ ) dλ , (5)and, due to conservation of energy, is trivially related to the intrinsic bolometricluminosity L ( t ) := (cid:82) ∞ L λ ( λ, t ) dλ(cid:93) by: f ( t, R, L ) = L ( t )4 πR . (6)We now introduce the concept of the redshift z , which is a measure of the relativevelocity between astrophysical objects through the observation of their spectral features[29, 30]: z := ( λ obs − λ em ) /λ em , where λ em is the wavelength of a spectral feature, as measured in its rest frame, and λ obs is the corresponding wavelength measured on Earth.In Appendix B we show an intuitive way to obtain the relation between flux andluminosity for a more general spacetime, taking z into account, which is (B.4) f λ ( λ, t, r, z, L λ ) = L λ ( λ/ (1 + z ) , t/ (1 + z ) )(1 + z ) πr , (7) (cid:93) Bolometric flux has the same units as band-limited flux: 1 erg/cm /s. t ( da ys ) λ ( ˚ A ) f λ ( − e r g / c m / s / ˚ A ) Figure 6.
Theoretical rest-frame spectral surface generated from Nugent’s template ofsynthetic spectra, at different epochs or phases, of a typical Branch-normal SN Ia.[28]We also show five typical spectra, projected onto a conveniently offset plane t = − λ = 13000 ˚A. where L λ ( λ/ (1 + z ) , t/ (1 + z ) ) is the specific luminosity in the source’s rest-frame.From specific flux measures in different wavelengths (or frequencies), in a givenepoch, we can construct a spectrum of an astrophysical object. In figure 7, left panel,we show spectra of three SNe Ia, SN1994D,[31] SN1998aq[32] and SN2003du,[33] takentwo days after maximum light (in B band, as we will see in the next sections), from thepublic database SUSPECT.[34] The characteristic shape of the spectral lines, known asP Cygni profile, indicates the presence of an expanding gas cloud. For a gas expandingwith spherical symmetry, part of the light that is emitted toward us is coming fromregions that are moving in our direction and is blueshifted , and the other part comes fromregions that are moving away from us, being therefore redshifted. Since different layersof the expanding gas move with different velocities, the resulting spectrum presents wideemission lines centered at the rest wavelenghth value. As an example of such lines, wecan see two SiII absorption lines with rest-frame wavelengths λ ≈ λ ≈ λ ≈ λ ≈ (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D (cid:72) z (cid:61) (cid:76) (cid:230) SN1998aq (cid:72) z (cid:61) (cid:76) (cid:244) SN2003du (cid:72) z (cid:61) (cid:76) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D (cid:72) z (cid:61) (cid:76) (cid:230) SN1998aq (cid:72) z (cid:61) (cid:76) (cid:244) SN2003du (cid:72) z (cid:61) (cid:76) Figure 7.
Left panel : spectra from Branch-normal SNe Ia 1994D, 1998aq and 2003dutaken two days after maximum, in the B band. Right panel : same spectra in theSN Ia rest frame (cf. subsection 3.2). The vertical solid (dashed) lines indicate thetypical rest-frame position of the absorption (emission) components for SiII, due tothe P Cygni profile. would have for a hypothetical observer at a distance of 10 parsecs †† and at rest withrespect to it ( z = 0), M X ( t ) := − . (cid:90) ∞ L λ ( λ, t )4 π (10 pc) S Xλ ( λ ) dλ (cid:90) ∞ λ =0 g Xλ ( λ ) S Xλ ( λ ) dλ . (8)We would like to call the reader’s attention to the fact that, by its very definition, itmakes no sense to refer to an absolute magnitude for z (cid:54) = 0, something that is not alwaysexplicit in the literature.We can also consider an ideal case, in which we could measure the flux of a sourcein all wavelengths with a perfect detector ( S Xλ ( λ ) = 1 , ∀ λ ), to define bolometricmagnitudes, m ( t, z ) := − . (cid:90) ∞ λ =0 f λ ( λ, t, z ) dλ (cid:90) ∞ λ =0 g λ ( λ ) dλ , (9) M ( t ) := − . (cid:90) ∞ λ =0 L λ ( λ, t )4 π (10 pc ) dλ (cid:90) ∞ λ =0 g λ ( λ ) dλ . (10) †† The parsec is a distance unit frequently used in astronomy and corresponds to approximately 3 . . × m. 1 parsec is the distance to an object with rest-frame size of 1 astronomicalunit and apparent angular size of 1 arc second. distance modulus : µ := m − M. (11)As we will discuss in Section 3, the observed spectrum of a source is modified withrespect to its intrinsic one by the redshift, and therefore the radiation emitted in a givenwavelength range in the source’s rest frame will be observed in a different range in theobserver’s frame. Also, since we simply cannot measure bolometric magnitudes, butonly magnitudes in some filters, it is useful to express the distance modulus in termsof filter magnitudes, which requires the introduction of the so called K -correction K XY defined as K XY := m Y − M X − µ. (12)A full discussion of K -corrections and their applications for cosmology are left bythe authors to another paper.
3. Dependence of specific flux on redshift and distance
It is important to note that even for a class of objects with the same intrinsic luminosity,which is approximately the case of SNe Ia (apart from the variations mentioned insection 1), their observed fluxes (both specific and bolometric) will differ mainly due tothe different redshifts and distances.From (B.4) we can see that, at a given time t and at a given wavelength λ , thespecific flux can vary with distance r to the source, with redshift z , and with thefunctional form of the specific luminosity L λ . Considering SNe Ia as standard candlesmeans that we will assume all events to have the same specific luminosity. We know,however, that there are variations in their luminosities that should be taken into accountand this will be considered in section 4. In the present section we will study how anarbitrary observed spectrum differs from the source’s rest-frame spectrum, as we change,independently, the distance r and the redshift z . To that end, we advise the reader torefer now to Appendix A, where we graphically remind what happens to a functionwhich is subjected to certain simple transformations that will be relevant in the nextsubsections. Let us analyze first the simpler effect, the one arising from distance changes only. From(B.4), we can see the dependence of the specific flux on the inverse square of the distance r . Thus, when r (cid:55)−→ r (cid:48) = cr ( c = const.) , (13)all other independent variables held constant, we have that f λ ( λ, t, r, z, L λ ) (cid:55)−→ f (cid:48) λ ( λ, t, r (cid:48) , z, L λ ) = c − f λ ( λ, t, r, z, L λ ) . (14)2Therefore, in a graph of the spectrum, as shown in figure 8 for SN1994D, we employ,in a linear scale (left panel), the vertical distortion of (A.3) and, in a logarithmic scale(right panel), the vertical translation of (A.1). The effect on the spectrum of a purechange only in distance is manifest in the logarithmic scale, where the rigid verticaltranslation is obvious. (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D, r (cid:61) d SN (cid:72) real spectrum (cid:76) (cid:230) SN1994D, r (cid:61) d SN (cid:244) SN1994D, r (cid:61) d SN (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D, r (cid:61) d SN (cid:72) real spectrum (cid:76) (cid:230) SN1994D, r (cid:61) d SN (cid:244) SN1994D, r (cid:61) d SN Figure 8.
Synthetic spectra simulating the effect of distance on the spectrum ofSN Ia 1994D, taken 2 days after maximum in the B band, in linear (left panel) andlogarithmic (right panel) scales. The same spectrum was divided by different constantsin order simulate different distances (cf. (B.4)). Let us analyze now the effect of the redshift, related to the relative motion betweensource and observer. Again, from (B.4), we can see the dependence of the specific fluxon the inverse cube of (1 + z ) and also modifying explicitly the independent variables λ , and t by factors of 1 / (1 + z ). Thus, when1 + z (cid:55)−→ z (cid:48) = c (1 + z ) ( c = const.) , (15)all other independent variables held constant, we have that f λ ( λ, t, r, z, L λ ) (cid:55)−→ f (cid:48) λ ( λ, t, r, z (cid:48) , L λ ) = c − f λ (cid:18) λc , tc , r, z, L λ (cid:19) . (16)Of course, referring to the Appendix, we see that this transformation of the specificflux involves the composition of a vertical distortion, (A.3), and a horizontal distortion,(A.4). To get a handle on it more intuitively, let us choose z = 0 so that the formerequation will provide the redshifted spectrum from the rest-frame one: f (cid:48) λ ( λ, t, r, z (cid:48) , L λ ) = 1(1 + z (cid:48) ) f λ (cid:18)
11 + z (cid:48) λ,
11 + z (cid:48) t, r, z = 0 , L λ (cid:19) , (17)3or vice versa, the rest-frame spectrum from the redshifted one: f λ ( λ, t, r, z = 0 , L λ ) = (1 + z (cid:48) ) f (cid:48) λ ( (1 + z (cid:48) ) λ, (1 + z (cid:48) ) t, r, z (cid:48) , L λ ) . (18)Now, to illustrate this redshifting effect in a most pristine situation, we apply (17) to atop-hat function. The result is shown in figure 9. In the left panel, we show that the Λ (cid:72) Å (cid:76) f Λ (cid:72) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) Rest frame (cid:72) z (cid:61) (cid:76) Redshifted frame (cid:72) z (cid:61) (cid:76) Λ (cid:72) Å (cid:76) f Λ (cid:72) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) Restframe (cid:72) z (cid:61) (cid:76) Redshiftedframe (cid:72) z (cid:61) (cid:76) Figure 9.
Effect of a variation in redshift on two top-hat spectra. The blue curveis the rest-frame spectrum ( z = 0) and the red one is same spectrum at a redshift z = 0 .
5, in the observer’s frame. total qualitative effect of the redshift is: (i) a vertical squeezing, due to the 1 / (1 + z (cid:48) ) pre-factor, and (ii) a horizontal stretch caused by the rescaling λ (cid:55)−→ λ/ (1 + z (cid:48) ) in thefirst argument of f λ . From this panel, the reader could naively be induced to regard thedisplacement towards greater wavelengths as a third, independent, effect; however, ascan be seen from the right panel of figure 9, such a displacement is in fact also due tothe horizontal stretch, which leaves the vertical y -axis ( λ = 0) fixed (cf. (A.4) and rightlower panel of figure A1).In figure 7, left panel, we showed observed spectra of three SNe Ia. In its rightpanel, we now show the corresponding rest-frame ( z = 0) spectra. We can notice thesmall horizontal displacement of the spectral lines (blueshifted, towards the left) but itis not possible to visualize the vertical displacement (upwards) due to the low valuesof the redshift involved. We can also see that, even after the redshift correction, thespectra do not coincide and this is because each SN is at a different distance from us.To explicitly reveal the redshifting effect on the spectrum of a concrete SN Ia,we show, in figure 10, three spectra of SN 1994D, the rest-frame one and two other(artificial) high redshift ones (left panel). In particular, the effect of the pre-factor1 / (1 + z (cid:48) ) in (17) can be best viewed using a logarithmic scale (right panel), in whichit becomes a simple vertical translation (cf. (A.1) and left upper panel of figure A1).4 (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D (cid:72) z (cid:61) (cid:76) (cid:230) SN1994D (cid:72) z (cid:61) (cid:76) (cid:244) SN1994D (cid:72) z (cid:61) (cid:76) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) Λ (cid:72) Å (cid:76) f Λ (cid:72) (cid:45) e r g (cid:144) c m (cid:144) s (cid:144) Å (cid:76) (cid:224) SN1994D (cid:72) z (cid:61) (cid:76) (cid:230) SN1994D (cid:72) z (cid:61) (cid:76) (cid:244) SN1994D (cid:72) z (cid:61) (cid:76) Figure 10.
Synthetic spectra simulating the effect of redshift on the spectrum of SNIa 1994D, taken 2 days after maximum in B band, for different values of redshift. Weuse a linear scale in the left panel and a logarithmic one in the right panel.
4. Light curve standardization
Although source-frame SNe Ia light curves are very similar, they are not identical. In thissection we will show that it is possible to make them even more similar by applying somesimple operations, which are dubbed standardization, and we will apply this procedureto a sample of real type Ia SNe. The process of standardization became possible afterthe discovery that intrinsically brighter SNe (at B band maximum) were also the oneswith wider light curves.[35, 36] Such a correlation rendered it possible to determine if agiven SN was brighter (fainter) than another one either because it was closer (further)or because it was intrinsically brighter (dimmer), just by looking at their light curves.The data used in this work are publically available,[37] and constitute the sampleof 85 low redshift SNe Ia observed by the Carnegie Supernova Project (CSP).[27, 38]Motivated by the higher uniformity of SNe Ia in the infra-red band, one of the maingoals of that project was to obtain particularly well sampled and well characterized lightcurves both in optical and near-infrared bands, which should improve the efficiency ofthe standardization process. We restricted ourselves to the subsample of only Branch-normal SNe Ia, which reduced the number of events to 71. The corrections that wewill present here were originally done simultaneously through a single fit that yieldsall the correction factors for each SN (cf. Goldhaber et al. [39]); however we chose toimplement them step by step in order to make clear the role of each one in the finalresult.5 (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231)(cid:231)(cid:231)(cid:231) (cid:231) (cid:231) (cid:231)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242) (cid:242) (cid:242)(cid:242) (cid:242) (cid:242)(cid:242)(cid:242)(cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242)(cid:242)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:224)(cid:224) (cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224) (cid:224) (cid:224)(cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42)(cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:180)(cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180) (cid:180) (cid:180) (cid:180) (cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180) (cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248)(cid:248) (cid:248)(cid:248)(cid:248) (cid:248) (cid:248)(cid:248)(cid:248)(cid:237)(cid:237) (cid:237)(cid:237)(cid:237)(cid:237) (cid:237) (cid:237)(cid:237) (cid:237)(cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237)(cid:236)(cid:236) (cid:236)(cid:236)(cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197) (cid:197)(cid:197)(cid:197) (cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:43)(cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196) (cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:45)
20 0 20 40 60 80 10022.21.20.19.18.17.16.15.14.13.12. (cid:72) t (cid:45) t (cid:76) (cid:72) days (cid:76) B (cid:231) SN2004ef (cid:230)
SN2004eo (cid:242)
SN2004ey (cid:243)
SN2005M (cid:224)
SN2005hc (cid:225)
SN2005iq (cid:244)
SN2005kc (cid:42)
SN2006X (cid:180)
SN2006ax (cid:248)
SN2006bh (cid:237)
SN2007af (cid:236)
SN2007le (cid:159)
SN2007on (cid:197)
SN2008bc (cid:43)
SN2008fp (cid:196)
SN2008gp (cid:158)
SN2008hv
Figure 11.
Apparent magnitude B band light curves of the 17 SNe Ia in our subsampleafter the time axis offset correction (cf. subsection 4.1). In the CSP light curve data, the epoch is expressed in Modified Julian Date (MJD). Inorder to compare them in a single plot, we need to define a common time scale t − t † ,where t is the epoch of maximum flux, traditionally considered in B band. We wrotea simple code to obtain t for each supernova in our subsample. Unfortunately, someof them were observed only after B band maximum and were thus excluded from oursubsample, which reduced considerably the number of SNe in the final subsample. Infact, we required our code to keep only the SNe that presented at least 3 observationstaken before maximum flux and at least one observation taken after 30 days frommaximum flux (the reason for this restriction will become clear in section 4.3). Thisleft us with a subsample of 17 SNe, whose names and redshifts are listed in table 2, andwhose time-offset-corrected light curves can be seen in figure 11. In order to properly standardize the light curves, we need to correct them for extrinsiceffects. As we have seen in section 3, two of them can be easily taken account of:distance and redshift. The latter entails a change of the time scale and an offset to themagnitude (or change of the flux normalization) whereas the former implies a simpleoffset to the magnitude. Thus the correction for both effects amounts to:(i) a (horizontal) dilation, cf. (A.4), of the time axis such that∆ t o ∆ t e = 1 + z, (19) † The time scale t − t is commonly called phase . Table 2.
Names, CMB-centric redshifts and stretch factors (see section 4.3) for all 17SNe Ia in the final subsample used to generate our simple light curve template.SN z CMB s s G where ∆ t o is a time interval in the observer’s frame and ∆ t e is the correspondinginterval in the source’s rest frame.(ii) a (vertical) rigid translation, cf. (A.1), of the magnitude axis.Notice that after the rigid vertical translations to correct for the redshift anddistance, it is possible that the peaks of the light curves still do not coincide, sincethere can be absolute magnitude differences among them. So, in order to make thepeaks coincide, a third vertical rigid translation is still needed. In our case, we do notknow the distances to the SNe in our sample, so what we actually did was to evaluate thepeak magnitude’s mean, and displace the light curves in order to make their magnitudesmatch this mean. This operation accounts for the rigid vertical translations due to boththe redshift and the distance corrections, and also to a third rigid translation to correctfor other differences in absolute magnitude.The resulting distance- and redshift-corrected light curves of our subsample areshown in figure 12. In order to display all SNe in their rest frame time, notice that wehave chosen to change the x -axis from t − t to ( t − t ) / (1 + z ). Because of this, a littlebit of care must be taken when comparing figure 12 and the following figures in thissection to the results presented in section 3 and Appendix A, where we are keeping the x -axis unchanged before and after a given transformation.7 (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231)(cid:231)(cid:231)(cid:231) (cid:231) (cid:231) (cid:231)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242) (cid:242) (cid:242)(cid:242) (cid:242) (cid:242)(cid:242)(cid:242)(cid:242) (cid:242) (cid:242) (cid:242) (cid:242) (cid:242)(cid:242)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243) (cid:243) (cid:243) (cid:243) (cid:243)(cid:243)(cid:243)(cid:224)(cid:224) (cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224) (cid:224) (cid:224)(cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42)(cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:180)(cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180)(cid:180) (cid:180) (cid:180)(cid:180) (cid:180) (cid:180) (cid:180) (cid:180) (cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180) (cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248)(cid:248) (cid:248)(cid:248)(cid:248) (cid:248) (cid:248)(cid:248)(cid:248)(cid:237)(cid:237) (cid:237)(cid:237)(cid:237)(cid:237) (cid:237) (cid:237)(cid:237) (cid:237)(cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237)(cid:236)(cid:236) (cid:236)(cid:236)(cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159) (cid:159)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197) (cid:197)(cid:197)(cid:197) (cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:43)(cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196) (cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:45)
20 0 20 40 60 80 10021.20.19.18.17.16.15. (cid:72) t (cid:45) t (cid:76) (cid:144) (cid:72) (cid:43) z (cid:76) (cid:72) days (cid:76) B (cid:43) o ff s e t (cid:231) SN2004ef (cid:230)
SN2004eo (cid:242)
SN2004ey (cid:243)
SN2005M (cid:224)
SN2005hc (cid:225)
SN2005iq (cid:244)
SN2005kc (cid:42)
SN2006X (cid:180)
SN2006ax (cid:248)
SN2006bh (cid:237)
SN2007af (cid:236)
SN2007le (cid:159)
SN2007on (cid:197)
SN2008bc (cid:43)
SN2008fp (cid:196)
SN2008gp (cid:158)
SN2008hv
Figure 12.
Apparent magnitude B band light curves + offset for the 17 SNe in oursubsample after the time axis offset (cf. subsection 4.1) and distance and redshiftcorrections (cf. sunsection 4.2). The stretch parameter s is related to the width of the light curve, i.e. it measureshow fast the supernova’s flux decreases (or its magnitude increases).[12, 39] In order tocalculate the stretch, we need to adopt a fiducial curve which, in our case, was chosento be simply the mean of all curves in the sample, and assign the value s = 1 to it. Acurve that declines slower (faster) than the fiducial one will have s > s < B bandmaximum but not necessarily at any other point. The stretch correction is designed sothat the curves also coincide at 15 source frame days after B band maximum. We showa sketch of this procedure in figure 13, in which we use a fiducial (red curve) and twoficticious light curves, 1 and 2, in blue.To obtain the stretch we need to solve for p i = ( t − t ,i ) / (1 + z i ) from the followingequation f i ( p i ) = m , (20)where f i is an interpolating function (in our case a spline) for the i -th SN Ia B bandlight curve, and m is the value of B (+ offset) of the mean light curve at p i = 15 days.Thus, the stretch can be written as s i = p i
15 days . (21)We can then divide all phases of a supernova by the obtained stretch so the curvescoincide at phase 15 days.As mentioned earlier, the width difference in the light curves is associated to theirintrinsic brightness (broader ↔ brighter). When we correct for the stretch we arecompensating the differences in intrinsic brightness between the supernovae.8 (cid:45)
10 0 10 20 30 4020.19.519.18.518.17.517.16.516.15.5 (cid:72) t (cid:45) t (cid:76) (cid:144) (cid:72) (cid:43) z (cid:76) (cid:72) days (cid:76) B (cid:43) o ff s e t m (cid:72) s (cid:60) (cid:76) (cid:72) s (cid:62) (cid:76) p p Figure 13.
Schematics for stretch calculation. The red solid curve is the fiduciallight curve, for which s = 1, by definition. The red dotted horizontal line indicatesthe position of m in the y -axis, which corresponds to the point in the fiducial curvewhere ( t − t ) / (1 + z ) = 15 days, marked with the red dotted vertical line. The bluedashed curves are two other fictitious light curves, upon which we want to apply thestretch correction. The two blue dotted lines show the values of p for each curve, andthe resulting range of the stretch factor is indicated near each curve. The result of the application of the stretch procedure to our sample can be seen infigure 14. Table 2 shows the values of the stretch s found for each SN, according to ourprocedure. For comparison, we also show the values of the same parameter, now dubbed s G , found using the method described in Goldhaber et al. ,[39] where the corrections areall done simultaneously and the fiducial curve is different from ours. We can see that theresults obtained with our step-by-step method agree quite well with the ones obtainedwith this more sophisticated fitting recipe. After we apply all the corrections discussed above we can construct what we can call a“rudimentary” template, a simple mean of all corrected curves, that can be compared tothe Nugent template, one of the most used in the literature.[28] We show the comparisonin figure 15. In this comparison we can not use the relative discrepancy between thesecurves ( ( B N − ¯ B ) /B N or ( B N − ¯ B ) / ¯ B ) because the normalizations of both are arbitrary .We can, on the other hand, compare the absolute difference shown in figure 15 (lowerpanel) to the range of the Nugent template in the depicted interval ([ − , .
5, and the discrepancy thus calculated is always less than 12%. Wecan see that, despite the simplified analysis performed here, our curve looks quite similarto the template, which shows the consistency between our template and the Nugent one.We show in figure 16 the standard deviation of our sample before and after thestretch correction. Again, we can not use relative discrepancies in the whole time9 (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231) (cid:231) (cid:231)(cid:231)(cid:231)(cid:231) (cid:231)(cid:231) (cid:231)(cid:231)(cid:231)(cid:231) (cid:231) (cid:231) (cid:231)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230) (cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:230) (cid:230)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242) (cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242)(cid:242) (cid:242)(cid:242)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243) (cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243) (cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:243)(cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224)(cid:224)(cid:224) (cid:224)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:224)(cid:224) (cid:224) (cid:224)(cid:225)(cid:225)(cid:225) (cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225)(cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225) (cid:225)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244)(cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:244) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42)(cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42)(cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42) (cid:42)(cid:42)(cid:180)(cid:180)(cid:180)(cid:180)(cid:180)(cid:180)(cid:180) (cid:180) (cid:180)(cid:180)(cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180) (cid:180) (cid:180) (cid:180) (cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180)(cid:180)(cid:180) (cid:180) (cid:248)(cid:248)(cid:248)(cid:248)(cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248) (cid:248) (cid:248) (cid:248)(cid:248) (cid:248)(cid:248) (cid:248)(cid:248)(cid:248) (cid:248) (cid:248)(cid:248)(cid:248)(cid:237)(cid:237) (cid:237)(cid:237)(cid:237)(cid:237) (cid:237) (cid:237)(cid:237) (cid:237)(cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237) (cid:237)(cid:236)(cid:236) (cid:236)(cid:236)(cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236) (cid:236)(cid:236) (cid:236) (cid:236)(cid:236)(cid:236) (cid:236) (cid:236)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159)(cid:159)(cid:159) (cid:159)(cid:159)(cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159) (cid:159)(cid:159)(cid:159)(cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:159) (cid:159)(cid:159) (cid:159) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197)(cid:197)(cid:197)(cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:197) (cid:43)(cid:43)(cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:43)(cid:43)(cid:43) (cid:43)(cid:43) (cid:43) (cid:43)(cid:43) (cid:43)(cid:43)(cid:43) (cid:43)(cid:43) (cid:43)(cid:43) (cid:43) (cid:43) (cid:43) (cid:43)(cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196) (cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196) (cid:196)(cid:196)(cid:196)(cid:196) (cid:196) (cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158) (cid:158) (cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158)(cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:158) (cid:45)
20 0 20 40 60 80 10021.20.19.18.17.16.15. (cid:72) t (cid:45) t (cid:76) (cid:144) (cid:64)(cid:72) (cid:43) z (cid:76) s (cid:68) (cid:72) days (cid:76) B (cid:43) o ff s e t (cid:231) SN2004ef (cid:230)
SN2004eo (cid:242)
SN2004ey (cid:243)
SN2005M (cid:224)
SN2005hc (cid:225)
SN2005iq (cid:244)
SN2005kc (cid:42)
SN2006X (cid:180)
SN2006ax (cid:248)
SN2006bh (cid:237)
SN2007af (cid:236)
SN2007le (cid:159)
SN2007on (cid:197)
SN2008bc (cid:43)
SN2008fp (cid:196)
SN2008gp (cid:158)
SN2008hv
Figure 14.
Apparent magnitude B band light curves + offset of the 17 SNe Ia inour subsample after the time axis offset (cf. subsection 4.1), distance and redshift(cf. subsection 4.2) and stretch corrections (cf. subsection 4.3). Note that the curvesstill present some dispersion, since the recipe imposes coincidence only at two points:( t − t ) / [(1 + z ) s ] = 0 ,
15 days. B (cid:43) o ff s e t (cid:45)
10 0 10 20 3000.10.20.3 00.10.20.3 (cid:72) t (cid:45) t (cid:76) (cid:144) (cid:64)(cid:72) (cid:43) z (cid:76) s (cid:68) (cid:72) days (cid:76) B N (cid:45) B Figure 15.
Comparison between our B band light curve template and Nugent’sone.[28] Upper panel: B band rest-frame magnitude (arbitrarily normalized) versusrest-frame stretched phase for our template (red, solid curve) and for Nugent’s one(blue, dashed curve). Lower panel: discrepancy between our template and Nugent’sone. interval to compare these curves because both the standard deviations are zero at t − t = 0 by construction (see subsection 4.2) and the stretch corrected one is alsonull at ( t − t ) / [ s (1 + z )] = 15. Nevertheless, the overall decreasing in the dispersion0 Σ after stretch correctionbefore stretch correction (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:72) t (cid:45) t (cid:76) (cid:144) (cid:72) (cid:43) z (cid:76) (cid:72) days (cid:76) Σ a f t e r (cid:45) Σ b e f o r e Figure 16.
The role of the stretch correction in diminishing the dispersion. Upperpanel: B band light curve standard deviation for the 17 SNe Ia after all corrections(blue, dashed curve), and with all but the stretch correction (red, solid curve). Lowerpanel: discrepancy between the standard deviations in magnitude of our subsampleafter all corrections (including the stretch one) and before the (last) stretch correction. after the maximum is clear from figure 16 . Since such gain is obtained through asimple linear transformation, with only one parameter, we can argue that it reflects thehomogeneity of the light curves in our sampleThe reader might note that the discrepancy between the standard deviation afterthe stretch correction (in figure 16) is greater before maximum. This feature reflectsthe fact that the dispersion of the SNe Ia is smaller before the maximum (see Hayden et al. [40]). The stretch is defined to decrease the dispersion after maximum but it isapplied to the whole light curve through (A.4) therefore, since the curves are moreuniform before the maximum, when we multiply their arguments by different numbersthe net result is an increasing of the dispersion in this interval.
5. Conclusion
This article had two main aims: (i) presenting and clarifying some fundamental conceptsand results related to the cosmological use of SNe Ia, and (ii) building a simple SN Ialight curve template.The first aim led us to introduce, in section 2, the specific flux or spectral energydistribution as the principal quantity characterizing the class of transient SNe Ia, andthe corresponding projections (spectra and specific light curves). In section 3, we studied1in particular its dependence on distance and redshift and the consequent impact on theobserved fluxes or magnitudes.To comply with the second aim cited above, in section 4, we built our naive lightcurve template, for didactic purposes, through a simplified version of the original stretchprocedure: we performed the determination of the three parameters of the method (theoverall normalization of the light curve, the epoch of maximum flux in B band andthe stretch itself) separately, instead of the simultaneous fit described in Goldhaber etal. .[39] We finally constructed a mean light curve after the application of the methodand compared it to a light curve template much used in the literature,[28] showingthat our simplified method is able to produce a template very similar to it. In fact,the discrepancy is less than 10 − for most of the phases in the interval of [ − , Acknowledgments
BBS would like to acknowledge financial support from the brazilian funding agencyCAPES-PNPD, grant number 2940/2011.
Appendix A. Basic function transformations
In this Appendix we investigate four simple transformations (of one real parameter c onan arbitrary function f : x (cid:55)−→ y = f ( x ) which bear upon the changes on the specificflux due to distance and redshift (cf. section 3). These are (cf. figure A1):(i) vertical translation T V,c : T V,c f : x (cid:55)−→ y := f ( x ) + c . (A.1)It always rigidly translates, along the vertical y -axis, the graph of the function f ,by c “units”: upwards, if c >
0, and downwards, if c < horizontal translation T H,c : T H,c f : x (cid:55)−→ y := f ( x + c ) . (A.2)It always rigidly translates, along the horizontal x -axis, the graph of the function f , by c “units”: left, if c >
0, and right, if c < vertical distortion D V,c : D V,c f : x (cid:55)−→ y := cf ( x ) . (A.3)It always distorts, along the vertical y -axis, the graph of the function f , keeping apoint with vanishing y coordinate fixed: if | c | >
1, it represents a dilation or stretch,the more so the larger | c | is, whereas if 0 < | c | <
1, it represents a contraction orcompression, the more so the smaller | c | is. Furthermore, if c <
0, this distortion isalso accompanied by a reflection of the graph with respect to the x -axis.(iv) horizontal distortion D H,c : D H,c f : x (cid:55)−→ y := f ( cx ) . (A.4)It always distorts, along the horizontal x -axis, the graph of the function f , keepinga point with vanishing x coordinate fixed: if | c | >
1, it represents a contractionor compression, the more so the larger | c | is, whereas if 0 < | c | <
1, it representsa dilation or stretch, the more so the smaller | c | is. Furthermore, if c <
0, thisdistortion is also accompanied by a reflection of the graph with respect to the y -axis. Appendix B. Obtaining the relation between specific flux and specificluminosity
Let us now proceed to the generalization of (5) to the Robertson-Walker spacetime,whose line element may be cast in the form: ds = − c dt + a ( t ) (cid:20) dr − kr + r (cid:0) dθ + sin θ dϕ (cid:1)(cid:21) , (B.1)where k is the spatial curvature and a ( t ) is the dimensionless scale factor. The coordinate r is variously called the comoving areal distance, transverse comoving distance or propermotion distance.[45, 46]We first deal with the traditional case in which source and detector are both inthe Hubble flow, so that their relative velocity is all due to the cosmic expansion,and is traditionally called a recession velocity. In this case, time intervals dt S in thesource’s rest-frame, such as the time between the emission of two consecutive photons,correspond to time intervals in the detector’s frame dt D = (1+¯ z ) dt S , where ¯ z is the usualcosmological redshift: 1 + ¯ z = 1 /a ( t ). The source-frame and detector-frame energies ofthe photon will also be related by a factor (1 + ¯ z ). Therefore, assuming conservation ofphotons, the bolometric flux of a source at cosmological redshift ¯ z can be written as f ( t, r, ¯ z, L ) = L ( t/ (1 + ¯ z ) )4 πr (1 + ¯ z ) , (B.2)3 − − − y = f ( x ) y = T V,c =2 f ( x ) := f ( x ) + 2 − − − y = f ( x ) y = T H,c =2 f ( x ) := f ( x + 2) − − − y = f ( x ) y = D V,c =2 f ( x ) := 2 f ( x ) − − − y = f ( x ) y = D H,c =2 f ( x ) := f (2 x ) Figure A1.
Basic transformations of a given arbitrary function f : the original graphis represented by a (blue) dashed line and the transformed graph by a (red) solid line.Left upper panel displays the effect of a vertical translation T V,c with c = 2. Rightupper panel displays the effect of a horizontal translation T H,c with c = 2. Lower leftpanel displays the effect of a vertical distortion D V.c with c = 2. Lower right paneldisplays the effect of a horizontal distortion D H.c with c = 2. where t is a time coordinate measured with respect to a reference time t R which, forsimplicity, we choose to be t R = 0 in both frames. In this way, a time interval t − t R = t in the observer’s frame corresponds to the interval ( t − t R ) / (1 + ¯ z ) = t/ (1 + ¯ z ) in thesource’s rest-frame.Finally, we state that an equation of this same form holds for an arbitrary motion(not in the Hubble flow) of the source and the detector, viz., f ( t, r, z, L ) = L ( t/ (1 + z ) )4 πr (1 + z ) , (B.3)where now z is the total redshift between the source and the detector, which is the reallyobserved one ‡ Since the specific flux is related to the bolometric one by (5), it is obvious ‡ This demonstration, besides some alternative definitions of luminosity distance, and its consequencesfor SNe Ia, will be explored in a forthcoming paper, in preparation. f λ ( λ, t, r, z, L λ ) = L λ ( λ/ (1 + z ) , t/ (1 + z ) )(1 + z ) πr , (B.4)where L λ ( λ/ (1 + z ) , t/ (1 + z ) ) is the specific luminosity in the source’s rest-frame.We can also obtain the frequency representation of the specific flux as f ν ( ν, t, r, z, L ν ) = L ν ( ν (1 + z ) , t/ (1 + z ) )(1 + z )4 πr . (B.5)We can obtain (6) by integrating either B.4) on λ or (B.5) on ν . References [1] Hoyle F and Fowler W A 1960 Nucleosynthesis in Supernovae
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