Near-Infrared K Corrections of Type Ia Supernovae and their Errors

Luis N. Boldt; Maximilian D. Stritzinger; Chris Burns; Eric Hsiao; M. M. Phillips; Ariel Goobar; G. H. Marion; Vallery Stanishev

doi:10.1086/676019

1. Introduction

Type Ia supernovae (hereafter SNe Ia) are standardizable distance indicators at optical wavelengths that provide critical constraints on cosmological parameters (see Goobar & Leibundgut 2011, and references therein). A number of groups have worked diligently to gather optical photometry of homogenous samples of low-, intermediate-, and high-z SNe Ia that, when combined, amount to well over 1000 objects. As the sample size has increased, systematic effects have come to dominate the final uncertainty in the measured value of the equation-of-state parameter of the Universe, w (e.g., Wood-Vasey et al. 2007; Astier et al. 2006; Freedman et al. 2009; Kessler et al. 2009; Folatelli et al. 2010; Conley et al. 2011; Suzuki et al. 2012).

A major systematic that plagues SN Ia cosmology is our inability to accurately estimate host galaxy dust extinction due to uncertainties in the reddening law and, in particular, variations in the value of the total-to-selective extinction, R_V, from object to object (see Phillips 2012, and references therein). This is further exacerbated by any systematic error in the relative zero points between the nearby and distance SNe Ia samples, thereby making the latter artificially redder (or bluer) than the former. A sensible way around these problems is to observe in rest-frame near-infrared (NIR) bands instead of rest-frame optical bands, because the effects of dust extinction are minimized and essentially independent of the adopted reddening law (Krisciunas et al. 2000).

Additional motivation is provided by empirical evidence indicating that the luminosities of SNe Ia show little or no dependence on decline rate⁹ in the NIR (Meikle 2000; Krisciunas, Phillips, & Suntzeff 2004; Wood-Vasey et al. 2008; Krisciunas et al. 2009; Mandel et al. 2009; Folatelli et al. 2010; Kattner et al. 2012). This translates to a reduced intrinsic dispersion in the NIR Hubble diagram, and at the same time evolutionary effects as a function of redshift on the progenitor populations are potentially minimized. These factors have provided significant impetus for future SN Ia cosmology studies to construct homogenous samples of low- and high-z SN Ia observed in the rest-frame NIR (e.g., Green et al. 2012).

Reaping the benefits afforded by observing SNe Ia in the NIR requires the development of tools to obtain rest-frame luminosities via K corrections (Oke & Sandage 1968). The K correction is defined as the difference in brightness between an object observed in its rest-frame with a given passband compared to its measured brightness with the same passband when observed at redshift z. Specifically, for a given passband, i, the K correction is defined as:

Here m_i is the SN Ia apparent magnitude observed on Earth, and M_i is its absolute magnitude. The magnitude difference encapsulated in the K term is explained by the shifting and stretching of an object's spectral energy distribution (SED), which is inherent to cosmological expansion.

The first calculations of SN Ia K corrections at optical wavelengths were published by Leibundgut (1990) and Hamuy et al. (1993). In the latter paper, optical observations of three nearby SNe Ia were used to construct a sequence of B- and V-band K corrections extending to a redshift of z = 0.5. The authors found that the temporal variation of the computed values largely mimicked the (B - V) color evolution, implying that the K correction is, to first order, driven by the color of the SN. Shortly thereafter, Kim, Goobar, & Perlmutter (1996) presented a method to compute cross-band K corrections which is particularly well-suited for SNe Ia located at z > 0.2. Nugent, Kim, & Perlmutter (2002) expanded upon these efforts by developing a set of SN Ia optical spectral templates that enabled K corrections to be computed as a function of temporal phase for any given optical bandpass. Later, Hsiao et al. (2007) constructed improved optical spectral templates based on a greatly expanded sample of nearby SN Ia. They found that besides color, spectral diversity also has a significant impact upon the magnitude of the K term. The Hsiao et al. (2007) template is now routinely used to K correct optical photometry of SNe Ia, and the statistical uncertainty associated with these corrections is reasonably well understood.

In comparison, our knowledge of NIR K corrections is still relatively crude. Krisciunas et al. (2004) presented the first calculations of the temporal evolution of the K term in the JHK passbands based on 11 NIR spectra of SN 1999ee (Hamuy et al. 2002). Their results indicated that NIR K corrections are nonnegligible even at relatively small redshifts. These initial findings emphasized the importance of accurately characterizing NIR K corrections and their uncertainties as a function of redshift and temporal phase. Five years later, the publication of 41 spectra by Marion et al. (2009) revolutionized the study of the NIR spectral characteristics of SNe Ia, and Hsiao (2009) used these data along with the other published spectra available at that time to extend his spectral template to include NIR wavelengths. In this paper, we refer to this template as the "Hsiao revised template".

In the present paper, we expand on this work by using the existing library of published NIR SNe Ia spectra to determine the uncertainties inherent to using the Hsiao revised spectral template to calculate NIR K corrections, particularly those due to intrinsic variations in spectral features. To do this, we first color match¹⁰ each observed spectrum to the template, and then calculate YJHK_s-band K corrections. An interpolation algorithm based on Gaussian Processes combined with Markov-Chain Monte-Carlo methodology is then used to produce mean K corrections as a function of temporal phase and redshift, along with estimates of both statistical and intrinsic sources of uncertainties. These errors, in turn, will provide important input to future studies utilizing the NIR light curves of SNe Ia to estimate cosmological parameters.

The organization of this article is as follows. In § 2 the concept of the K correction is briefly reviewed; § 3 introduces the data used in our calculations, along with the adopted passbands and methods used to interpolate our computed K corrections; § 4 contains the results; and finally, § 5 presents our conclusions.

2. THE K CORRECTION

In this study we limit ourselves to the discussion of single-band K corrections rather than cross-band K corrections, which were the subject of papers by Nugent, Kim, & Perlmutter (2002) and Hsiao et al. (2007). Given a SED, f(λ), and a transmission curve of a particular passband, S_i(λ), the K term is computed following equation (2) of Oke & Sandage (1968):

The first "bandwidth" term of equation (2) is independent of wavelength and accounts for the narrowing of the observed filter as the SED is stretched as a function of redshift (Sandage 1995). The second term is (for a given passband) the ratio of the response of the SED at the rest wavelength compared to that measured at a redshift z, and accounts for the effects of doppler shifting. This term includes a factor of λ in both the numerator and denominator because most modern photometric systems count photons rather than energy (for details see Nugent, Kim, & Perlmutter [2002]).

In what follows we adopt for S_i(λ) the YJHK_s passbands of the Carnegie Supernova Project (CSP; Hamuy et al. 2006). Currently the CSP relies on modeled NIR instrumental passbands which have been constructed by multiplying together the factory-measured filter transmissivities with the transmission functions of two generic mirror reflections, various optical elements, a HAWAII–1 1024 × 1024 pixel HgCdTe detector response curve and a telluric absorption spectrum (Hamuy et al. 2006). The modeled passbands are electronically available on the CSP webpage,¹¹ and are plotted in Figure 1, along with NIR spectra of the normal Type Ia SNe 2005am and 2001bg obtained +4 and +10 days past B-band maximum (hereafter T[B_max]), respectively. This comparison highlights the prevalent emission features that coincide with the position of the rest-frame H band, which emerge and increase to maximum strength within the first two weeks past maximum. To illustrate the effect of cosmological redshift each passband is plotted at rest and at the positions corresponding to the wavelength regions of the SED they sample at z = 0.08.

**Fig. 1.** NIR spectra of the normal Type Ia SNe 2005am and 2001bg obtained +4 and +10 days relative to T(B_max), respectively. Also plotted as black lines are the rest-frame CSP natural system NIR transmission functions. In addition, these passbands are plotted as dotted lines at the locations which cover the SED they sample when observing a SN Ia at z = 0.08. The vertical bands correspond to regions containing prevalent telluric absorption.

3. Methods

3.1. NIR Spectral Data

Our analysis is based on 75 published NIR spectra of 33 SNe Ia that cover the YJHK_s bands. The temporal range spanned is from -14.6 to +53.8 days relative to T(B_max). The majority of the data are drawn from the Marion et al. (2009) catalog, which were obtained between 2000 and 2005 with the NASA Infrared Telescope Facility (IRTF) equipped with SpeX, a medium-resolution NIR spectrograph and imager. Additional spectra include the published sequences of SN 1999ee (Hamuy et al. 2002), SN 2003du (Stanishev et al. 2007), SN 2005cf (Gall et al. 2012), and SN 2011fe (Hsiao et al. 2013). Table 1 lists each spectrum sorted by phase along with its date of observation, the telescope used to make the observations, the redshift of each host galaxy as given by NED, and an estimate of Δm₁₅(B) for each object. When possible, the reported phase of a spectrum was estimated with respect to T(B_max) measured from its B-band light curve. However, there are a fews cases where no light curve information is available; we therefore estimated T(B_max) by cross-correlating an optical spectrum to a library of SN Ia spectra using the Supernova Identification (SNID) code (Blondin & Tonry 2007). In these instances, the estimated temporal phase with respect to T(B_max) has a realistic uncertainty of ± 3 days (Blondin & Tonry 2007).

3.2. Assigning Proper Errors

Before K corrections can be calculated, the errors associated with the spectra must be quantified. These arise from two principal sources: signal-to-noise ratio of the spectrum, and improperly removed telluric features, which are especially important to account for at NIR wavelengths. In the first case, we used the associated error spectrum when this was available. Otherwise, an estimate of the signal-to-noise ratio as a function of wavelength was made for each spectrum, typically amounting to 10% of the signal. Due to the fact that this is uncorrelated noise, its effect on the uncertainty of the K corrections is relatively small.

Errors in the telluric corrections can arise from several factors, but typically are due to rapid changes in the water vapor conditions. Also, for some features, the absorption reaches 100% of the continuum. These errors become important at higher redshifts when the features are shifted into the filter bandpasses. Unlike the errors due to signal-to-noise ratio, these telluric correction errors can be highly correlated if improperly subtracted. Fortunately, most spectra showed no sign of any systematic over- or under-subtraction, in which case we treated the error as white noise. Only in the cases where removal of the features left a clear imprint on the corrected spectrum did we simulate an additional error using a telluric template spectrum.

These two errors were then propagated using Monte Carlo simulations. In brief, 100 artificial spectra were created by introducing noise at a level consistent with the dispersion of the above-described errors, resulting in a sample of 100 K corrections. The standard deviation was then computed and adopted as the final error in the K correction for the particular redshift being calculated.

3.3. Extending the Wavelength Coverage of the Input Spectra

The goal of this paper is to quantify the errors in the K corrections due to variations in spectral features (cf. Hsiao et al. 2007). Our approach is to color-match the sample spectra to observed colors derived from the Hsiao revised template at a range of redshifts. The procedures used for the color-matching are those described by Hsiao et al. (2007) and Burns et al. (2011).

Ideally, when calculating K corrections for a particular supernova, multiple observed colors will be available spanning an extended wavelength range so as to properly anchor the color-matching function. However, in the present paper, we are dealing with spectral observations with limited wavelength coverage. Hence, it is necessary to extend the sample spectra to bluer and redder wavelengths. First, for each observed spectrum, the Hsiao revised template is color-matched to synthetic colors calculated from the observed spectrum. The color-matched template spectrum was then scaled to match the integrated flux under the bluest (or reddest, in the case of the K_s band) filter covered by the observed spectrum. The template and observed spectra are then combined utilizing an overlapping 300 Å region with appropriate weighting. Note that the extended sections are not used in the K-correction calculations, but serve only for color matching.

An example of an original and color-matched spectrum of a normal SN Ia that has been extrapolated in the blue is shown in Figure 2.

**Fig. 2.** *(top panel)* Comparison of the maximum light spectrum of SN 2011fe (*solid black line*) to its altered version (*dotted red line*) that is matched to the observed colors derived from the Hsiao revised spectral template at z = 0.03. Overplotted in the top panel are the CSP *YJHK*_s passbands at rest wavelength. (*bottom panel*) The spline function used to perform the color matching of the input spectrum to match the colors calculated from the Hsiao revised spectral template. .

3.4. Gaussian Process and MCMC Modeling

We have adopted a Bayesian formalism to compute an interpolated function of the K term as a function of light-curve phase, redshift, and filter, along with a robust estimate of its uncertainty, which is more accurate than what is obtained by just simply binning residuals, particularly in the presence of outliers. In this manner the computed error snake is a combination of (1) observational error, (2) the differing number of points constraining the interpolator at any particular phase and redshift, and (3) intrinsic differences between individual SN Ia. These last two errors are particularly difficult to disentangle, prompting us to opt for a combined Gaussian Process and Markov-Chain Monte-Carlo (MCMC) method to obtain an empirically-based model and an associated error estimate.

Gaussian Processes (Rasmussen & Williams 2006) are a generalization of probability distributions for functions. Instead of a mean value, one solves for a mean function, F(t), and in addition, a covariance function, [C(t₁,t₂)]. The covariance function is a generalization of the standard deviation of a Gaussian distribution and encapsulates the uncertainty in the mean function. While the mean function is nonparametric, we elected to use the Matérn covariance function (Banerjee, Carlin, & Gelfand 2003), which has four parameters: scale (s), amplitude (θ), the degree of differentiability (d), and an intrinsic variance (σ_var). The value of s sets the scale over which departures from the mean function are coherent, while θ determines the size of these departures, d controls the degree of differentiability (or smoothness) of the function, and σ_var characterizes any additional uncorrelated dispersion of the interpolated function that exceeds the uncertainty of the K corrections. To implement this approach, the python package PyMC (Patil, Huard, & Fonnesbeck 2010) was used to compute the best values for s, θ, d, and σ_var, as well as F(t) via the MCMC method.

The advantage of MCMC modeling is that it allows one to marginalize over the adopted assumptions and missing information, and at the same time provides a robust error snake. At each MCMC step, s, θ, and d are proposed and a Gaussian Process is computed which gives an interpolation between the observed K corrections. The residuals between the observed points and the interpolated function are then determined. If the residuals are consistent with the errors of the individual K corrections, then the MCMC process has found a higher likelihood state. If the residuals are larger than the errors, a lower likelihood state is obtained. The Metropolis-Hastings algorithm is used to propose each new MCMC step, randomly walking through parameter space, though converging to higher probability states. There are two states where the likelihood is locally maximized: (1) s and d are small, θ is large, and the interpolator fits the noise; and (2) s and d are larger, θ smaller, and the interpolator smoothly fits the data, though with larger scatter than the errors in the data should allow. Introducing the extra noise term σ_var gives the second case a higher probability and also gives us an estimate of the intrinsic scatter in the K corrections.

Once the Markov chain has converged, all the states are tallied, and for each state, i, the interpolating function, F_i(t), is computed. The median of all F_i(t) represents the final F(t), while the standard deviation of the interpolations gives the "error in the mean" function, which we refer to in the following as σ_stat. Due to the varying density of data points used to constrain the interpolation, the value of σ_stat varies with both redshift and temporal phase. The computed value of σ_var is held constant for each redshift bin since, given the size of the current library of NIR spectra, there simply is not enough data to obtain an estimate as a function of temporal phase. We are unable to directly attribute σ_var to any one variable, but this parameter most likely reflects intrinsic SN-to-SN differences such as spectral line diversity. It is our view that a combination of both noise terms (σ_stat and σ_var) represents the most appropriate error snakes to adopt if one intends to include the K-term uncertainty in the overall error budget of cosmological parameters derived from NIR observations of SNe Ia.

4. Results

Armed with the YJHK_s passbands shown in Figure 1 and our adopted library of NIR SN Ia spectra (see Table 1), K corrections were computed using equation (2) for redshifts of z = 0.03, 0.05, 0.08. The arrays of K corrections for each of these redshift intervals were then fed into our MCMC algorithm, from which smooth mean functions were computed. The definitive K-correction interpolations and their associated values of σ_stat are provided in Table 2. Listed in Table 3 are the computed values of σ_var, which dominate the total errors for all four passbands near maximum light.

Illustrations of the computed K corrections and their mean interpolated functions as a function of temporal phase are presented in Figure 3 for the three adopted redshifts. In each panel the points correspond to the individual K corrections derived from equation (2). The solid black lines are the MCMC mean functions, and the shaded regions are confidence levels of these functions. The darker of the two shaded regions corresponds to σ_stat, while the less dark regions are obtained through the summation in quadrature of σ_stat and σ_var. As expected, the overall evolution of the K terms in Y, J, and H as a function of phase, mimic to first order the evolution of the (Y - J), (J - H), and (H - K) color curves.

**Fig. 3.** K corrections in the *YJHK*_s bandpasses (*blue points*) computed from 75 NIR SN Ia spectra at z = 0.03, 0.05, and 0.08. Solid black line corresponds to MCMC interpolated functions, while dark and light shaded regions correspond to σ_stat, and the summation in quadrature of σ_stat and σ_var, respectively. .

We would emphasize that the K terms given in Table 2 are listed for reference only, and should not be used blindly to correct NIR photometry for any given SN. Rather, we recommend a two-step process. The chosen spectral template should first be color-matched to the SN photometry and then used to calculate K corrections for the filter bandpasses employed. The uncertainties given in Tables 2 and 3 should then be interpolated to the redshift of the SN and propagated as part of the K-correction process. As mentioned in § 3.4, we recommend summing in quadrature the values of σ_stat and σ_var to estimate the total error in the K correction.

In the remainder of this section, we discuss results for each of the four CSP filter bandpasses.

4.1. Y and J Filters

Figure 3 shows that, of the four CSP bandpasses, Y yields the lowest K-correction uncertainties. This is due in large part to the fact that this wavelength region does not develop any strong spectral features until ∼30 days after T(B_max) (see Fig. 9 of Marion et al. [2009]). This is reflected in the relatively small increase σ_var from ± 0.02 to ± 0.04 mag between z = 0.03 and 0.08 (see Table 3 and Fig. 4). Moreover, the Y band is not affected by strong telluric absorption, and is generally characterized by a higher signal-to-noise ratio than any of the other NIR bandpasses. The small K-correction uncertainties combined with an essentially negligible dependence of absolute magnitude on decline rate (Kattner et al. 2012) and low sensitivity to dust extinction make the Y band an interesting option for future cosmological studies. Note, however, that to obtain the highest precision in the Y band, it is important that the photometric coverage of the SN extend to at least the i (or I) filter since this is essential for proper color-matching of the spectral template.

**Fig. 4.** Plot of the uncertainties most likely due to spectral diversity, σ_var, versus redshift.

The K-correction models indicate that the J passband is also characterized by relatively small uncertainties, although not quite at the low levels observed for the Y band. The statistical errors of the J-band K terms are slightly higher than those for Y and, as indicated in Table 3 and illustrated in Figure 4, the uncertainties due to intrinsic spectral variations are also ∼50% greater. Nevertheless, the J filter clearly offers considerable promise for cosmological studies.

4.2. H Filter

Kasen (2006) has argued from theoretical grounds that the spread in the absolute magnitudes of SNe Ia should decrease steadily from the optical to the NIR, reaching a minimum dispersion of ∼ ± 0.1 mag at H, and Wood-Vasey et al. (2008) concluded the same based on observations of 21 SNe Ia in in JHK. Nevertheless, Figure 3 shows that the dispersion in the H-band K corrections increases much more sharply with redshift than for the Y and J filters, with the error snakes indicating that the increases occur in both σ_stat and σ_var. The increase in σ_stat is predictable since the library spectra all correspond to low-redshift (z ≲ 0.03) SNe. As illustrated in Figure 1, the wavelength region of the library spectra affected by the telluric absorption band at 1.34–1.41 μm is shifted more and more into the blue edge of the H passband at increasing redshift, introducing increasing levels of noise into the K-correction calculation due to errors in the telluric corrections. These statistical errors can be beaten down as NIR spectra are obtained of SNe Ia at a larger range of redshifts.

As shown in Figure 4, the increase in σ_var with redshift for the H band is greater than that suffered by any of the other CSP NIR filters, reaching a value of nearly ± 0.1 mag at z = 0.08. This behavior is most likely the result of intrinsic diversity in the strong iron-peak emission features that strengthen dramatically in the H band beginning only a few days after T(B_max) (Wheeler et al. 1998). Hsiao et al. (2013) recently studied the strength of the "break" at ∼1.5 μm, and found evidence that it correlates with Δm₁₅(B). If confirmed, it should be possible to eventually decrease σ_var by creating NIR spectral templates in the H band that are a function of Δm₁₅(B). This, however, will require a much larger library of NIR spectral observations.

4. 3. K_s Filter

In principle, the K_s bandpass also offers significant advantages for measuring cosmological distances. Dust extinction is lowest at this wavelength, and the increase of σ_var with redshift is only ± 0.01 mag greater than derived for the J band (see Table 3 and Figure 4). However, the signal-to-noise ratio typically achieved in the K_s band is less than in the other NIR filters (see Fig. 9 of Marion et al. [2009]). This is due to both the faintness of the SNe and the much stronger sky emission encountered at these longer wavelengths. Secondly, similar to the H band, K-correction calculations for the K_s filter suffer from redshift-dependent noise introduced by the strong telluric absorption band at 1.8–1.9 μm (see Fig. 1) although, again, it should be possible to minimize the impact of this effect as NIR spectra of SNe Ia are obtained over a larger range of redshifts. A final disadvantage of the K_s bandpass is the difficulty of properly color-matching it to a spectral template since observations are not generally available in a longer-wavelength filter to help anchor the correction.

5. Conclusions

Using a library of publicly available SNe Ia NIR spectra, we have computed a set of K corrections in the CSP YJHK_s filters at redshifts of z = 0.03, 0.05, and 0.08. The individual spectra were first color-matched to the Hsiao revised spectral template before calculating the K corrections. A combined Gaussian Process and MCMC method was then employed to derive an empirically-based model of the K terms as a function of temporal phase and redshift. This procedure returns uncertainties in the K-correction model due to both statistical noise and intrinsic diversity in the features of the input spectra.

Krisciunas et al. (2004) published a table of NIR K corrections computed using spectra of SN 1999ee, and a set of filters functions similar to those adopted in this study. A comparison between the results of Krisciunas et al. to those obtained with our expanded set of spectra and more advanced color-matching technique provides good agreement in the J and H bands at z = 0.03, while for the K_s band we find differences of up to ± 0.08 mag. The discrepancy in the K_s is not surprising owing the the dearth of spectra included in the work of Krisciunas et al., as well as the difficulties highlighted in this study concerning the problems inherent to computing robust K_s-band K corrections.

Our emphasis in this paper has not been to provide "lookup" tables of NIR K corrections, but rather to derive accurate estimates of the uncertainties inherent in the use of spectral templates (like the Hsiao revised template) constructed from the NIR sample of SNe Ia spectra available today. We find that the Y and J bands currently afford the greatest precision in K-correction calculations due to the general weakness of the spectral features and the minimal effect of telluric corrections. The Y band is particularly noteworthy, with the uncertainty due to spectral feature variations increasing from only ± 0.02 to ± 0.04 mag from z = 0.03–0.05. The H band is currently more problematic due to noise introduced by telluric corrections and the appearance just after maximum light of strong Fe-peak emission features. Diversity in the strength of this emission most likely explains the strong increase we find in the intrinsic component of the K-correction error from ± 0.02 to ± 0.10 mag over the redshift range z = 0.03–0.05. This result is a departure from pervious theoretical (Kasen 2006) and observational (Wood-Vasey et al. 2008) findings, which have identified the H band as having potentially the least dispersion among the NIR passbands. Finally, the uncertainties in the K terms for the K_s filter due to intrinsic spectral diversity are similar to those found for the J band, but the currently-available library of spectra covering this wavelength region suffers from both low signal-to-noise ratios and telluric absorption.

Further progress in reducing the uncertainties in the K corrections for NIR filter bandpasses demands many more spectral observations over a wider range of redshifts. Figure 5 shows a histogram of the decline rates of the SNe Ia in our sample with temporal phases within ± 3 days of T(B_max). For reference, the same figure includes a plot of the decline rates of the spectra employed by Hsiao et al. (2007) to create their optical spectral template. The discrepancy between these two histograms dramatically illustrates the limitations of the currently-available library of NIR spectra. To address this problem, we have embarked on a four-year, six-months-a-year SN Ia followup program that builds upon the legacy of the CSP. This project, designated CSP-II, is designed to obtain optical and NIR light-curves of ∼100 SNe Ia in the smooth Hubble flow (0.03 < z < 0.08). Complementary to these observations, frequent NIR spectroscopy is being carried out of nearby SNe Ia, mainly with the Folded-port InfraRed Echellette (FIRE) spectrograph mounted on the 6.5-m Magellan Baade telescope. Particular emphasis is being placed on obtaining spectral sequences for a subsample of the SNe that span a range in decline rate and phase, which are essential to determining correlated errors in the K corrections. We are confident that such observations will not only allow us to more precisely characterize NIR K-correction uncertainties, but also to decrease these errors to the low levels now obtained at optical wavelengths.

**Fig. 5.** Histogram of the decline rates of the published NIR spectra of SNe Ia in our sample obtained within ± 3 days of T(B_max). A comparable histogram for the optical spectra in the Hsiao et al. (2007) template over the same range of epochs is shown for comparison.

In order to ascertain the contribution of NIR K-correction errors to the final error budget of cosmological parameters one must determine whether they are systematic in temporal phase, and therefore propagate to the peak magnitude of a SN Ia. On the other hand, if the K-correction errors are random in phase then the final uncertainty in the peak magnitude will be reduced by template fitting. To determine which of these is the case is, however, beyond the scope of this paper due to the lack of a statistical sample of SNe Ia with good spectral time series. In the near future we will be able to tackle this issue with the high-fidelity spectral series currently being obtained.

M. D. S. gratefully acknowledges generous support provided by the Danish Agency for Science and Technology and Innovation realized through a Sapere Aude Level 2 grant. We also acknowledge support from the USA's NSF through grants AST–0306969, AST–0607438, and AST–1008343. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.

Near-Infrared K Corrections of Type Ia Supernovae and their Errors

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Abstract

1. Introduction

2. THE K CORRECTION