Models¶

By design, the specification of models is done in its own namespace: model. All model classes are subclasses of model.model. In the following sections, I detail what each model can be used for and the mathematics behind it.

SNooPy now has two different parametrizations for the shape of the light-curves. One is the traditional $\Delta m_{15}$ of [Phillips 1999]_ and the newer ‘color-stretch’ introduced in [Burns+2011], which more consistently deals with fast-declining events. Note that the EBV_model will only work with $\Delta m_{15}$ -based templates.

EBV_model¶

This light-curve model is a variation of that given by [Prieto+2006]. It is most useful for determining a distance using the Phillips relation and correcting for dust extinction in the host. We compare the observed data in $N$ filters to the following model:

$\begin{eqnarray*} m_{X}\left(t\right) & = & T_{Y}\left(t^{\prime},\Delta m_{15}\right)+M_{Y}\left(\Delta m_{15}\right)+\mu+R_{X}E\left(B-V\right)_{gal}+\\ & & R_{Y}E\left(B-V\right)_{host}+K_{X,Y} \end{eqnarray*}$

where $m_{X}$ is the observed magnitude in band $X$ , $t$ is the observed time relative to B maximum, $t^\prime$ is the de-redshifted time relative to B maximum, $\Delta m_{15}$ is the decline rate parameter [Phillips1999], $M_{Y}$ is the absolute magnitude in filter $Y$ in the rest-frame of the supernova, $E(B-V)_{gal}$ and $E(B-V)_{host}$ are the reddening due to galactic foreground and host galaxy, respectively, $R_{X}$ and $R_{Y}$ are the total-to-selective absorptions for filters $X$ and $Y$ , respectively, $K_{XY}$ is the cross-band k-correction from rest-frame $X$ to observed filter $Y$ . Note that the k-corrections depend on the epoch and can depend on the host and galaxy extinction (as these modify the shape of the spectral template). In the fitting, one has the choice of the spectral template of [Nugent+2002] or [Hsiao+2007]. The latter is the default. You can choose to allow the K-corrections to vary during the fit, keep them fixed, or not use them at all. See the parameters for the fit() function.

The template $T\left(t,\Delta m_{15}\right)$ can be generated from the code of citet{2006ApJ...647..501P} or citet{Burns2011}. In the former case, you will be fitting to rest-frame $B_{s}V_{s}R_{s}I_{s}$ [1] while in the latter case, you can fit to $uBVgriYJH$ from the CSP data [Contreras+2010]. You can mix and match which template generator you use: it is all a matter of which filter you choose for the restbands instance variable (see Doing the Fit). In all, this model fits 4 parameters: Tmax, dm15, EBVhost, and DM.

Note that to determine the host reddening, you need to fit at least two distinct restframe filters. For now, I’ve left the galactic and host reddening laws ( $R_{X}$ and $R_{Y}$ ) as member variables rather than parameters to be fit [2]. This could change in the future.

EBV_model2¶

This is the same model as EBV_model, except that it can fit both $\Delta m_{15}$ - and $s_{BV}$ -based templates. And instead of the calibration presented in [Folatelli+2010], the calibration from the upcoming CSP analysis paper [Burns+2011] is used. When using the choose_model function to select this model, you can specify stype='dm15' or stype='st' to select the two different light-curve parameters.

max_model¶

An alternative to assuming some kind of relationship between the different filters (i.e., some kind of reddening law as is done in the EBV_model), one can simply fit the maximum magnitude of each light-curve. This is how the max_model model works. It uses the same light-curve templates as EBV_model, but instead fits the following model:

$m_{X}\left(t\right) = T_{Y}\left(t^\prime,\Delta m_{15}\right)+m_{Y}+R_{X}E\left(B-V\right)_{gal}+K_{X,Y}$

where $m_{Y}$ , the peak magnitude in filter $Y$ is now a parameter. Of course, depending on the number of filters you fit, you will have that many $m_{Y}$ parameters. Note that K-corrections and Milky-way extinction are performed exactly as in the EBV_model case. Therefore, for $N$ filters, max_model will have $N+2$ parameters: Tmax, dm15, and $N$ emph{f}``max``, where emph{f} is the rest-band filter name (so if, for instance, you fit an observed light-curve to restframe $B$ , there will be a Bmax parameter). As with the EBV_model2 model, you can choose which light-curve parameter you wish to use by specifying the stype argument to choose_model().

NOTE: Please be aware that any non-linear fitter will only find a emph{local} minimum in $\chi^{2}$ . It is up to you, the user, to try different starting points in parameter space and see which one gives the overall best-fit. After each fit, the member variable rchisq is updated with the reduced- $\chi^{2}$ . You can therefore use this to discriminate between different solutions.

color_model¶

In this model, intrinsic colors from [Burns+2014] are used to infer the amout of extinction $E(B-V)$ as well as the shape reddening law $R_{V}$ . Mathematically, the model being fit to the light-curves is:

$\begin{eqnarray*} m_{X}\left(t\right) & = & T_{Y}\left(t^{\prime},\Delta m_{15}\right)+B_{max}+(X-B)\left(s_{BV}\right)+R_{X}E\left(B-V\right)_{gal}+R_{Y}\left(R_{V}\right)E\left(B-V\right)_{host} + \\ & & K_{X,Y} \end{eqnarray*}$

where $B_{max}$ is the de-reddened and K-corrected $B$ maximum (treated as a free parameter) and $(X-B)\left(s_{BV}\right)$ is the intrinsic $X-B$ color, which is a function of $s_{BV}$ . In [Burns+2014] it is modeled as a 2nd degree polynomial in $\left(s_{BV}-1\right)$ . All other variables have the same meaning as in previous models. The model has 5 free parameters: $s_{BV}$ , $T_{max}$ , $B_{max}$ , $E(B-V)$ , and $R_{V}$ . Note that the distance modulus is not included in this model.

A major complication of this model is that two parameters, $R_{V}$ and $E(B-V)$ , appear as multiplicative factors in the same term. At best, they will be highly covariant. At worst (for low values of $E(B-V)$ ), the model becomse insensitive to $R_{V}$ . For this reason, it may be necessary to impose priors on $R_{V}$ . Two such priors were introduced in [Burns+2014]: a Gaussian mixture model that applies to all SNeIa, and a binned prior, where a separate Gaussian prior is applied to the SNIa depending on its value of $E(B-V)$ . Because SNooPy uses the LM least-squares algorithm by default, there is no natural way to incorporate these priors using the standard fit() routine. Instead, use the fitMCMC() routine, which fits the light-curves using Markov Chain Monte Carlo and allows priors to be specified on parameters. In this case, use the keyword argument rvprior='mix' for the Gaussian mixture model, or rvprior='bin' for the binned prior. See section Doing the Fit for more details.

[Phillips1999]

Phillips, M.M., AJ, 118, 1766 (1999) http://adsabs.harvard.edu/abs/1999AJ....118.1766P

[Nugent+2002]

Nugent et al., PASP, 114, 803 (2002). http://adsabs.harvard.edu/abs/2002PASP..114..803N

[Hsiao+2007]

Hsiao et al., ApJ, 663, 1187 (2007). http://adsabs.harvard.edu/abs/2007ApJ...663.1187H

[Stritzinger+2005]

Stritzinger et al., PASP, 117, 810 (2005) http://adsabs.harvard.edu/abs/2005PASP..117..810S

[Contreras+2010]

Contreras et al., AJ, 139, 519 (2010). http://adsabs.harvard.edu/abs/2010AJ....139..519C

[Burns+2011]

(1, 2) Burns et al., AJ, 141, 19B (2011). http://adsabs.harvard.edu/abs/2011AJ....141...19B

[Burns+2014]

(1, 2, 3) Burns et al., ApJ, 789, 32B (2014). http://adsabs.harvard.edu/abs/2014ApJ...789...32B

[1]	The ‘s’ subscript refers to ‘standard’, which is to say the Bessel filters from [Stritzinger+2005]

[2]	So far, data I’ve analyzed hasn’t been good enough to distinguish between reddening laws.

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Models¶

EBV_model¶

EBV_model2¶

max_model¶

color_model¶