This is a project to infer type Ia supernova (SN Ia) spectrophotometric templates based on filter photometry only. It is partly an academic exercise in Bayesian inference, but should also have concrete applications in the era of wide-area transient surveys, such as LSST.
SNe Ia are used in astronomy to measure distances to faraway galaxies, since they are luminous (as bright as the galaxy in which they appear) and because their luminosity can be predicted with 10-15% precision. They result from thermonuclear explosions of carbon/oxygen white dwarfs, brightening and fading over timescales of a few weeks. Distances are measured using the inverse square law, comparing the known luminosity of a supernova with its apparent brightness as measured. This technique enables the accurate measurement of the expansion history of the Universe: as the Universe expands by a scale factor a(t), the wavelength of all light within it is redshifted by a factor z = 1/a - 1. Given that light travels at a constant speed c, measuring redshift z as a function of distance d is thus equivalent to measuring a as a function of time t = d/c. Measurements based on SNe Ia were used to discover the accelerating expansion of the Universe, which won the 2011 Nobel Prize in Physics. The underlying cause of the expansion (commonly called "dark energy") is still unknown, and SN Ia cosmology is still a very active area of research.
Measurements of the dark energy density in the Universe require observations from a large number of SNe Ia spread across a wide range of redshifts. Observations are usually made in optical wavelengths using glass filters, which capture only the average flux within some wavelength window at a specific point in time; a time series of these points is called a light curve. The spectrum of a SN Ia is complex, with many broad atomic lines that also vary in time, so that the observed brightness depends strongly on the redshift of the SN, the time since explosion, and the filter used. Combining observations of SNe Ia for cosmology can therefore be done only using a full model of the time-evolving spectral energy distribution, which forms the core of any parametrized light curve fitting procedure.
The current gold standard for SN Ia light curve fitters, SALT2 ([Guy et al. 2007] (http://adsabs.harvard.edu/abs/2007A%26A...466...11G), [2010] (http://adsabs.harvard.edu/abs/2010A%26A...523A...7G)), uses a combination of photometric and spectroscopic data for training. The photometry captures only average fluxes over intervals. In principle a spectrum can be integrated over a set of filter transmission curves to produce photometric fluxes (synthetic photometry), but flux calibration of spectra is usually highly uncertain, and so only the relative depths of atomic line features are reliable. Systematic errors therefore creep in due to uncertain calibration of the spectra to match the filter photometry, along with uncertainty in the photometry itself, and the need for spectroscopic coverage in wavelength ranges that can be difficult or expensive to obtain (e.g. infrared spectra, which requires long exposures using large telescopes at very stable sites, and ultraviolet spectra, which must be obtained from space). SALT2 and other light curve fitters are also currently formulated as linear models, but a great deal of rich structure in the underlying high-dimensional distribution of the data may remain untapped.
[Guy et al. 2007] (http://adsabs.harvard.edu/abs/2007A%26A...466...11G) suggest in their paper the future possibility of deconvolving the full SN Ia spectrophotometric time series. Photometric datasets of SNe Ia are now so extensive, and computing power sufficiently cheap and advanced, that we can now contemplate doing this. The advantages include:
- A more advanced model structure, using Gaussian processes in place of binned spectra and capturing full covariances of all observations rather than just mean square errors.
- A fully Bayesian treatment of all aspects of the model, including the potential to self-calibrate by treating the filter transmissions and calibration spectra as random variables.
- Removal of limitations based on spectroscopic coverage, and inference of mean spectroscopic behavior at all wavelengths for subclasses of SNe Ia with sufficient photometric coverage.
- Inference over latent parameters, giving a data-driven, low-dimensional summary of SN Ia spectroscopic behavior. Complex latent relationships such as graphical models can be considered here, extending the current simple linear regression models used. Such representations may also be used to mine SN Ia physics when compared with numerical simulations.
The current plan is to use a Gaussian process (GP; [Rasmussen & Williams 2006] (http://gaussianprocess.org/)) over wavelength and time to represent the (latent) spectral time series. This latent time series is linked to the observations by transformations of the GP prior ([Murray-Smith & Pearlmutter 2005] (http://link.springer.com/chapter/10.1007/11559887_7)) corresponding to synthetic photometry; the filter transmission curves can be compressed via Bayesian quadrature to make this more tractable (O'Hagan 1991; Huszar & Duvenaud 2012). Eventually the GP parameters themselves will admit additional hyperpriors (such as a graphical model) to extract a compressed representation of the full spectrophotometric time series. The main parameter will correspond to the light curve decline rate, as noted extensively in the SN Ia literature (e.g. (Phillips 1993)[http://adsabs.harvard.edu/abs/1993ApJ...413L.105P], (Nugent et al. 1995)[http://adsabs.harvard.edu/abs/1995ApJ...455L.147N]), but further parameters are likely to be of greater interest physically, and could result in improved luminosity distances overall.