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Register a new userSpectrophotometric Redshifts. A New Approach to the Reduction of Noisy Spectra and its Application to GRB090423
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We have developed a new method, close in philosophy to the photometric redshift technique, which can be applied to spectral data of very low signal-to-noise ratio. Using it we intend to measure redshifts while minimising the dangers posed by the usual extraction techniques. GRB afterglows have generally very simple optical spectra over which the separate effects of absorption and reddening in the GRB host, the intergalactic medium, and our own Galaxy are superimposed. We model all these effects over a series of template afterglow spectra to produce a set of clean spectra that reproduce what would reach our telescope. We also model carefully the effects of the telescope-spectrograph combination and the properties of noise in the data, which are then applied on the template spectra. The final templates are compared to the two-dimensional spectral data, and the basic parameters (redshift, spectral index, Hydrogen absorption column) are estimated using statistical tools. We show how our method works by applying it to our data of the NIR afterglow of GRB090423. At z ~ 8.2, this was the most distant object ever observed. We use the spectrum taken by our team with the Telescopio Nazionale Galileo to derive the GRB redshift and its intrinsic neutral Hydrogen column density. Our best fit yields z=8.4^+0.05/-0.03 and N(HI)<5x10^20 cm^-2, but with a highly non-Gaussian uncertainty including the redshift range z [6.7, 8.5] at the 2-sigma confidence level. Our method will be useful to maximise the recovered information from low-quality spectra, particularly when the set of possible spectra is limited or easily parameterisable while at the same time ensuring an adequate confidence analysis.
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The Gaussian mixture distribution is important in various statistical problems. In particular it is used in the Gaussian-sum filter and smoother for linear state-space model with non-Gaussian noise inputs. However, for this method to be practical, an efficient method of reducing the number of Gaussian components is necessary. In this paper, we show that a closed form expression of Pearson chi^2-divergence can be obtained and it can apply to the determination of the pair of two Gaussian components in sequential reduction of Gaussian components. By numerical examples for one dimensional and two dimensional distribution models, it will be shown that in most cases the proposed criterion performed almost equally as the Kullback-Libler divergence, for which computationally costly numerical integration is necessary. Application to Gaussian-sum filtering and smoothing is also shown.
A new approach is given for the implementation of boundary conditions used in solving the Mukhanov-Sasaki equation in the context of inflation. The familiar quantization procedure is reviewed, along with a discussion of where one might expect deviations from the standard approach to arise. The proposed method introduces a (model dependent) fitting function for the z/z and a/a terms in the Mukhanov-Sasaki equation for scalar and tensor modes, as well as imposes the boundary conditions at a finite conformal time. As an example, we employ a fitting function, and compute the spectral index, along with its running, for a specific inflationary model which possesses background equations that are analytically solvable. The observational upper bound on the tensor to scalar ratio is used to constrain the parameters of the boundary conditions in the tensor sector as well. An overview on the generalization of this method is also discussed.
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