36x48 vertical poster template

1
DUST EXTINCTION MODEL
For source frame x 1µm/? lt 1.82, we use the CCM
extinction model where y x
1.82. For 3.3 lt x lt 10.96, we use the FM
extinction model For 1.82 lt x lt 3.3, we
use a weighted average of the two models. For x
gt 10.96 (Lyman limit), absorption due to neutral
hydrogen in the host galaxy is total (A? 8).
In the AMP dust extinction model, c2, c4 and AV
are free parameters c1, UV bump height c3/?2,
and RV are constrained by priors on the
correlation model parameters (see figures). ? and
x0 are constrained by Gaussian priors whose
widths s? and sxo themselves have Gaussian
priors AMP dust extinction model
curve (Reichart 2001)
STATISTICS
Given a two dimensional data set of N points in
the x-y plane with intrinsic uncertainties in
both dimensions xn,yn sx,n,sy,n, Bayes
theorem allows us to compute the probability of
a hypothetical model distribution, described by a
curve defined by M parameters yc(x?m) and
extrinsic sample variance (sx,sy), given any
prior constraints on the parameters. If we
assume the intrinsic errors and sample variance
have Gaussian probability distributions, and the
prior distributions are flat, the best-fit model
is found by maximizing the likelihood where

and For a given data point, the path
integral along the curve yc is approximated by
finding the point at which the curve is tangent
to an ellipse with axes proportional to Sx, Sx,
and integrating along a line through that point
with slope mt (bottom right figure). The result
of this integral depends upon the choice of the
differential element ds. DAgostini (2005) D05
uses dsdx. Reichart (2001) R01 uses
We present a
new statistic TRF09 where the differential
element is projected onto a line perpendicular
to the segment connecting the data point and the
tangent point of the curve to the intrinsic
error ellipse the angle between this line and
the tangent line is f, where the tangent line has
slope mt' tan?' (top right figure). Two
criteria motivate our choice 1) The statistic
should invertible i.e., if xn ? yn and sxn ?
syn the best-fit model parameters should
describe the curve xc (y) yc-1 (x). 2) The
statistic should reduce to the traditional 1D ?2
statistic when sx 0 or sy 0. Below is a
summary of the properties of the D05, R01 and
TRF09 statistics for the case of a linear model
yc mx b. Note that for a linear model, mt
m't m.
Top model curve distribution and tangent line
with respect to the intrinsic error ellipse of
the data point. Bottom After adding intrinsic
and extrinsic errors in quadrature, the tangent
line changes. Shaded areas indicate 1, 2 and 3s
confidence regions. The blue line provides the
differential element for the likelihood function
path integration.
H2 ABSORPTION MODEL
Linear fits to a symmetric Gaussian random
distribution of width s1 in (x, y) of 1000 data
points with small, random errorbars sx ? sy
0.01. Left Fit to x vs. y using D05, R01 and
TRF09 statistics. Center Fits to y vs. x for
the same data set. Note that the D05 and R01
fits are invertible (mxy1/myx), while D05 is
biased towards m0 in both fits. Right
Probability distribution of ? tan-1(m) for
ensembles of fits to Gaussian random clouds of
N1000 points with symmetric errorbars sx sy.
pR01(?)pTRF09(?)constant, while pD05(?) ? cosN?.
Lya FOREST MODEL
Theoretical absorption spectra due to
rovibrationally excited molecular hydrogen for
column densities NH2 1016, 1018 and 1020 cm-2,
for clouds fully shielded from (left) and exposed
to (right) Lyman continuum radiation from the GRB
(Draine 2000). For computational simplicity, we
approximate these spectra with broken line
absorption profiles (dashed lines). AMP models
H2 absorption by interpolating between these
broken lines, with column density NH2 and
fractional Lyman continuum illumination 0 ? XLyc
? 1 as free parameters.
Lya transmission T vs. absorber redshift z.
Points are flux deficits for 64 QSOs, measured in
binned regions of their spectra redward of Lya in
the source frame (Fan et al. 2006 Songaila
2004). The solid line is the best-fit to the
model Shaded regions are the 1, 2 and 3s
confidence intervals of the fit. AMP models Lya
forest absorption using the parameters a, b, c, d
and sample variance sz with Gaussian priors,
along with a prior source redshift zGRB from
spectral or other observations. IGM absorption
model priors are The redshifts z1 3.8826
and z2 6.2279 were chosen to minimize
correlations among fitted parameters. Note the
onset of the Gunn-Peterson trough at z 6. The
AMP model also includes a damped Lyman absorber
profile at the source redshift, whose shape is
parameterized by the neutral hydrogen column
density NH, and total absorption at wavelengths
shorter than the Lyman limit ? lt 912Å in the
source frame. NH may be fit as a free parameter,
or with prior constraints from X-ray or other
observations, when available.
REFERENCES
Cardelli, J. A., Clayton, G. C. Mathis, J. S.
1989. The Relationship between Infrared, Optical,
and Ultraviolet Extinction. Astrophysical Journal
345245-256. D'Agostini, G. 2005. Fits, and
Especially Linear Fits, with Errors on Both Axes,
Extra Variance of the Data Points and Other
Complications. arXivphysics/0511182.
D05 Draine, B. T. 2000. Gamma-Ray Bursts in
Molecular Clouds H2 Absorption and Fluorescence.
Astrophysical Journal 532273-280. Fan, X., et
al. 2006. Constraining the Evolution of the
Ionizing Background and the Epoch of Reionization
with z6 Quasars. II. A Sample of 19 Quasars.
Astronomical Journal 132117-136. Fitzpatrick, E.
L. Massa, D. 1988. An Analysis of the Shapes of
Ultraviolet Extinction Curves. II - The Far-UV
extinction. Astrophysical Journal
328734-746. Fitzpatrick, E. L. Massa, D. 1990.
An Analysis of the Shapes of Ultraviolet
Extinction Curves. III - An Atlas of Ultraviolet
Extinction Curves. Astrophysical Journal
Supplement Series 72163-189. Gordon, K. D.,
Clayton, G. C., Misselt, K. A., Landolt, A. U.
Wolff, M. J. 2003. A Quantitative Comparison of
the Small Magellanic Cloud, Large Magellanic
Cloud, and Milky Way Ultraviolet to Near-Infrared
Extinction Curves. Astrophysical Journal
594279-293. Reichart, D. E. 2001. Dust
Extinction Curves and Lya Forest Flux Deficits
for Use in Modeling Gamma-Ray Burst Afterglows
and All Other Extragalactic Point Sources.
Astrophysical Journal 553235-253.
R01 Schlegel, David J., Finkbeiner, D. P.
Davis, M. 1998. Maps of Dust Infrared Emission
for Use in Estimation of Reddening and Cosmic
Microwave Background Radiation Foregrounds.
Astrophysical Journal 500525. Songaila, A. 2004.
The Evolution of the Intergalactic Medium
Transmission to Redshift 6. Astronomical Journal
1272598-2603. Valencic, L. A., Clayton, G. C.
Gordon, K. D. 2004. Ultraviolet Extinction
Properties in the Milky Way. Astrophysical
Journal 616912-924.