RXJ1131-1231
Model requires external shear, consistent with nearby foreground cluster. Include shear in the ray tracing κ ext analysis
12 Suyu et al.
Fig. 5.— 11.5!×10.5!R-band image obtained from stacking 60 hours of the best-quality images in the COSMOGRAIL monitoring. The lens system is marked by the box near the center. Galaxies (stars) in the field are indicated by solid (dashed) circles. The radius of the solid circle is proportional to the flux of the galaxy. X-ray map from Chartas et al. (2009) are overlaid on the image within the dashed box. The concentrations of mass structures to the east of the lens are consistent with the modeled external shear and convergence gradient directions.
counts in a 45
!!aperture having I-band magnitudes be-tween 18.5 and 24.5. These provide samples for the PDF P (κ
ext, γ
ext, d
env|MS). We assume that the constructed PDF is applicable to strong-lens lines of sight, following Suyu et al. (2010) who showed that the distribution of κ
extfor strong lens lines of sight are very similar to that for all lines of sight.
Structures in front of the lens distort the time delays and the images of the lens/source, while structures be-hind the lens further affect the time delays and images of the source. However, to model simultaneously the mass distributions of the strong lens galaxies and all structures along the line of sight is well beyond the current state of the art. In practice, the modeling of the strong lens galaxies is performed separately from the description of line-of-sight structures, and we approximate the effects of the lines-of-sight structures into the single correction term κ
ext, whose statistical properties we estimate from the Millennium Simulation.
By selecting the lines of sight in the Millennium Sim-ulation that match the properties of RXJ1131−1231, we can obtain P (d
env|κ
ext, γ
ext, MS) P (κ
ext) and simultane-ously marginalize over γ
extin Equation (10). We as-sumed a uniform prior for γ
extin the lensing analysis, such that P (γ
ext) is a constant. The lensing likelihood is the only other term that depends on γ
ext, and from Sec-tion 6.4, the lensing likelihood provides a tight constraint on γ
extthat is approximately Gaussian: 0.089 ± 0.006.
We can therefore simplify part of Equation (10) to
! dγ
extP (d
ACS, ∆t|D
∆t, γ
!, θ
E, γ
ext, κ
ext)
·P (d
env|κ
ext, γ
ext, MS)
" P (d
ACS, ∆t|D
∆t, γ
!, θ
E, κ
ext)
·P (d
env|κ
ext, γ
ext= 0.089 ± 0.006, MS), (24) where the above approximation, i.e., neglecting the co-variance between γ
extand the other parameters in the lensing likelihood and then marginalizing γ
extseparately,
Cosmological constraints from time-delay lenses 13
all
all, weigh ted 1.35 ! n r " 1.45
1.35 ! n r " 1.45, weigh ted
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0
5 10 15
Κ ext pd f! Κ ex t "
Fig. 6.— The effective prior probability distribution for the ex-ternal convergence κ ext from combining ray tracing through the Millennium Simulation with the relative galaxy counts of 1.4 and the modeled external shear of 0.089 ± 0.006. Solid line: the vergence distribution for all lines of sight; Dotted line: the con-vergence distribution for lines of sight with relative galaxy count n r = 1.4 ± 0.05; Dashed line: the convergence distribution for all lines of sight weighted by the likelihood for γ ext from the lens model; Dot-dashed line: the γ ext -weighted convergence distribu-tion for lines of sight with n r = 1.4 ± 0.05. The effective prior for κ ext used in the final cosmological parameter inference is described by this, most informative, distribution.
is conservative since we would gain in precision by includ-ing the covariances with other parameters. Furthermore, by Bayes’ rule,
P (d env |κ ext , γ ext = 0.089 ± 0.006, MS)P (κ ext )
∝ P (κ ext |d env , γ ext = 0.089 ± 0.006, MS), (25) which is precisely the PDF of κ ext by selecting the sam-ples in P (κ ext , γ ext , d env |MS) that satisfies d env with a relative galaxy count within 1.4 ± 0.05, and subsequently weighting these samples by the Gaussian likelihood for γ ext . This effective prior PDF for κ ext that is con-structed from the weighted samples, P (κ ext |d env , γ ext = 0.089 ± 0.006, MS), is shown by the dot-dashed line in Figure 6.
TODO (SHS): Update with figure from Stefan.
8. TIME-DELAY DISTANCE OF RXJ1131−1231
We combine all the PDFs obtained in the previous sec-tions to infer the time-delay distance D ∆t .
8.1. Cosmological priors
As written above, we could infer the time delay dis-tance D ∆t directly, given a uniform prior. However, we are primarily interested in the cosmological information contained in such a distance measurement, so prefer to infer these directly. The posterior probability distribu-tion on D ∆t can then be obtained by first calculating the posterior PDF of the cosmological parameters π through the marginalizations in Equations (11) and (10), and then changing variables to D ∆t . Such transformations are of course straightforward when working with sam-pled PDFs.
In Table 2, we consider the following five cosmological world models, each with its own prior PDF P (π):
• UH 0 : Uniform prior PDF for H 0 between 0 and 150 km s −1 Mpc −1 in a ΛCDM cosmology with Ω Λ = 1 − Ω m = 0.73. This is similar to the typical
priors that were assumed in most of the early lens-ing studies, which sought to constrain H 0 at fixed cosmology.
• UwCDM: Uniform priors on the parameters {H 0 , Ω de , w} in a flat wCDM cosmology, where w is time-independent and Ω m = 1 − Ω de .
• WMAP7wCDM: The prior PDF for the parameters {H 0 , Ω de , w} is taken to be the posterior PDF from the WMAP 7-year data set (Komatsu et al. 2011), assuming a flat wCDM cosmology, where w is time-independent and Ω m = 1 − Ω de .
• WMAP7oΛCDM: The prior PDF for the parame-ters {H 0 , Ω Λ , Ω k } is taken to be the posterior PDF from the WMAP 7-year data set, assuming an open (or rather, non-flat) cosmology, with dark energy described by Λ and Ω k = 1 − Ω Λ − Ω m as the cur-vature parameter.
• WMAP7owCDM: The prior PDF for the parame-ters {H 0 , Ω de , w, Ω k } is taken to be the posterior PDF from the WMAP 7-year data set, assum-ing an open wCDM cosmology, where w is time-independent and Ω k = 1−Ω de −Ω m is the curvature parameter.
8.2. Posterior sampling
We sample the posterior PDF by weighting samples drawn from the prior PDF with the joint likelihood func-tion evaluated at those points (Suyu et al. 2010). We generate samples of the cosmological parameters π from the priors listed in Table 2. We then join these to samples of κ ext drawn from P (κ ext ) from Section 7.2 and shown in Figure 6, and to uniformly distributed samples of γ "
within [−1.5, 2.5] and r ani within [−0.5, 5] R eff . Rather than generating samples of θ E from the uniform prior, we obtain samples of θ E directly from the Gaussian approx-imation to the lensing and time-delay likelihood since θ E is quite independent of other model parameters (as shown in Figure 3). This boosts sampling efficiency, and the θ E samples are only used to evaluate the kinematics likelihood.
For each sample of {π, κ ext , γ " , r ani , θ E }, we obtain the weight (or importance) as follows: (1) we determine D ∆t from π via Equation (2), (2) we calculate D model ∆t via Equation (7), (3) we evaluate P (d ACS , ∆t|D model ∆t , γ " ) based on the Gaussian approximation shown in Figure 3 for D model ∆t and γ " , (4) we compute P (σ|π, γ " , κ ext , θ E , r ani ) via Equation (23), and (5) we weight the sam-ple by the product of P (d ACS , ∆t|D ∆t model , γ " ) and P (σ|π, γ " , κ ext , θ E , r ani ) from the previous two steps.
The projection of these weighted samples onto π or D ∆t effectively marginalizes over the other parameters.
8.3. Blind analysis in action
As a brief illustration of our blind analysis approach,
we show in the left panel of Figure 7 the blinded plot of
the time-delay distance measurement. For all
cosmolog-ical parameters such as D ∆t , D ∆t model , H 0 , w, Ω m , etc.,
we always plotted their probability distribution with
re-spect to the median during the blind analysis.
There-fore, we could use the shape of the PDFs to check our
Let
(all model parameters)
(cosmological parameters)
We are after the posterior PD F for given the data, marginalised over the nuisance parameters :
where
3-dataset likelihood
Prior
Method: importance sample from WMAP5 Pr() and Millenium Simulation Pr(κ ext ), using 3-dataset likelihood
Inferring cosmological parameters
在文檔中
with Time Delay
(頁 52-55)