Mapping cosmological fields using weak lensing and Fourier...
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Mapping cosmological fields using weak lensing and Fourier phase information Rafał Szepietowski (ICG, Portsmouth / JBCA, Manchester) [email protected] with David Bacon (ICG, Portsmouth), Jörg Dietrich (LMU, Munich), Michael Busha (UZurich), Risa Wechsler (Stanford), Peter Melchior (Ohio) b) d) References: [1] Bartelmann M., Narayan R., Seitz S., Schneider P., 1996,ApJ, 464, L115+ [2] Amara A. et al., 2012, MNRAS, 424, 553 [3] Simon P., 2012, arXiv:1203.6205 [4] Chiang L.-Y., 2001, MNRAS, 325, 405 b) d) a) b) b) c) Context: galaxies are a high resolution but biased tracer of mass, while weak lensing is unbiased but, due to low signal-to-noise ratio, has poor resolution. Aims: reconstructing the projected density field using the complementarity of weak lensing and galaxy positions. Method: a maximum-probability reconstruction of the 2D lensing convergence with a likelihood term for shear data and a prior on the Fourier phases constructed from the galaxy positions. TRUTH LENSING ALONE OUR METHOD Maximum-Probability reconstruction In our method we perform a forward reconstruction of the 2D convergence field [1] by seeking to find a hypothesis field which has the maximum probability of accounting for the observed data. Using Bayes theorem we write , ∝ where gal is the prior information on the Fourier phases of the field obtained from galaxy positions. Differential phase distribution To combine the information from weak lensing and galaxy positions [2,3] we make a prediction of the convergence g based on galaxy count overdensities g and assuming that the matter overdensity =b -1 g , where b is linear bias. However, then the value of the linear bias becomes an additional assumption in the reconstruction. To avoid this problem we consider only the information about the phases of the Fourier modes of g , neglecting their amplitudes. This preserves the information about the morphology of the field [4], simultaneously allowing lensing to determine the amplitude of the density fluctuations. Figure 1 shows the relation between the phases of the true convergence and count convergence g found in DES mock catalogue v4.02. The phase difference between these fields follows a wrapped Cauchy p.d.f. (∆) ∝ 1− 2 1+ 2 − 2 cos(∆) where relates to the half-width of the distribution. For this study we have used the mock galaxy catalogues created for the Dark Energy Survey based on the algorithm Adding Density Determined GAlaxies to Lightcone Simulations (ADDGALS; Wechsler et al 2013, in preparation; Busha et al 2013, in preparation). For the simulation of gravitational lensing, weak lensing shear at each galaxy position was computed using the multiple plane ray tracing code CALCLENS (Becker 2012). Figure 1. Left: The distributions of the phases of the true convergence field (solid line) and those obtained from the galaxy distribution g (dotted line) are close to uniform, as expected for close-to-Gaussian random fields. Overlaid (peaked curves), the histogram of the phase difference between and g (solid line). The distribution is well approximated by a wrapped Cauchy distribution (dashed line) and shows a strong correlation between the phases of the two fields. Right: Median absolute deviation (MAD) of the phase difference between the and g as a function of l. It is to be expected that will be a function of l, with the phase differences between galaxies and dark matter for large scale modes being more constrained than for small scale ones.
Transcript of Mapping cosmological fields using weak lensing and Fourier...
Mapping cosmological fields using weak lensing and Fourier phase information Rafał Szepietowski (ICG, Portsmouth / JBCA, Manchester) [email protected]
with David Bacon (ICG, Portsmouth), Jörg Dietrich (LMU, Munich), Michael Busha (UZurich), Risa Wechsler (Stanford), Peter Melchior (Ohio)
b) d)
References: [1] Bartelmann M., Narayan R., Seitz S., Schneider P., 1996,ApJ, 464, L115+ [2] Amara A. et al., 2012, MNRAS, 424, 553 [3] Simon P., 2012, arXiv:1203.6205 [4] Chiang L.-Y., 2001, MNRAS, 325, 405
b) d)
a) b)
b) c)
Context: galaxies are a high resolution but biased tracer of mass, while weak lensing is unbiased but, due to low signal-to-noise ratio, has poor resolution.
Aims: reconstructing the projected density field using the complementarity of weak lensing and galaxy positions.
Method: a maximum-probability reconstruction of the 2D lensing convergence with a likelihood term for shear
data and a prior on the Fourier phases constructed from the galaxy positions.
TRUTH LENSING ALONE OUR METHOD
Maximum-Probability reconstruction
In our method we perform a forward reconstruction of the 2D convergence field [1] by seeking to find a hypothesis field which has the maximum probability of accounting for the observed data. Using Bayes theorem we write
𝑃 𝜅 𝛾, 𝛼𝑔𝑎𝑙 ∝ 𝐿 𝛾 𝜅 𝑃 𝜅 𝛼𝑔𝑎𝑙
where gal is the prior information on the Fourier phases of the field obtained from galaxy positions.
Differential phase distribution
To combine the information from weak lensing and galaxy positions [2,3] we make a prediction of the convergence g based on galaxy count overdensities g and assuming that the matter overdensity =b-1g, where b is linear bias. However, then the value of the linear bias becomes an additional assumption in the reconstruction. To avoid this problem we consider only the information about the phases of the Fourier modes of g, neglecting their amplitudes. This preserves the information about the morphology of the field [4], simultaneously allowing lensing to determine the amplitude of the density fluctuations. Figure 1 shows the relation between the phases of the true convergence and count convergence g found in DES mock catalogue v4.02. The phase difference between these fields follows a wrapped Cauchy p.d.f.
𝑃(∆𝛼) ∝1 − 𝜌2
1 + 𝜌2 − 2𝜌 cos(∆𝛼)
where relates to the half-width of the distribution.
For this study we have used the mock galaxy catalogues created for the Dark Energy Survey based on the algorithm Adding Density Determined GAlaxies to Lightcone Simulations (ADDGALS; Wechsler et al 2013, in preparation; Busha et al 2013, in preparation). For the simulation of gravitational lensing, weak lensing shear at each galaxy position was computed using the multiple plane ray tracing code CALCLENS (Becker 2012).
Figure 1. Left: The distributions of the phases of the true convergence field (solid line) and those obtained from the galaxy distribution g (dotted line) are close to uniform, as expected for close-to-Gaussian random fields. Overlaid (peaked curves), the histogram of the phase difference between and g (solid line). The distribution is well approximated by a wrapped Cauchy distribution (dashed line) and shows a strong correlation between the phases of the two fields. Right: Median absolute deviation (MAD) of the phase difference between the and g as a function of l. It is to be expected that will be a function of l, with the phase differences between galaxies and dark matter for large scale modes being more constrained than for small scale ones.
