weak lensing 3
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Weak Lensing 3. Tom Kitching. Introduction. Scope of the lecture Power Spectra of weak lensing Statistics. Recap. Lensing useful for Dark energy Dark Matter Lots of surveys covering 100’s or 1000’s of square degrees coming online now. Recap. Lensing equation Local mapping - PowerPoint PPT PresentationTRANSCRIPT
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Weak Lensing 3
Tom Kitching
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Introduction• Scope of the lecture
• Power Spectra of weak lensing
• Statistics
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Recap• Lensing useful for • Dark energy• Dark Matter
• Lots of surveys covering 100’s or 1000’s of square degrees coming online now
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Recap• Lensing equation
• Local mapping • General Relativity relates this to the gravitational
potential
• Distortion matrix implies that distortion is elliptical : shear and convergence
• Simple formalise that relates the shear and convergence (observable) to the underlying gravitational potential
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Recap • Observed galaxies have instrinsic ellipticity and
shear • Reviewed shape measurement methods • Moments - KSB • Model fitting - lensfit
• Still an unsolved problem for largest most ambitous surveys
• Simulations • STEP 1, 2 • GREAT08 • Currently LIVE(!) GREAT10
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Part V : Cosmic Shear
• Introduction to why we use 2-point stats
• Spherical Harmonics
• Derivation of the cosmic shear power spectra
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• When averaged over sufficient area the shear field has a mean of zero
• Use 2 point correlation function or power spectra which contains cosmological information
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• Correlation function measures the tendency for galaxies at a chosen separation to have pre- ferred shape alignment
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Spherical Harmonics• We want the 3D power spectrum for cosmic
shear • So need to generalise to spherical harmonics for
spin-2 field
• Normal Fourier Transform
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• Want equivalent of the CMB power spectrum
• CMB is a 2D field• Shear is a 3D field
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Spherical Harmonics
Describes general transforms on a sphere for any spin-weight quantity
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Spherical Harmonics• For flat sky approximation and a scalar field
(s=0)
• Covariances of the flat sky coefficients related to the power spectrum
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Derivation of CS power spectrum• The shear field we can observe is a 3D spin-2
field
• Can write done its spherical harmonic coefficients • From data :
• From theory :
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Derivation of CS power spectrum• How to we theoretically predict ( r )?
• From lecture 2 we know that shear is related to the 2nd derivative of the lensing potential
• And that lensing potential is the projected Netwons potential
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Derivation of CS power spectrum• Can related the Newtons potential to the
matter overdensity via Poisson’s Equation
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Derivation of CS power spectrum• Generate theoretical shear estimate:
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• Simplifies to
• Directly relates underlying matter to the observable coefficients
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Derivation of CS power spectrum• Now we need to take the covariance of this to
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Large Scale Structure
Geometry
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Tomography• What is “Cosmic Shear Tomography” and how
does it relate to the full 3D shear field?
• The Limber Approximation • (kx,ky,kz) projected to (kx,ky)
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Tomography• Limber ok at small scales
• Very useful Limber Approximation formula (LoVerde & Afshordi)
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Tomography
• Limber Approximation (lossy)
• Transform to Real space (benign)
• Discretisation in redshift space (lossy)
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• Tomography • Generate 2D shear correlation in redshift bins• Can “auto” correlate in a bin• Or “cross” correlate between bin pairs• i and j refer to redshift bin pairs
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z
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Part VI : Prediction
• Fisher Matrices
• Matrix Manipulation
• Likelihood Searching
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What do we want?• How accurately can we estimate a model
parameter from a given data set?
• Given a set of N data point x1,…,xN
• Want the estimator to be unbiased • Give small error bars as possible
• The Best Unbiased Estimator
• A key Quantity in this is the Fisher (Information) Matrix
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What is the (Fisher) Matrix?• Lets expand a likelihood surface about the
maximum likelihood point
• Can write this as a Gaussian
• Where the Hessian (covariance) is
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What is the Fisher Matrix?• The Hessian Matrix has some nice properties
• Conditional Error on
• Marginal error on
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What is the Fisher Matrix?• The Fisher Matrix defined as the expectation of
the Hessian matrix
• This allows us to make predictions about the performance of an experiment !
• The expected marginal error on
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Cramer-Rao • Why do Fisher matrices work?
• The Cramer-Rao Inequality : • For any unbiased estimator
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The Gaussian Case• How do we calculate Fisher Matrices in
practice?
• Assume that the likelihood is Gaussian
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The Gaussian Case
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derivativematrix identity
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derivative
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How to Calculate a Fisher Matrix
• We know the (expected) covariance and mean from theory
• Worked example y=mx+c
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Adding Extra Parameters• To add parameters to a Fisher Matrix
• Simply extend the matrix
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Combining Experiments• If two experiments are independent then the
combined error is simply
Fcomb=F1+F2
• Same for n experiments
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Fisher Future Forecasting• We now have a tool with which we can predict
the accuracy of future experiments!
• Can easily • Calculate expected parameter errors• Combine experiments• Change variables• Add extra parameters
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• For shear the mean shear is zero, the information is in the covariance so (Hu, 1999)
• This is what is used to make predictions for cosmic shear and dark energy experiments
• Simple code available http://www.icosmo.org
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Weak Lensing Surveys• Current and on going surveys
05 10 15 20
CFHTLenS**
Pan-STARRS 1**
25
KiDS*
DES
Euclid
LSST
** complete or surveying * first light
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Dark Energy• Expect constraints of 1% from Euclid
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things we didn’t cover• Systematics • Photometric redshifts • Intrinsic Alignments
• Galaxy-galaxy lensing • Can use to determine galaxy-scale properties and
cosmology
• Cluster lensing• Strong lensing• Dark Matter mapping • ….• ….
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Conclusion• Lensing is a simple cosmological probe • Directly related to General Relativity • Simple linear image distortions
• Measurement from data is challenging • Need lots of galaxies and very sophisticated
experiments
• Lensing is a powerful probe of dark energy and dark matter