Measuring cosmic shear and birefringence using resolved radio sources
MMon. Not. R. Astron. Soc. , 1–19 (2015) Printed 5 November 2018 (MN L A TEX style file v2.2)
Measuring cosmic shear and birefringence using resolved radiosources
Lee Whittaker, Richard A. Battye & Michael L. Brown
Jodrell Bank Centre for Astrophysics, School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL
ABSTRACT
We develop a new method of extracting simultaneous measurements of weak lensing shearand a local rotation of the plane of polarization using observations of resolved radio sources.We show that the direction of polarization is statistically linked with that of the gradient ofthe total intensity field, and this provides the basis of our method. Using a number of sourcesspread over the sky, this method allows constraints to be placed on cosmic shear and birefrin-gence, and it can be applied to any resolved radio sources for which such a correlation exists.Assuming that the rotation and shear are constant across the source, we use this relationshipto construct a quadratic estimator and investigate its properties using simulated observations.We develop a calibration scheme using simulations based on the observed images to mitigatea bias which occurs in the presence of measurement errors and an astrophysical scatter on thepolarization. The method is applied directly to archival data of radio galaxies where we mea-sure a mean rotation signal of (cid:104) ω (cid:105) = − . ◦ ± . ◦ and an average shear compatible withzero using 30 reliable sources. This level of constraint on an overall rotation is comparablewith current leading constraints from CMB experiments and is expected to increase by at leastan order of magnitude with future high precision radio surveys, such as those performed bythe SKA. We also measure the shear and rotation two-point correlation functions and estimatethe number of sources required to detect shear and rotation correlations in future surveys. Key words: gravitational lensing: weak - methods: analytical - methods: statistical - cosmol-ogy: theory
Cosmic shear is the coherent distortion of the shapes of backgroundgalaxies by the large scale distribution of foreground matter. Themeasurement of the cosmic shear signal is now considered a pow-erful probe of cosmology due to its sensitivity to the integratedmass along the line of sight. In recent years, weak lensing sur-veys, such as the Canada-France-Hawaii Telescope Lensing Survey(CFHTLenS) (Heymans et al. 2013), the Dark Energy Survey Sci-ence Verification (DES SV) (Abbott et al. 2016), and using ∼
450 sqdegrees of the Kilo-Degree Survey (KiDS-450) (Hildebrandt et al.2017), have placed constraints on the total matter density parame-ter and the amplitude of the linear matter power spectrum, and fu-ture surveys, such as those performed by
Euclid , the Large Synop-tic Survey Telescope (LSST), and the Wide-Field Infrared SurveyTelescope (WFIRST) aim to place tight constraints on the dark en-ergy equation of state (e.g. Schrabback et al. 2010; Kilbinger et al.2013). http://sci.esa.int/euclid http://wfirst.gsfc.nasa.gov The intrinsic shapes of galaxies are not known, therefore thestandard method for measuring cosmic shear assumes that thegalaxies are intrinsically randomly orientated. Under this assump-tion, the measured shape of a galaxy provides an unbiased estimateof the shear at that position on the sky. The intrinsic randomnessin galaxy shapes contributes a shot noise error to estimates of theshear. Furthermore, the assumption that the galaxies are randomlyorientated is expected to break down for galaxies that share an evo-lutionary history. For such galaxies, tidal effects can cause galaxiesto become intrinsically aligned with the local large-scale structure(Heavens et al. 2000; Croft & Metzler 2000; Crittenden et al. 2001;Catelan et al. 2001). This alignment biases shear estimates and con-sequently estimates of the cosmological parameters (Heymans et al.2013). For a given scale, errors on estimates of the cosmic shearsignal, resulting from shape noise and errors on the shape measure-ments, are typically beaten down by averaging over the shapes of alarge number of galaxies. The number density of source galaxies ina survey therefore plays a key role in determining the power withwhich the survey can constrain cosmology. Current state-of-the-artsurveys are in the optical waveband and are detecting galaxies ata number density of ∼
10 arcmin − over an area of ∼ ,
000 deg . c (cid:13) a r X i v : . [ a s t r o - ph . C O ] S e p Lee Whittaker, Richard A. Battye & Michael L. Brown
Euclid aims to push the number density of observed galaxies up to
30 arcmin − over an area of ∼ ,
000 deg .Detections of cosmic shear have also been made the in ra-dio band (Chang et al. 2004), and by cross-correlating optical andradio observations using the SDSS and the VLA FIRST survey(Demetroullas & Brown 2016). However, radio weak lensing iscurrently uncompetitive with optical weak lensing due to the lowsource density. This position is likely to change in the future withthe advent of a whole new raft of facilities. The SKA, for exam-ple, is expected to perform surveys competitive with optical sur-veys, achieving number densities ∼
10 arcmin − over an area of ∼ ,
000 deg (Brown et al. 2015).Recent work by Brown & Battye (2011) and Whittaker et al.(2015) proposed statistical approaches for using polarization in-formation from radio surveys to reduce the effects of shot noiseand intrinsic alignments. These ideas rely on the hypothesis thatthe integrated polarization position angle is anti-correlated withthe structural position angle for star-forming galaxies, which willbe the dominant population at the flux densities needed to achievesource densities ∼
10 arcmin − . Such a relationship has been es-tablished in the local Universe (Stil et al. 2009), and this is expectedto be universal at some level, providing unbiased estimates of theintrinsic position angles of the galaxies.In this paper, we develop a related but different approach to de-tecting weak lensing using radio observations. We propose to usehigh signal-to-noise and high resolution observations to measurethe local shear field in the direction of individual resolved radiogalaxies. We show that there exists a statistical alignment betweenthe gradient of the total intensity field and the polarization positionangle for a sample of well studied galaxies. This alignment willbe broken in the presence of a shear and/or rotation of the planeof polarization, and by mapping the gradient and polarization po-sition angles across a well resolved source, one can use this breakin alignment to estimate the signal. Using a large number of sourcegalaxies ( ∼ ), the shear field can then be mapped over a signifi-cant portion of the sky. The ultimate aim is to apply this techniqueto data from the SKA, but proof of principle on surveys of smallerareas should be possible with the SKA progenitors. In fact, a simi-lar method has already been used by Kronberg et al. (1991), Kron-berg et al. (1996), and Burns et al. (2004) to establish the existenceof a lensing galaxy in front of the giant radio source 3C9. Theirapproach was to search for the effects of individual objects. Whatwe describe here is more general and aims to detect the extendeddistribution of matter across a large area of the sky.In addition to measuring cosmic shear, an estimate of the ro-tation signal can be used to probe cosmic birefringence. Cosmicbirefringence is the hypothesized rotation of the plane of polariza-tion of electromagnetic radiation due to a parity violating modifica-tion of Maxwell’s laws (e.g. (Carroll et al. 1990; Harari & Sikivie1992; Colladay & Kosteleck´y 1998; Kosteleck´y & Mewes 2009)).A rotation of the plane of polarization does not change the observedshape of a source which makes this effect different to that of a shearsignal. This difference between the effects of shear and rotationallows the two signals to be separated for a sufficiently resolvedsource. There has been a previous claim of a detection of cosmicbirefringence using the integrated polarization from extra-galacticradio sources (Nodland & Ralston 1997); however, this claim hassince been shown to be statistically insignificant (Carroll & Field1997; Loredo et al. 1997), and today the strongest constraints on anoverall global rotation come from cosmic microwave background (CMB) data (di Serego Alighieri et al. 2014; Ade et al. 2015; Grup-puso et al. 2016). Despite the strong constraints that come from theCMB, there is interest in a spatially correlated rotation (e.g. Cald-well et al. 2011; Lee et al. 2015; Namikawa 2017), and a largesurvey of sources could allow this to be measured.In Section 2 we derive the estimator and discuss the assump-tion that the linear polarization vector is aligned with the gradientof the total intensity distribution. In Section 3 we simulate observa-tions of Fanaroff and Riley Class II (FRII)-like sources using sumsof Gaussian profiles and use these simulations to test the perfor-mance of our estimator; although we note that the estimator shouldwork for any type of source which is resolved. We also developa method to correct for the biases that are introduced in the pres-ence of measurement errors and an astrophysical scatter betweenthe polarization vector and the gradient.In Section 4 we describe the VLA FRII data used to measurethe shear and rotation two point correlation functions in Section 5.Finally, in Section 6 we constrain the number of sources requiredfor a survey performed using the SKA to make a detection of theshear signal using this approach. We conclude in Section 7. The effect of gravitational lensing on any Stokes parameter distri-bution can be described by S obs ( x ) = S ( x (cid:48) ) = S ( x − ∇ ψ ) ,where S = I, Q, U , and ψ is the lensing potential. I ( x ) describesthe total intensity distribution of the source, and Q ( x ) and U ( x ) describe the linear polarization. S obs is the observed distributionof the Stokes parameters in the image plane and S is the intrinsicdistribution in the source plane. Assuming that the properties of thelens vary slowly compared to the size of the source, we can performa Taylor expansion on ∇ ψ , so that x (cid:48) i = x i − ∇ ψ = x i − ∇ i ψ | − x j ∇ j ∇ i ψ | , (1)which can be written as x (cid:48) = d + A x . (2)The term d = − ∇ ψ | is an undetectable offset that, for simplicity,we assume to be zero for the remainder of this paper. The matrix A i,j = δ i,j − ∇ i ∇ j ψ is the Jacobian of the transformation. Theconvergence is defined in terms of the lensing potential as κ = ∇ ψ , and the shear, γ , is defined as γ = 12 (cid:0) ∇ − ∇ (cid:1) ψ,γ = ∇ ∇ ψ, (3)so that the Jacobian matrix becomes A = (cid:18) − κ − γ − γ − γ − κ + γ (cid:19) . (4)In addition to the effects of the shear, an overall rotation ofthe Stokes distributions can be described by defining the rotationmatrix R = (cid:18) cos ω − sin ω sin ω cos ω (cid:19) , (5)where ω is the rotation angle of the plane of polarization. Assum-ing a zero shear signal and a non-zero rotation, a Stokes parameterdistribution is transformed as S obs ( x ) = S ( x (cid:48) ) = S (cid:0) R T x (cid:1) . If c (cid:13) , 1–19 easuring cosmic shear and birefringence we now assume a non-zero shear signal and rotation before lensing,the corresponding coordinate transformation is x (cid:48) = R T A x , (6)whereas if the rotation occurs after lensing, x (cid:48) = AR T x . (7)To first order in ω , κ and γ , it can be shown that both of the trans-formations given in equations (6) and (7) are identical. That is, for ω, κ, γ (cid:28) , ˜ A x ≈ R T A x ≈ AR T x , (8)with ˜ A = (cid:18) − κ − γ − γ + ω − γ − ω − κ + γ (cid:19) . (9)Assuming both a rotation and lensing, we can therefore approxi-mate the transformation of a distribution of Stokes parameters tofirst order in γ and ω as S obs ( x ) = S (cid:16) ˜ A x (cid:17) . For convenience,we now redefine the matrix A as A ≡ ˜ A .We can express Stokes Q and U in polar form: Q ( x ) = P ( x ) cos [2 α ( x )] ,U ( x ) = P ( x ) sin [2 α ( x )] , (10)where P ( x ) is the polarized intensity distribution and α ( x ) is thedistribution of polarization position angles. The factor of two is in-cluded as the linear polarization is described by a spin-2 vector,which is invariant under rotations of ◦ . The fact that the distri-butions of Stokes Q and U transform as Q obs ( x ) = Q ( x (cid:48) ) and U obs ( x ) = U ( x (cid:48) ) implies that ˆ n P obs ( x ) ≡ (cid:18) cos ( α obs ( x ))sin ( α obs ( x )) (cid:19) = ˆ n P (cid:0) x (cid:48) (cid:1) ≡ (cid:18) cos ( α ( x (cid:48) ))sin ( α ( x (cid:48) )) (cid:19) . (11). It can be shown that the effect of shear and rotation on thegradient of I obs ( x ) is ∇ x I obs ( x ) = A T ∇ x (cid:48) I (cid:0) x (cid:48) (cid:1) . (12)Defining β ( x ) as the local direction of ∇ x I obs ( x ) , we have ˆ n I ( x ) ≡ ∇ x I ( x ) | ∇ x I ( x ) | = (cid:18) cos ( β ( x ))sin ( β ( x )) (cid:19) . (13)Let us assume, for the moment, that the linear polarization isperfectly aligned with the gradient of the intensity distribution ˆ n I ( x ) = ± ˆ n P ( x ) , (14)where the plus or minus accounts for the quadrant of the gradientvector as α is only defined on the range − ◦ (cid:54) α < ◦ , whereas β is defined over − ◦ (cid:54) β < ◦ .For this ideal case, we can relate the gradient of the observedintensity distribution to the measured polarization position angles, ˆ n I obs ( x ) = ∇ x I obs ( x ) | ∇ x I obs ( x ) | = K A T ˆ n P obs ( x ) , (15)with K = ± | ∇ x (cid:48) I ( x (cid:48) ) || ∇ x I obs ( x ) | , (16)which contains the unobservable | ∇ x (cid:48) I ( x (cid:48) ) | . Expanding equation (15) we have cos β obs ( x ) = K ( A cos α obs ( x ) + A sin α obs ( x )) , sin β obs ( x ) = K ( A sin α obs ( x ) + A cos α obs ( x )) , (17)and then taking the ratio of the two equations in equation (17), wecan eliminate K , giving cos β obs ( x ) ( A sin α obs ( x ) + A cos α obs ( x ))= sin β obs ( x ) ( A cos α obs ( x ) + A sin α obs ( x )) . (18)If we now consider a source image consisting of N pixels with asignal-to-noise above a desired threshold level and with a measure-ment of α obs and β obs for each pixel, we can define the χ χ = N (cid:88) k =1 (cid:20) cos β ( k )obs (cid:16) A sin α ( k )obs + A cos α ( k )obs (cid:17) − sin β ( k )obs (cid:16) A cos α ( k )obs + A sin α ( k )obs (cid:17)(cid:21) . (19)Minimizing this χ with respect to γ and ω , we recover theestimator ˆ Γ = B − d , (20)where Γ = ωγ γ . (21)This estimator is similar in form to the shear estimator derived byBrown & Battye (2011) which also includes polarization informa-tion. The matrix B is given as B = 1 N N (cid:88) k =1 C ( k ) − S ( k ) α + S ( k ) β − (cid:16) C ( k ) α + C ( k ) β (cid:17) S ( k ) α + S ( k ) β − C ( k )+ − S ( k )+ − (cid:16) C ( k ) α + C ( k ) β (cid:17) − S ( k )+ C ( k )+ , (22)and the vector d is d = 1 N N (cid:88) k =1 − S ( k ) − C ( k ) α − C ( k ) β S ( k ) α − S ( k ) β , (23)where C ( k ) α = cos (cid:16) α ( k )obs (cid:17) , C ( k ) β = cos (cid:16) β ( k )obs (cid:17) ,S ( k ) α = sin (cid:16) α ( k )obs (cid:17) , S ( k ) β = sin (cid:16) β ( k )obs (cid:17) ,C ( k ) − = cos (cid:104) (cid:16) α ( k )obs − β ( k )obs (cid:17)(cid:105) , C ( k )+ = cos (cid:104) (cid:16) α ( k )obs + β ( k )obs (cid:17)(cid:105) ,S ( k ) − = sin (cid:104) (cid:16) α ( k )obs − β ( k )obs (cid:17)(cid:105) , S ( k )+ = sin (cid:104) (cid:16) α ( k )obs + β ( k )obs (cid:17)(cid:105) . (24)So far we have assumed that the correlation between α and β is exact, that is, α − β is zero at all points in space. However, whilethere are physical reasons to expect a close correlation between thetwo, it is unlikely to be perfect. We will model the deviations from aperfect correlation by Gaussian random fluctuations added to eachof the Stokes parameters, Q and U , with standard deviations, σ Q and σ U , respectively, which will of course be an approximation towhat is a highly complicated situation. c (cid:13) , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
In order to understand the expected distribution of α − β , letus first consider the angle distribution created by random fluctu-ations about zero with standard deviations in Stokes parameters Q = P cos θ and U = P sin θ where − π (cid:54) θ < π ignoringfor the moment the spin-two nature of the polarization. One canshow that this is given by P ( θ ) dθ = sinh (cid:0) η (cid:1) dθ π [cosh (2 η ) − cos 2 θ ] , (25)where cosh (cid:0) η (cid:1) = σ Q + σ U | σ Q − σ U | , sinh (cid:0) η (cid:1) = 2 σ Q σ U | σ Q − σ U | . (26)This is a 2D projected Normal distribution P N ( , Σ) with zeromean and covariance matrix Σ = diag( σ Q , σ U ) . There is a wellknown equivalence between such a distribution and the WrappedCauchy distribution W C (0 , ρ ) . In particular, if θ ∼ P N ( , Σ) then θ ∼ W C (0 , ρ ) where cosh (cid:0) η (cid:1) = 1 + ρ ρ , sinh (cid:0) η (cid:1) = 1 − ρ ρ , (27)and the parameter η describes the dispersion of θ in units of radians.However, the situation we are interested in is slightly differentto this. In particular we are dealing with a position angle spin-twopolarization position angle, − π/ (cid:54) α < π/ , and a real spaceposition angle, − π (cid:54) β < π , but any deviations from a perfectcorrelation will be modelled by standard deviations in the Stokesparameters as above. Taking this into account one can model α − β using the distribution in equation (25) above with the transforma-tion θ = α − β for − π/ (cid:54) θ < π/ , α − β = θ + π for − π (cid:54) θ < − π/ and α − β = θ − π for π/ (cid:54) θ < π . Hence,we deduce that P ( α − β ) d ( α − β ) = sinh (cid:0) η (cid:1) d ( α − β ) π (cosh (2 η ) − cos[2( α − β )] , (28)for − π/ (cid:54) α − β < π/ . We find that this distribution fits thedata well and we use this distribution in the subsequent analysisto model the deviations form a perfect correlation in the simula-tions used to estimate errors in our birefringence/shear estimates.It should be noted that there is no justification from first principlesfor modeling the distributions of Q and U as Gaussian; however,we find that this assumption provides a good fit to the distributionof measured α − β via equation (28). There will be a random scat-ter in the orientation of the polarization vectors due to turbulence inthe magnetic fields, and hence if one assumes the central limit the-orem, one might expect the observed Q and U to be approximatelyGaussian distributed. We have tested the estimator given in equation (21) using simula-tions where we assumed a simple model for a jet/lobe dominatedradio source. This model consists of three Gaussian distributions -a highly elliptical Gaussian for the jet, and one circular Gaussianfor each of the two lobes. Using this toy model, we created noisy I , Q and U maps which included both a shear and rotation sig-nal. We do not claim that this is an accurate representation of radiosources, just that it is similar to sources we will study in subse-quent sections. We then tested the performance of the estimator byrecovering estimates of the input signals from the simulated maps. Figure 1.
An example of the simulated high signal-to-noise total intensityimages used to test the estimator. The simulation is constructed from threeGaussian components using equation (29). Equivalent maps were also con-structed for the Stokes Q and U distributions. For this simple model, the total intensity distribution for thesource is I obs ( x ) = I ( x ) + I ( x ) + I ( x ) ,I ( x ) = ¯ I exp (cid:20) −
12 ( A x − ¯ x ) T M ( A x − ¯ x ) (cid:21) ,I ( x ) = ¯ I exp (cid:20) −
12 ( A x − ¯ x ) T M ( A x − ¯ x ) (cid:21) ,I ( x ) = ¯ I exp (cid:20) −
12 ( A x − ¯ x ) T M ( A x − ¯ x ) (cid:21) , (29)where the subscripts denote the individual components of the simu-lated image, ¯ I n is the peak intensity of component n , and ¯ x n is thecentroid of component n . For our simulations, the zeroth compo-nent was used to model the jet and the first and second componentswere used to model the lobes of the source. The matrices M n arequadrupole matrices describing the shapes of the components, with M ( n )11 = cos θ n a n + sin θ n b n ,M ( n )22 = sin θ n a n + cos θ n b n ,M ( n )12 = M ( n )21 = (cid:18) a n − b n (cid:19) sin θ n cos θ n . (30)Here, a n and b n are the major and minor axes of the componentsrespectively, and θ n are the corresponding position angles. An ex-ample of the simulated sources used to test the method is shownin Figure 1. This image consists of a
256 pixel ×
256 pixel grid.We set a = 15 . , b = 1 . , θ = 45 . ◦ , a = b = a = b = 5 . and θ = θ = 0 . . The centroids of the components are ¯ x = [128 . , . , ¯ x = [64 . , . , and ¯ x = [192 . , . .The background noise was assumed to be Gaussian, and we set thedispersion to unity. We began by testing the method on high signal-to-noise realizationsof the source. For the initial tests, we set the peak intensities to c (cid:13) , 1–19 easuring cosmic shear and birefringence ¯ I = 20 . , ¯ I = ¯ I = 1000 . to give a significant number of highsignal-to-noise pixels.To complete our simulations, we simulated images of theStokes Q and U distributions by assuming that the polarized in-tensity distribution was identical to the total intensity distribution;that is, we initially set P ( x ) = I ( x ) . The polarization positionangles were derived analytically using the assumption of equation(14). Letting z n = A x − ¯ x n , the unit vector for the polarization is n P ( x ) = ∇ x I ( x ) | ∇ x I ( x ) | = − M ( z ) I ( x ) + M ( z ) I ( x ) + M ( z ) I ( x ) | M ( z ) I ( x ) + M ( z ) I ( x ) + M ( z ) I ( x ) | . (31)We created images of Q ( x ) and U ( x ) at the same pixel scale as I ( x ) .Once these intrinsic I , Q and U maps had been constructed,a constant shear and rotation signal were applied. For these tests,we assumed an input shear of γ = 0 . and γ = − . , andincluded a rotation of ω = − . ◦ .For these first simple simulations, we ignored the effects ofbeam convolution and assumed Gaussian errors on the componentsof the gradient vector, with the dispersion equal to | ∇ I obs | / √ ;this gives an error on the position angle that is dependent on themodulus of the gradient and is consistent with using the finite dif-ferencing method discussed later in this section but avoids the sys-tematic effects of pixelization. The polarization is assumed to beperfectly aligned with the gradient, and the errors on Q obs and U obs were assumed to be Gaussian with zero mean and the dispersionequal to unity.For a constant level of background noise, the error on the mea-surement of α obs depends on the polarized intensity, P obs . In orderto avoid complications when calibrating for noise bias – discussedlater in this section – we chose to make a very high signal-to-noisecut on the Stokes Q and U maps. For these tests, we concentratedour analysis on regions of the images where P obs ( x ) > σ p ,where σ p is the dispersion of background noise in the Q and U maps (for these tests σ p = 1 , as discussed above). Similarly, theerror on the measurement of β obs depends on the magnitude of thegradient, | ∇ I obs | . We therefore made a signal-to-noise cut so thatwe consider only the regions where | ∇ I obs | ( x ) > σ b / √ ,where σ b is the background noise level for the total intensity im-age. The factor of √ reflects the fact that each component of thegradient calculated using the finite differencing method discussedin the following subsection considers the difference between twopixel values separated by a distance of two pixels, each with an in-dependent noise realization. The number of pixels above the signal-to-noise threshold in the noise free maps was N pix = 998 ; how-ever, in all tests conducted in this paper, the signal-to-noise cutswere made on the noisy images, and hence the precise number ofpixels considered in each realization varies somewhat due to noise.We recovered shear estimates from noise realizations,with the results presented in the top three panels of Figure 2.The mean estimates were (cid:104) ˆ γ (cid:105) = 0 . , (cid:104) ˆ γ (cid:105) = − . , (cid:104) ˆ ω (cid:105) = − . ◦ . The error bars on the mean estimates were < . of the recovered mean signal in all three cases, and the mean es-timates were all consistent with the input signals. This shows thatthe estimator in Section 2 is unbiased for this simple test. Next we tested the method on more realistic simulations, whichincluded realistic signal-to-noise levels and considered an approachto measuring the total intensity gradient which one might use inreality.The signal-to-noise cut on the polarized intensity for thesetests was made at P obs = 5 σ P , and the signal-to-noise cut on thegradient was made at | ∇ I obs | = 5 σ b / √ . This is identical to thesignal-to-noise cut made on the VLA data discussed in Section 4.As explained in the previous section, errors on the measurements of β depend on | ∇ I obs | . However, the signal-to-noise cut on | ∇ I obs | selects regions of the total intensity images where the contours areclosely packed and can ignore regions where the signal-to-noise ofthe intensity images are high, such as the centres of the sources.The simulated source was constructed using the same model as de-scribed in Section 3.1, but we now set the peak intensities of thecomponents of the source to ¯ I = 4 and ¯ I = ¯ I = 20 . These peakintensity levels were selected to provide ∼ α , and the to-tal intensity gradient position angle, β , was drawn randomly fromthe distribution given in equation (28), with a mean of zero and adispersion of η = 40 ◦ .In anticipation of using real data, we measured the gradient ofthe total intensity distribution using a method of finite differencing: ∇ I obs ( x i , y j ) =12 [ I ( x i +1 , y j ) − I ( x i − , y j ) , I ( x i , y j +1 ) − I ( x i , y j − )] , (32)where the subscripts on x and y denote the pixel index.We recovered the estimated shear from noise realizationsof the simulations discussed above. The results of this test areshown in the three middle panels of Figure 2. The mean estimateswere (cid:104) ˆ γ (cid:105) = (0 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − , (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ .Here we see that there is clear evidence for a bias in ˆ γ and ˆ ω , with the bias being largest for the rotation estimate. This biasis a result of the measurement errors on the position angles, thecontribution of an astrophysical scatter, and the finite differencingmethod used to measure the gradient. To see how these effects con-tribute a bias to the estimates, we can write the measured positionangles in terms of the true position angles and an error, which mayhave a contribution from both a random component, due to noise,and a systematic offset; in this case, a systematic offset is caused bythe finite differencing method, but it could also have a contributionfrom beam convolution, as is the case when we consider real datain Section 5. The observed position angles are then ˆ α obs = α obs + δα obs , ˆ β obs = β obs + δβ obs . (33)For convenience we will assume that the true intrinsic linear po-larization vector is aligned with the intrinsic gradient of I , so thatthe error term δα obs in equation (33) contains both the contributionfrom measurement errors on Q and U , and the astrophysical scatter.The error term δβ obs has a contribution from measurement errorsonly. These errors on the angles propagate a bias into the mean sineand cosine terms in equations (22) and (23). To see why this is thecase, we can follow the approach outlined in Whittaker et al. (2014) c (cid:13)000
256 pixel grid.We set a = 15 . , b = 1 . , θ = 45 . ◦ , a = b = a = b = 5 . and θ = θ = 0 . . The centroids of the components are ¯ x = [128 . , . , ¯ x = [64 . , . , and ¯ x = [192 . , . .The background noise was assumed to be Gaussian, and we set thedispersion to unity. We began by testing the method on high signal-to-noise realizationsof the source. For the initial tests, we set the peak intensities to c (cid:13) , 1–19 easuring cosmic shear and birefringence ¯ I = 20 . , ¯ I = ¯ I = 1000 . to give a significant number of highsignal-to-noise pixels.To complete our simulations, we simulated images of theStokes Q and U distributions by assuming that the polarized in-tensity distribution was identical to the total intensity distribution;that is, we initially set P ( x ) = I ( x ) . The polarization positionangles were derived analytically using the assumption of equation(14). Letting z n = A x − ¯ x n , the unit vector for the polarization is n P ( x ) = ∇ x I ( x ) | ∇ x I ( x ) | = − M ( z ) I ( x ) + M ( z ) I ( x ) + M ( z ) I ( x ) | M ( z ) I ( x ) + M ( z ) I ( x ) + M ( z ) I ( x ) | . (31)We created images of Q ( x ) and U ( x ) at the same pixel scale as I ( x ) .Once these intrinsic I , Q and U maps had been constructed,a constant shear and rotation signal were applied. For these tests,we assumed an input shear of γ = 0 . and γ = − . , andincluded a rotation of ω = − . ◦ .For these first simple simulations, we ignored the effects ofbeam convolution and assumed Gaussian errors on the componentsof the gradient vector, with the dispersion equal to | ∇ I obs | / √ ;this gives an error on the position angle that is dependent on themodulus of the gradient and is consistent with using the finite dif-ferencing method discussed later in this section but avoids the sys-tematic effects of pixelization. The polarization is assumed to beperfectly aligned with the gradient, and the errors on Q obs and U obs were assumed to be Gaussian with zero mean and the dispersionequal to unity.For a constant level of background noise, the error on the mea-surement of α obs depends on the polarized intensity, P obs . In orderto avoid complications when calibrating for noise bias – discussedlater in this section – we chose to make a very high signal-to-noisecut on the Stokes Q and U maps. For these tests, we concentratedour analysis on regions of the images where P obs ( x ) > σ p ,where σ p is the dispersion of background noise in the Q and U maps (for these tests σ p = 1 , as discussed above). Similarly, theerror on the measurement of β obs depends on the magnitude of thegradient, | ∇ I obs | . We therefore made a signal-to-noise cut so thatwe consider only the regions where | ∇ I obs | ( x ) > σ b / √ ,where σ b is the background noise level for the total intensity im-age. The factor of √ reflects the fact that each component of thegradient calculated using the finite differencing method discussedin the following subsection considers the difference between twopixel values separated by a distance of two pixels, each with an in-dependent noise realization. The number of pixels above the signal-to-noise threshold in the noise free maps was N pix = 998 ; how-ever, in all tests conducted in this paper, the signal-to-noise cutswere made on the noisy images, and hence the precise number ofpixels considered in each realization varies somewhat due to noise.We recovered shear estimates from noise realizations,with the results presented in the top three panels of Figure 2.The mean estimates were (cid:104) ˆ γ (cid:105) = 0 . , (cid:104) ˆ γ (cid:105) = − . , (cid:104) ˆ ω (cid:105) = − . ◦ . The error bars on the mean estimates were < . of the recovered mean signal in all three cases, and the mean es-timates were all consistent with the input signals. This shows thatthe estimator in Section 2 is unbiased for this simple test. Next we tested the method on more realistic simulations, whichincluded realistic signal-to-noise levels and considered an approachto measuring the total intensity gradient which one might use inreality.The signal-to-noise cut on the polarized intensity for thesetests was made at P obs = 5 σ P , and the signal-to-noise cut on thegradient was made at | ∇ I obs | = 5 σ b / √ . This is identical to thesignal-to-noise cut made on the VLA data discussed in Section 4.As explained in the previous section, errors on the measurements of β depend on | ∇ I obs | . However, the signal-to-noise cut on | ∇ I obs | selects regions of the total intensity images where the contours areclosely packed and can ignore regions where the signal-to-noise ofthe intensity images are high, such as the centres of the sources.The simulated source was constructed using the same model as de-scribed in Section 3.1, but we now set the peak intensities of thecomponents of the source to ¯ I = 4 and ¯ I = ¯ I = 20 . These peakintensity levels were selected to provide ∼ α , and the to-tal intensity gradient position angle, β , was drawn randomly fromthe distribution given in equation (28), with a mean of zero and adispersion of η = 40 ◦ .In anticipation of using real data, we measured the gradient ofthe total intensity distribution using a method of finite differencing: ∇ I obs ( x i , y j ) =12 [ I ( x i +1 , y j ) − I ( x i − , y j ) , I ( x i , y j +1 ) − I ( x i , y j − )] , (32)where the subscripts on x and y denote the pixel index.We recovered the estimated shear from noise realizationsof the simulations discussed above. The results of this test areshown in the three middle panels of Figure 2. The mean estimateswere (cid:104) ˆ γ (cid:105) = (0 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − , (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ .Here we see that there is clear evidence for a bias in ˆ γ and ˆ ω , with the bias being largest for the rotation estimate. This biasis a result of the measurement errors on the position angles, thecontribution of an astrophysical scatter, and the finite differencingmethod used to measure the gradient. To see how these effects con-tribute a bias to the estimates, we can write the measured positionangles in terms of the true position angles and an error, which mayhave a contribution from both a random component, due to noise,and a systematic offset; in this case, a systematic offset is caused bythe finite differencing method, but it could also have a contributionfrom beam convolution, as is the case when we consider real datain Section 5. The observed position angles are then ˆ α obs = α obs + δα obs , ˆ β obs = β obs + δβ obs . (33)For convenience we will assume that the true intrinsic linear po-larization vector is aligned with the intrinsic gradient of I , so thatthe error term δα obs in equation (33) contains both the contributionfrom measurement errors on Q and U , and the astrophysical scatter.The error term δβ obs has a contribution from measurement errorsonly. These errors on the angles propagate a bias into the mean sineand cosine terms in equations (22) and (23). To see why this is thecase, we can follow the approach outlined in Whittaker et al. (2014) c (cid:13)000 , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 2.
Recovered estimates of the shear and rotation from noise realizations. The top panels are the results for the high-signal-to-noise tests described in Section 3.1, which appears to give unbiased reconstructions of γ and ω . The middle row of panels are the resultsfor the realistic noise tests, without noise bias correction, discussed in Section 3.2, for which there is a bias in the reconstruction, mostobvious for ω . The bottom row shows the same set of realizations as in the middle panels but with the bias correction applied. In eachpanel the vertical black line shows the mean recovered estimate and the dashed vertical red line shows the input signal. The red curves inthe bottom panels are Gaussian distributions with the means equal to the mean recovered estimates, and the dispersions calculated usingequation (40). and write the expectation values of the cosine and sine of α obs as (cid:18) (cid:104) cos 2 α obs (cid:105)(cid:104) sin 2 α obs (cid:105) (cid:19) = 1 ξ C + ξ S (cid:18) C (cid:48) ξ C + S (cid:48) ξ S S (cid:48) ξ C − C (cid:48) ξ S (cid:19) , (34) where C (cid:48) = (cid:104) cos 2ˆ α obs (cid:105) − cov (cos 2 α obs , cos 2 δα obs )+ cov (sin 2 α obs , sin 2 δα obs ) ,S (cid:48) = (cid:104) sin 2ˆ α obs (cid:105) − cov (sin 2 α obs , cos 2 δα obs ) − cov (cos 2 α obs , sin 2 δα obs ) ,ξ C = (cid:104) cos 2 δα obs (cid:105) ,ξ S = (cid:104) sin 2 δα obs (cid:105) . (35)A similar analysis can be applied to the means of all the cosine andsine terms in B and d of equations (22) and (23). From equations(34) and (35), we see that the means of the trigonometric functions c (cid:13) , 1–19 easuring cosmic shear and birefringence are biased by terms that depend on the distributions of errors on theangles, and on the covariances between the errors and the true ob-served position angles (i.e. the observed position angles assumingzero errors and no astrophysical scatter between the polarizationand the gradient of the total intensity). As the bias terms depend on α obs , the level of bias will depend on the shape of the source. As weare interested in only the high signal-to-noise regions of the galax-ies, we propose a method whereby we use the observed source as atemplate for a set of calibration simulations that provide estimatesof the ξ and covariance terms corresponding to each of the meansin B and d . Here we present the steps followed when creating ourcalibration simulations:(i) Create an image of P using the Q and U images of the lensedsource.(ii) Measure the gradient of the total intensity image using finitedifferencing, and set the polarization position angles, α ( i )obs , to beequal to the recovered gradient position angles, β ( i )obs , thereby en-forcing the assumption that the shear and rotation signals are zeroin the calibration simulations.(iii) Reconstruct the I , Q and U maps by adding noise at thesame level as used in the original images; this includes the contri-bution from astrophysical scatter.(iv) Create a suite of Monte-Carlo noise realizations and esti-mate all of the required ξ and covariance terms by applying thesame signal-to-noise cuts to the calibration simulations and mea-suring the gradient using finite differencing.For these tests, the calibration simulations were created fromthe simulated lensed maps prior to the addition of backgroundnoise. This was done to reduce the number of calibration simu-lations required to test for biases at this level. For calibration, werequire that the overall structure of the simulated source resemblesthe intrinsic structure of the source in order to recover an estimateof the ξ and covariance terms in equation (34) (and equivalently forthe terms corresponding to the other mean cosine and sine termsin B and d ). As we are only interested in the high signal-to-noiseregions of the source, we expect the contribution from backgroundnoise in the maps to the estimates of the ξ and the covariance termsto be small and that the dominant contribution will be from astro-physical scatter.Using the approach described above, we recovered estimatesof the ξ and covariance terms from independent Monte-Carlosimulations. We then reanalyzed the simulations used to producethe middle panels in Figure 2, but we calculated the mean trigono-metric functions using equation (34). To clarify, the measured val-ues of the position angles, ˆ α obs and ˆ β obs , in equation (34) wererecovered from the same simulations used in the middle panels ofFigure 2, but the ξ and covariance terms were estimated from the independent Monte-Carlo simulations under the assumption ofzero input shear and rotation signals, as described above. The re-sults of this test are shown in the bottom panels of Figure 2. Thenumerical results of this test were (cid:104) ˆ γ (cid:105) = (0 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − , (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ . Hence,we see that the bias in the estimates has been greatly reduced andis now consistent with zero. However, the errors on the shear es-timates have increased by a factor of 2.65, and the errors on therotation estimates have increased by a factor of 1.86. It should benoted that it is likely that equation (32) is a sub-optimal method formeasuring β and a more accurate method may increase the preci-sion of the estimator. To gain some insight into the nature of the errors on estimatesof the shear and rotation, we can make progress by assuming thatthe source is approximately circular. For a large number of reliable(i.e. meeting the signal-to-noise requirements) pixels, the matrix B can then be approximated as B ≈ ξ α ξ β , (36)where ξ α = (cid:104) cos 2 δα obs (cid:105) ,ξ β = (cid:104) cos 2 δβ obs (cid:105) . (37)The estimator is then ˆ ω = − ξ α ξ β N N (cid:88) k =1 sin 2 (cid:16) ˆ α ( k )obs − ˆ β ( k )obs (cid:17) , ˆ γ = 1 ξ α ξ β N N (cid:88) k =1 (cid:104) ξ β cos 2ˆ α ( k )obs − ξ α cos 2 ˆ β ( k )obs (cid:105) , ˆ γ = 1 ξ α ξ β N N (cid:88) k =1 (cid:104) ξ β sin 2ˆ α ( k )obs − ξ α sin 2 ˆ β ( k )obs (cid:105) . (38)Using equations (14) and (15), we can write the trigonometric func-tions of α obs ( x ) and β obs ( x ) in terms of β ( x (cid:48) ) to first order inthe shear and rotation as cos 2 α obs ( x ) = cos 2 β (cid:0) x (cid:48) (cid:1) − γ − γ cos 4 β (cid:0) x (cid:48) (cid:1) − γ sin 4 β (cid:0) x (cid:48) (cid:1) , sin 2 α obs ( x ) = sin 2 β (cid:0) x (cid:48) (cid:1) − γ − γ sin 4 β (cid:0) x (cid:48) (cid:1) + γ cos 4 β (cid:0) x (cid:48) (cid:1) , cos 2 β obs ( x ) = cos 2 β (cid:0) x (cid:48) (cid:1) − γ − ω sin 2 β (cid:0) x (cid:48) (cid:1) , sin 2 β obs ( x ) = sin 2 β (cid:0) x (cid:48) (cid:1) − γ + 2 ω sin 2 β (cid:0) x (cid:48) (cid:1) . (39)Substituting equation (39) into equation (38) and assuming a cir-cular source, so that (cid:104) cos 2 β ( x (cid:48) ) (cid:105) = (cid:104) sin 2 β ( x (cid:48) ) (cid:105) = 0 , it can beshown that < ˆ Γ > = Γ . Furthermore, as the estimator in equation(38) is linear in the trigonometric functions, if the shear and rota-tion vary across the source, the recovered estimates are the meansof the variable signals.The linearity of the estimator in equation (38) in terms of themean cosine and sine terms also implies that the errors on the esti-mates will be approximately Gaussian distributed with the disper-sions σ ω = 18 ξ α ξ β N (1 − ξ ) ,σ γ = σ γ = 12 ξ α ξ β N (cid:0) ξ α + ξ β − ξ α ξ β (cid:1) , (40)where ξ = (cid:104) cos 4 δα obs (cid:105) (cid:104) cos 4 δβ obs (cid:105) . (41)We see that the correction terms, ξ , correct for biasing due tothe measurement errors by boosting the means of the trigonomet-ric functions. This boosting effect propagates into the errors on theshear and rotation estimates, increasing the errors as compared withthe uncorrected estimators. If one assumes that the difference be-tween the angles α and β is dominated by an astrophysical scattercomponent with a distribution described by equation (28), then itcan be shown that ξ α = e − η , ξ β = 1 , ξ = e − η . (42) c (cid:13)000
Recovered estimates of the shear and rotation from noise realizations. The top panels are the results for the high-signal-to-noise tests described in Section 3.1, which appears to give unbiased reconstructions of γ and ω . The middle row of panels are the resultsfor the realistic noise tests, without noise bias correction, discussed in Section 3.2, for which there is a bias in the reconstruction, mostobvious for ω . The bottom row shows the same set of realizations as in the middle panels but with the bias correction applied. In eachpanel the vertical black line shows the mean recovered estimate and the dashed vertical red line shows the input signal. The red curves inthe bottom panels are Gaussian distributions with the means equal to the mean recovered estimates, and the dispersions calculated usingequation (40). and write the expectation values of the cosine and sine of α obs as (cid:18) (cid:104) cos 2 α obs (cid:105)(cid:104) sin 2 α obs (cid:105) (cid:19) = 1 ξ C + ξ S (cid:18) C (cid:48) ξ C + S (cid:48) ξ S S (cid:48) ξ C − C (cid:48) ξ S (cid:19) , (34) where C (cid:48) = (cid:104) cos 2ˆ α obs (cid:105) − cov (cos 2 α obs , cos 2 δα obs )+ cov (sin 2 α obs , sin 2 δα obs ) ,S (cid:48) = (cid:104) sin 2ˆ α obs (cid:105) − cov (sin 2 α obs , cos 2 δα obs ) − cov (cos 2 α obs , sin 2 δα obs ) ,ξ C = (cid:104) cos 2 δα obs (cid:105) ,ξ S = (cid:104) sin 2 δα obs (cid:105) . (35)A similar analysis can be applied to the means of all the cosine andsine terms in B and d of equations (22) and (23). From equations(34) and (35), we see that the means of the trigonometric functions c (cid:13) , 1–19 easuring cosmic shear and birefringence are biased by terms that depend on the distributions of errors on theangles, and on the covariances between the errors and the true ob-served position angles (i.e. the observed position angles assumingzero errors and no astrophysical scatter between the polarizationand the gradient of the total intensity). As the bias terms depend on α obs , the level of bias will depend on the shape of the source. As weare interested in only the high signal-to-noise regions of the galax-ies, we propose a method whereby we use the observed source as atemplate for a set of calibration simulations that provide estimatesof the ξ and covariance terms corresponding to each of the meansin B and d . Here we present the steps followed when creating ourcalibration simulations:(i) Create an image of P using the Q and U images of the lensedsource.(ii) Measure the gradient of the total intensity image using finitedifferencing, and set the polarization position angles, α ( i )obs , to beequal to the recovered gradient position angles, β ( i )obs , thereby en-forcing the assumption that the shear and rotation signals are zeroin the calibration simulations.(iii) Reconstruct the I , Q and U maps by adding noise at thesame level as used in the original images; this includes the contri-bution from astrophysical scatter.(iv) Create a suite of Monte-Carlo noise realizations and esti-mate all of the required ξ and covariance terms by applying thesame signal-to-noise cuts to the calibration simulations and mea-suring the gradient using finite differencing.For these tests, the calibration simulations were created fromthe simulated lensed maps prior to the addition of backgroundnoise. This was done to reduce the number of calibration simu-lations required to test for biases at this level. For calibration, werequire that the overall structure of the simulated source resemblesthe intrinsic structure of the source in order to recover an estimateof the ξ and covariance terms in equation (34) (and equivalently forthe terms corresponding to the other mean cosine and sine termsin B and d ). As we are only interested in the high signal-to-noiseregions of the source, we expect the contribution from backgroundnoise in the maps to the estimates of the ξ and the covariance termsto be small and that the dominant contribution will be from astro-physical scatter.Using the approach described above, we recovered estimatesof the ξ and covariance terms from independent Monte-Carlosimulations. We then reanalyzed the simulations used to producethe middle panels in Figure 2, but we calculated the mean trigono-metric functions using equation (34). To clarify, the measured val-ues of the position angles, ˆ α obs and ˆ β obs , in equation (34) wererecovered from the same simulations used in the middle panels ofFigure 2, but the ξ and covariance terms were estimated from the independent Monte-Carlo simulations under the assumption ofzero input shear and rotation signals, as described above. The re-sults of this test are shown in the bottom panels of Figure 2. Thenumerical results of this test were (cid:104) ˆ γ (cid:105) = (0 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − , (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ . Hence,we see that the bias in the estimates has been greatly reduced andis now consistent with zero. However, the errors on the shear es-timates have increased by a factor of 2.65, and the errors on therotation estimates have increased by a factor of 1.86. It should benoted that it is likely that equation (32) is a sub-optimal method formeasuring β and a more accurate method may increase the preci-sion of the estimator. To gain some insight into the nature of the errors on estimatesof the shear and rotation, we can make progress by assuming thatthe source is approximately circular. For a large number of reliable(i.e. meeting the signal-to-noise requirements) pixels, the matrix B can then be approximated as B ≈ ξ α ξ β , (36)where ξ α = (cid:104) cos 2 δα obs (cid:105) ,ξ β = (cid:104) cos 2 δβ obs (cid:105) . (37)The estimator is then ˆ ω = − ξ α ξ β N N (cid:88) k =1 sin 2 (cid:16) ˆ α ( k )obs − ˆ β ( k )obs (cid:17) , ˆ γ = 1 ξ α ξ β N N (cid:88) k =1 (cid:104) ξ β cos 2ˆ α ( k )obs − ξ α cos 2 ˆ β ( k )obs (cid:105) , ˆ γ = 1 ξ α ξ β N N (cid:88) k =1 (cid:104) ξ β sin 2ˆ α ( k )obs − ξ α sin 2 ˆ β ( k )obs (cid:105) . (38)Using equations (14) and (15), we can write the trigonometric func-tions of α obs ( x ) and β obs ( x ) in terms of β ( x (cid:48) ) to first order inthe shear and rotation as cos 2 α obs ( x ) = cos 2 β (cid:0) x (cid:48) (cid:1) − γ − γ cos 4 β (cid:0) x (cid:48) (cid:1) − γ sin 4 β (cid:0) x (cid:48) (cid:1) , sin 2 α obs ( x ) = sin 2 β (cid:0) x (cid:48) (cid:1) − γ − γ sin 4 β (cid:0) x (cid:48) (cid:1) + γ cos 4 β (cid:0) x (cid:48) (cid:1) , cos 2 β obs ( x ) = cos 2 β (cid:0) x (cid:48) (cid:1) − γ − ω sin 2 β (cid:0) x (cid:48) (cid:1) , sin 2 β obs ( x ) = sin 2 β (cid:0) x (cid:48) (cid:1) − γ + 2 ω sin 2 β (cid:0) x (cid:48) (cid:1) . (39)Substituting equation (39) into equation (38) and assuming a cir-cular source, so that (cid:104) cos 2 β ( x (cid:48) ) (cid:105) = (cid:104) sin 2 β ( x (cid:48) ) (cid:105) = 0 , it can beshown that < ˆ Γ > = Γ . Furthermore, as the estimator in equation(38) is linear in the trigonometric functions, if the shear and rota-tion vary across the source, the recovered estimates are the meansof the variable signals.The linearity of the estimator in equation (38) in terms of themean cosine and sine terms also implies that the errors on the esti-mates will be approximately Gaussian distributed with the disper-sions σ ω = 18 ξ α ξ β N (1 − ξ ) ,σ γ = σ γ = 12 ξ α ξ β N (cid:0) ξ α + ξ β − ξ α ξ β (cid:1) , (40)where ξ = (cid:104) cos 4 δα obs (cid:105) (cid:104) cos 4 δβ obs (cid:105) . (41)We see that the correction terms, ξ , correct for biasing due tothe measurement errors by boosting the means of the trigonomet-ric functions. This boosting effect propagates into the errors on theshear and rotation estimates, increasing the errors as compared withthe uncorrected estimators. If one assumes that the difference be-tween the angles α and β is dominated by an astrophysical scattercomponent with a distribution described by equation (28), then itcan be shown that ξ α = e − η , ξ β = 1 , ξ = e − η . (42) c (cid:13)000 , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Hence, if we further assume that η (cid:28) , then equation (40) sim-plifies to σ ω = η N , σ γ = σ γ = 2 η N . (43)The error on estimates of the shear due to galaxy shape noisewhen using the standard approach of averaging over galaxy shapesis σ γ = σ (cid:15) /N gal , where σ (cid:15) is the dispersion in intrinsic galaxyshapes and N gal is the number of source galaxies. Hence, we seethat the contribution from astrophysical scatter to the error on theshear estimates using our approach ( η in equation (43)) plays thesame role as the intrinsic shape dispersion in the standard shapebased approach.In the bottom panels of Figure 2, the red curves show Gaussiandistributions using the means from the simulations and the disper-sions given in equation (40). These plots show that, for the sourcemodel used, the distribution of errors on the estimates are well de-scribed by the Gaussian assumption and the dispersions predictedby equation (40).The dominant contribution to the errors is expected to comefrom the astrophysical scatter as we only consider the high signal-to-noise regions of the sources. It is expected that the magnitude ofthe astrophysical scatter will have some dependence on the resolu-tion at which the objects are observed. Hence, it may be possible toidentify an optimal scale on which to perform this analysis. If theoriginal image is made at a high resolution, one is free to smooththe image to the desired resolution if this smoothing step is also ap-plied consistently to the calibration simulations when calculatingthe bias correction terms.Equation (34) was derived assuming an exact knowledge ofthe bias correction terms. In general these terms have to be es-timated using both the data and the calibration simulations, andhence there will also be an error on the estimates of these terms.In order to gain some insight into the impact of errors on thebias correction terms, let us assume that the distribution of α obs − β obs is dominated by an astrophysical scatter described by equation(28). In this case, ξ β is given by equation (42). If we attempt to fitequation (28) to the data, we will have an error on our estimate of η due to the finite number of high signal-to-noise pixels used inthe analysis. If we now assume that errors on η are much less than η and are drawn randomly from a Gaussian distribution with zeromean and variance σ η , then errors on estimates of ξ β will follow alognormal distribution. Ignoring any systematic effects associatedwith the details of the simulations, the estimator will be biased dueto the error on η , so that ˆ Γ = Γ e σ η . (44)For the analysis in Section 5, the contribution from this bias is ex-pected to be well below the percent level. Errors on the estimateswill also increase by a factor of e σ η , which again leads toa fractional difference below the percent level for the analysis inSection 5. Therefore, for the remainder of this paper, we will ig-nore contributions from errors on estimates of the bias correctionterms. Here we describe the data used in the analysis performed in the fol-lowing section, where we apply our estimator to observations madewith the VLA and MERLIN array. The specific sources we use have FRII morphology, but this is not necessary and, as already pointedout, the technique can be applied to any well resolved source. Weuse these sources since the data is available. They were taken forother reasons, and hence are not necessarily optimal for this pur-pose.The data consists of polarization and total intensity maps for58 sources with redshifts in the range < z < . The majority ofthe data was retrieved from the Mullin et al. (2008) database , withthe rest of the sources from data presented in Leahy et al. (1997)and Black et al. (1992). The important physical properties of thefull sample considered are given in Table 1. Where possible, thepositions of the objects were obtained from the Mullin et al. (2008)database, and the remainder were found in Tabara & Inoue (1980).For our analysis, we began by first inspecting the source im-ages by eye and, where possible, masking the regions in the im-ages containing the central source AGN; these regions have a highsignal-to-noise, but they are unresolved. We also selected regionsof the image where the radio emission from the source is expectedto be negligible so that these regions could be used to measure thenoise in the total intensity and polarization maps. We then mea-sured the gradient of the total intensity images and selected the re-gions of the maps which satisfied the signal-to-noise requirementsdiscussed in Section 3.1. The distribution of α obs − β obs from the58 sources is shown in left-panel of Figure 3. The red curve is equa-tion (28) fitted to the data, with best-fit parameter η = 42 . ◦ . Theright-panel is the distribution of recovered means of α obs − β obs for each source. Numerical values of the means of α obs − β obs for the sources considered in Section 5 are given in table 2 alongwith estimates of the uncertainties. Three examples of the total in-tensity images used are shown in Figure 4 along with the corre-sponding distribution of α obs − β obs . These three examples wereselected as they demonstrate a range of different dispersions be-tween α obs − β obs .In order to estimate the contribution from astrophysical scat-ter, we adopted a method of forward modeling using simulationsconstructed from the data via the approach discussed in Section 3.1.We produced a set of simulated I , Q and U maps assuming Gaus-sian noise, with the noise levels set to match the levels measuredfrom the images. An assumed level of astrophysical scatter wasthen added to the polarization position angle using the distributiongiven in equation (28) with a constant dispersion, η astro . The simu-lated maps were smoothed with a Gaussian beam, and the width ofthe beam matched the width of the CLEAN beam applied when cre-ating the images. The convolution of the noisy simulated map witha Gaussian beam produces noise correlations between neighboringpixels. The detail of noise in radio images is more complicated thanthe noise in our calibration simulations, but as we are only inter-ested in the high signal-to-noise regions, we expect our simulationsto be sufficient.The gradient of the total intensity images and the polarizationposition angles were recovered from the high signal-to-noise re-gions of these maps for noise realizations. A histogram wasconstructed using the α obs − β obs values recovered from the sim-ulations and a binsize of ◦ , and equation (28) was fitted to thishistogram providing a corresponding dispersion parameter for thetotal contribution of errors and astrophysical scatter, η total . Thiswas repeated for a range of values of η astro . The estimated value of η astro for each source was taken to be the one which provided the zl1.extragalactic.info c (cid:13) , 1–19 easuring cosmic shear and birefringence Source Resolution (arcsec) Pixel width (arcsec) RA (B1950) Dec (B1950) Redshift Frequency (GHz)3C6.1 0.25 0.065 0 13 34.4 +79 0 11.1 0.84 8.463C15 2.3 0.67 0 34 30.6 -1 25 40.4 0.073 8.463C20 0.22 0.050 0 40 20.1 +51 47 10.2 0.17 8.413C22 0.25 0.050 0 48 4.71 +50 55 45.4 0.94 8.463C34 0.40 0.070 1 7 32.5 +31 31 23.9 0.69 4.873C41 0.20 0.050 1 23 54.7 +32 57 38.3 0.79 8.463C47 1.3 0.20 1 33 40.4 +20 42 10.2 0.43 4.893C55 0.40 0.10 1 54 19.5 +28 37 4.80 0.74 4.853C67 0.050 0.015 2 21 18.0 +27 36 37.2 0.31 4.993C105 0.25 0.050 4 4 47.8 +3 32 49.7 0.089 8.413C111.1.6 1.6 0.20 4 15 1.10 +37 54 37.0 0.049 8.353C123 0.23 0.050 4 33 55.2 +29 34 12.6 0.22 8.443C132 0.220 0.050 4 53 42.2 +22 44 43.9 0.21 8.443C153 0.26 0.050 6 5 44.4 +48 4 48.8 0.28 8.413C175 0.25 0.060 7 10 15.4 +11 51 24.0 0.77 8.453C184 0.35 0.070 7 33 59.0 +70 30 1.10 0.99 4.863C184.1 2.5 0.80 7 34 25.0 +80 33 24.1 0.12 8.473C192 0.80 0.25 8 2 35.5 +24 18 26.4 0.060 8.233C196 0.35 0.075 8 9 59.4 +48 22 7.57 0.87 4.863C197.1 0.25 0.080 8 18 1.06 +47 12 8.30 0.13 8.463C207 0.35 0.060 8 38 1.72 +13 23 5.57 0.68 4.863C215 0.37 0.10 9 3 44.1 +16 58 16.1 0.41 4.893C216 0.25 0.065 9 6 17.3 +43 5 58.6 0.67 8.213C217 0.35 0.060 9 5 41.4 +38 0 29.9 0.90 4.863C220.1 0.25 0.050 9 26 31.9 +79 19 45.4 0.61 8.443C220.3 0.35 0.060 9 31 10.5 +83 28 55.0 0.69 4.863C223 0.25 0.080 9 38 18.0 +39 58 20.2 0.11 8.473C225B 0.050 0.015 9 39 32.2 +13 59 33.3 0.58 4.993C226 0.20 0.050 9 41 36.2 +10 0 3.80 0.82 8.463C227 2.5 0.70 9 45 7.80 +7 39 9.00 0.086 8.473C234 0.30 0.060 9 58 57.4 +29 1 37.4 0.19 8.443C247 0.35 0.060 10 56 8.38 +43 17 30.6 0.75 4.863C249.1 0.35 0.10 11 0 27.3 +77 15 8.62 0.31 4.893C254 0.35 0.070 11 11 53.3 +40 53 41.5 0.73 4.893C263 0.35 0.065 11 37 9.30 +66 4 27.0 0.66 4.863C263.1 0.35 0.070 11 40 49.2 +22 23 34.9 0.82 4.893C265 0.40 0.070 11 42 52.4 +31 50 29.1 0.81 4.853C277.2 0.40 0.070 12 51 4.20 +15 58 51.2 0.77 4.863C280 0.35 0.070 12 54 41.7 +47 36 32.7 1.0 4.893C284 0.90 0.20 13 8 41.3 +27 44 2.60 0.24 8.063C289 0.35 0.070 13 43 27.4 +50 1 32.0 0.97 4.893C299 0.41 0.050 14 19 6.29 +41 58 30.2 0.37 4.863C319 0.90 0.20 15 22 43.9 +54 38 38.4 0.19 8.443C325 0.35 0.070 15 49 14.0 +62 50 20.0 0.86 4.863C336 0.35 0.060 16 22 32.2 +23 52 2.00 0.93 4.863C340 0.40 0.060 16 27 29.4 +23 26 42.6 0.78 4.863C349 2.9 0.80 16 58 4.44 +47 7 20.3 0.21 8.443C352 0.35 0.080 17 9 18.0 +46 5 6.00 0.81 4.713C381 0.25 0.060 18 32 24.5 +47 24 39.0 0.16 8.443C401 0.27 0.050 19 39 38.8 +60 34 33.5 0.20 8.443C403 0.75 0.20 19 49 44.1 +2 22 41.5 0.059 8.473C433 0.25 0.080 21 21 30.5 +24 51 33.0 0.10 8.473C438 0.23 0.050 21 53 45.5 +37 46 12.8 0.29 8.443C441 0.35 0.065 22 3 49.3 +29 14 43.8 0.71 4.863C452 0.25 0.080 22 43 32.8 +39 25 27.3 0.081 8.463C455 0.40 0.050 22 52 34.5 +12 57 33.5 0.54 4.864C14.11 0.23 0.070 4 11 40.9 +14 8 48.3 0.21 8.444C74.16 0.30 0.065 10 9 49.7 +74 52 29.8 0.81 8.47
Table 1.
Physical properties of the FRII sample used in this paper. Details of where the data were obtained from are given in the maintext.c (cid:13) , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 3.
Left-panel : The distribution of α obs − β obs from the high signal-to-noise regions of the 58 FRII sources. For each source,the mean value of α obs − β obs has been subtracted from each value of α to remove any systematic astrophysical effects; this step wascarried out for presentation purposes only and was not included when applying the estimator to the data in Section 5. The total numberof angles included in the histogram is 119742. The vertical black line indicates zero difference. The red curve is the distribution givenin equation (28), with best-fit dispersion parameter η total = 42 . ◦ . Right-panel : The distribution of the means of α obs − β obs for the58 sources, which were removed from the left-hand figure. The vertical black line is the mean, weighted by the number of pixels used ineach source. The mean value is − . ◦ . Numerical values of the means of α obs − β obs for the sources included in the results discussedin Section 5 are given in table 2. value of η total in the fit to the histogram of simulated α obs − β obs that matched that of the true maps. This procedure was repeated foreach source in table 1.The distribution of η total fitted to the 58 sources in the sampleis shown in Figure 5. The mean value is (cid:104) η total (cid:105) = 43 . ◦ . This is aweighted mean, with the weights taken to be w i = N ( i )eff = 2 ln 2 π N i ( W i /P i ) , (45)where we define N ( i )eff as the effective number of reliable pixels forsource i . Here, N i is the number of pixels in the maps above thesignal-to-noise threshold for the i th source , W i is the beam width(FWHM), and P i is the pixel width. The ratio W i /P i has been in-cluded as an estimate of the contribution of the beam to the numberof effective pixels in the sample. The exact effect of the beam onthe number of effective pixels is difficult to quantify when consid-ering the errors on angle measurements, as the errors depend on themorphology of the Stokes parameter distributions. For this analy-sis we assume that the errors on the angles scale proportionally tothe area of the beam, which, given a constant level of noise, de-scribes how the beam effects the number of effective pixels whenmeasuring and .The effects of the beam (the effects of pixelization will be sub-dominant as the pixel width is less than the beam width) on themean sines and cosines in equations (22) and (23) are captured bythe ξ and covariance terms in equation (34) which we derive usingcalibration simulations, as discussed in Section 3.2. Therefore, tocalibrate for both noise bias and the bias introduced by beam con-volution, we created a suite of calibration simulations for eachsource using the approach discussed in Section 3.2. As discussedabove, we assume a Gaussian error distribution for the noise on the I , Q and U maps, and we also assume a Gaussian beam whenconvolving the maps with the beam. The beam convolution step isapplied directly after the noise is added.Beam convolution has two undesirable effects when creatingthe calibration simulations. Firstly, it smooths the noise, and hencereduces the apparent values of σ P and σ b in the convolved maps.To ameliorate this problem, we scaled the noise so that the noiseapplied to the calibration simulations is σ calibb = σ b (cid:112)(cid:80) i K ( x i ) ,σ calib P = σ P (cid:112)(cid:80) i K ( x i ) , (46)where K is the Kernel describing the shape of the beam. This scal-ing provides the correct level of noise in the beam convolved sim-ulated maps. The other effect is that the beam also reduces the ap-parent intensity of the maps. This reduction in intensity means thatregions of the calibration images which are included in the analysisare omitted from the calibration simulations as they do not meetthe signal-to-noise requirements. This propagates as a bias into es-timates of the calibration terms. For this analysis, we multiplied thesimulated images by a scaling factor so that the peak intensities ofthe smoothed calibration simulations, before the addition of noise,matched the peak intensities of the true images. At some level, thisstep attempts to correct for the fact that the calibration images areconstructed from images that have already been convolved with thebeam.Ideally, the calibration simulations would be constructed fromthe unconvolved images and include the correct shear, rotation andsystematics. However, in reality this is obviously not possible. Wehave confirmed that calibrating the mean cosines and sines using c (cid:13) , 1–19 easuring cosmic shear and birefringence Figure 4.
The total intensity distributions for the three sources 3C192, 3C196, and 3C340 are shown in the left-hand panels. In red weshow the pixels which satisfy the signal-to-noise cuts. The right-hand panels show the distributions of α obs − β obs in the red regionsfor each source. The red curves show equation (28) with the dispersion parameters, η total , fitted to the data and given in Table 2. Thegreen squares show masked regions which contain the source AGNs.c (cid:13) , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 5.
The distribution of η total fitted to histograms of the measured α obs − β obs from the image data for the 58 sources in the sample. Thevertical line shows the mean value. equation (34) completely corrects for biases due to beam convolu-tion if the ideal scenario is assumed. However, there is some resid-ual biasing if one assumes the more realistic approach describedabove due to the imperfect calibration simulations. This bias, how-ever, was found to be much smaller than the errors on the shear androtation in all of the tests performed.From these calibration simulations, we estimate the ξ and co-variance terms in equation (34), using finite differencing to mea-sure the gradients and including the correct signal-to-noise cuts,and correct the mean trigonometric functions accordingly. Here we discuss the results of applying the estimator, with the cal-ibration steps described in the previous section, to the 58 objectsshown in table 1. We present results for the estimated shear and ro-tation, and also estimate the shear and rotation two point correlationfunctions (2PCFs).We began by applying the estimator to each of the objectsand recovering shear and rotation estimates. In order to quantifythe errors on the estimates, we created simulated images using themethod used to produce the calibration simulations, and assumingzero input shear and rotation signals. We then applied the estimator,including the bias corrections, to simulated noise realizationsof each image and recovered the estimates. The recovered estimatesfrom the real data and the simulations corresponding to the threesources shown in Figure 4 are presented in Figure 6.For some of the objects, we find that there are substantial out-liers in the estimates recovered from the simulations used to deter-mine the errors. These were due to catastrophic failures in the simu-lation process. This leads to a calculated dispersion of the estimateswhich is not representative of the bulk of the distribution. In orderto recover a more accurate estimate of the error, we chose to makecuts on the simulated distributions. First, we cut the distributions of the simulated shears and rotations so that we excluded any realiza-tions where either component of the shear lies more than 3 σ fromthe corresponding mean value. We then compared the dispersionsof the cut shear distributions with the original dispersions. If the dif-ference between either of the two pairs of dispersions was greaterthan 0.05 of the original value (for an underlying Gaussian distri-bution, the corresponding difference would be . σ , we chose . σ to allow for deviations from Gaussianity), the suite of sim-ulations was replaced by the cut suite. This procedure was iterateduntil the cut distributions were consistent with the uncut distribu-tions.Two further cuts were made to create the final sample of shearand rotation estimates. The first was a cut on the sample so thatwe removed any source with N eff < , where the definition of N eff is given in equation (45). This cut was made as the shear androtation estimates were deemed unreliable for low numbers of re-liable pixels due to large dispersions and substantial outliers in thesimulated distributions. The second cut was to remove any sourcefor which η total > . . Large dispersions between α obs and β obs due to measurement errors and astrophysical scatter produce largebiases in the estimates. These biases, in turn, require large bias cor-rections which again produce large outliers in the shear estimates;therefore, the cut on η total is made to avoid this problem. The cutswere made so that all shear and rotations deemed problematic wereexcluded from the sample. Ultimately such estimates would havebeen downweighted in subsequent use of the results, but it seemedprudent to seclude them from the beginning using well motivatedcuts. For future surveys, where a larger number of sources will beavailable, a more objective method of cutting the sample must bedevised.The final list of derived quantities for the cut sample is given intable 2. The shear and rotation estimates in table 2 have been recov-ered under the assumption that the contribution from Faraday rota-tion is negligible. If one assumes that the observed Faraday rotationsignal across a source is constant, one would expect the recoveredrotation estimate for that source to be the sum of the Faraday ro-tation signal and the sought after cosmological rotation signal. Inreality, however, the observed Faraday rotation signal varies acrossthe sources, and our method will be sensitive to these variations.It should also be noted that systematic effects may arise due to thefact that we only use high signal-to-noise regions of the sources;however, we attempt to calibrate for these effects by mimicking thecuts in the calibration simulations.Figure 7 shows the estimated rotation signal for the 30 sourcespresented in table 2 plotted against ∆ α , where we define ∆ α = RM × λ , (47)and where RM is the best-fit rotation measure for an observedsource (Simard-Normandin et al. 1981) and λ obs is the wavelengthof the observation. Where possible, the rotation measures used inFigure 7 were obtained from Simard-Normandin et al. (1981), andthe rest are from Tabara & Inoue (1980). If we assume that the Fara-day rotation across each source is constant and that there is zerocontribution from other systematics and cosmological rotation, onewould expect to see a one-to-one relationship between ˆ ω and ∆ α inFigure 7; however, variations in Faraday rotation across each sourceat the pixel level reduces this correlation. Therefore, in order to re-move contamination from Faraday rotation in future applicationsof our method, rotation measures should be mapped across eachsource and corrections for Faraday rotation made at the pixel level; c (cid:13) , 1–19 easuring cosmic shear and birefringence Source number Source ˆ γ ˆ γ ˆ ω (degs) η total (degs) (cid:104) α obs − β obs (cid:105) N eff . ± . − . ± .
148 0 . ± .
73 34 . ± . . ± . . ± . − . ± . − . ± .
47 38 . ± . − . ± . − . ± . − . ± . − . ± .
43 37 . ± . − . ± . − . ± . − . ± . − . ± . . ± . − . ± . . ± .
291 0 . ± .
296 3 . ± .
48 45 . ± . − . ± . − . ± . − . ± . − . ± .
09 34 . ± . − . ± . . ± . − . ± . − . ± .
56 44 . ± . − . ± . . ± . − . ± .
168 10 . ± . . ± . − . ± . − . ± .
318 0 . ± .
310 5 . ± .
52 43 . ± . . ± . . ± . − . ± . − . ± .
13 41 . ± . − . ± . . ± .
133 0 . ± . − . ± .
87 29 . ± . − . ± . − . ± .
425 0 . ± . − . ± . . ± . − . ± . − . ± . − . ± . − . ± . . ± . − . ± . − . ± . − . ± .
244 0 . ± .
37 42 . ± . − . ± . . ± . − . ± .
108 4 . ± .
06 32 . ± . − . ± . − . ± . − . ± . − . ± .
91 43 . ± . . ± . − . ± .
237 0 . ± .
254 5 . ± .
78 37 . ± . . ± . . ± .
150 0 . ± .
145 3 . ± .
39 34 . ± . . ± . − . ± .
114 0 . ± . − . ± . . ± . − . ± . . ± . − . ± .
174 7 . ± .
75 36 . ± . . ± . − . ± .
188 0 . ± .
186 12 . ± . . ± . . ± . − . ± . − . ± .
131 8 . ± .
18 35 . ± . . ± . . ± .
283 0 . ± . − . ± .
94 43 . ± . . ± . . ± . − . ± .
106 0 . ± .
90 31 . ± . − . ± . . ± . − . ± .
277 2 . ± .
21 39 . ± . . ± . . ± .
178 0 . ± .
207 13 . ± . . ± . − . ± . . ± .
159 0 . ± . − . ± . . ± . − . ± . − . ± .
302 0 . ± . − . ± .
94 46 . ± . . ± . . ± .
175 0 . ± . − . ± .
33 42 . ± . . ± . − . ± . − . ± . − . ± . . ± . − . ± . Table 2.
Derived properties from the FRII sample. The source number identifies the source with the estimates displayed in Figure 8.The columns labeled ˆ γ , ˆ γ , and ˆ ω show the estimates for the two components of the shear and the rotation respectively for each ofthe sources. The column labeled η total shows the total dispersion parameters fitted to the distributions of α obs − β obs , (cid:104) α obs − β obs (cid:105) shows the means of the measured α obs − β obs for each source (the error bars are estimates as η total / √ N eff ), and N eff shows thenumber of effective pixels used in the analysis of each source. uncertainties on these corrections must also be included at the pixellevel in the calibration simulations.In table 2, we see that there are a number of sources whichshow a significant rotation detection. The source of this signal isexpected to be dominated by Faraday rotation, and at present wedo not have the power to discriminate between Faraday rotationand birefringence. The most striking example is the source 3C441,which shows a ∼ σ detection of a rotation signal. However, forthis source we see a large value of <α obs − β obs > which we alsofound to be approximately equal to ∆ α , and so we expect that theestimated rotation signal for this source is dominated by Faradayrotation.The recovered shear and rotation estimates are displayed inFigure 8. Using the inverse variance of each estimate as a weight-ing, the means of recovered shear and rotation estimates are (cid:104) ˆ γ (cid:105) = (2 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − ,and (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ .For this analysis we have assumed a perfect knowledge of thebias correction terms; however, as these terms are estimated usingsimulations based on the observed images, there will be additionaluncertainties due to these terms. Assuming an accurate model ofthe beam and that the dispersion of α obs − β obs is dominated byastrophysical scatter, it is expected that errors in the bias correction terms will primarily come from errors in the astrophysical scattermodel assumed which, in turn, is dominated by errors on the fit tothe distribution of α obs − β obs . Assuming that the distribution of α obs − β obs follows equation (28), we can estimate the errors onthe estimates of η total by assuming Poisson errors on the values ineach bin of the histogram of measured α obs − β obs to which themodel is fitted and assuming that the fitted histogram is the truemodel. Using this approach, we estimated the errors on the fittedvalues of η total ; these errors are shown in table 2. We find that thefractional errors on η total are less than 10% in all cases.Using the estimates presented in table 2 and the source po-sitions from table 1, we calculated the two-point correlation func-tions (2PCFs) for the sample. The 2PCFs considered were ξ ± ( θ ) = (cid:80) w a w b [ γ t ( x a ) γ t ( x b ) ± γ x ( x a ) γ x ( x b )] (cid:80) w a w b ,ξ ω ( θ ) = (cid:80) w ω a w ω b ω ( x a ) ω ( x b ) (cid:80) w ω a w ω b ,ξ γω ( θ ) = (cid:80) (w a w ω b + w ω a w b ) ( | γ | ( x a ) ω ( x b ) + ω ( x a ) | γ | ( x b ))2 (cid:80) (w a w ω b + w ω a w b ) , (48)where the weights were taken to be the inverse mean square vari- c (cid:13)000
Derived properties from the FRII sample. The source number identifies the source with the estimates displayed in Figure 8.The columns labeled ˆ γ , ˆ γ , and ˆ ω show the estimates for the two components of the shear and the rotation respectively for each ofthe sources. The column labeled η total shows the total dispersion parameters fitted to the distributions of α obs − β obs , (cid:104) α obs − β obs (cid:105) shows the means of the measured α obs − β obs for each source (the error bars are estimates as η total / √ N eff ), and N eff shows thenumber of effective pixels used in the analysis of each source. uncertainties on these corrections must also be included at the pixellevel in the calibration simulations.In table 2, we see that there are a number of sources whichshow a significant rotation detection. The source of this signal isexpected to be dominated by Faraday rotation, and at present wedo not have the power to discriminate between Faraday rotationand birefringence. The most striking example is the source 3C441,which shows a ∼ σ detection of a rotation signal. However, forthis source we see a large value of <α obs − β obs > which we alsofound to be approximately equal to ∆ α , and so we expect that theestimated rotation signal for this source is dominated by Faradayrotation.The recovered shear and rotation estimates are displayed inFigure 8. Using the inverse variance of each estimate as a weight-ing, the means of recovered shear and rotation estimates are (cid:104) ˆ γ (cid:105) = (2 . ± . × − , (cid:104) ˆ γ (cid:105) = ( − . ± . × − ,and (cid:104) ˆ ω (cid:105) = − . ◦ ± . ◦ .For this analysis we have assumed a perfect knowledge of thebias correction terms; however, as these terms are estimated usingsimulations based on the observed images, there will be additionaluncertainties due to these terms. Assuming an accurate model ofthe beam and that the dispersion of α obs − β obs is dominated byastrophysical scatter, it is expected that errors in the bias correction terms will primarily come from errors in the astrophysical scattermodel assumed which, in turn, is dominated by errors on the fit tothe distribution of α obs − β obs . Assuming that the distribution of α obs − β obs follows equation (28), we can estimate the errors onthe estimates of η total by assuming Poisson errors on the values ineach bin of the histogram of measured α obs − β obs to which themodel is fitted and assuming that the fitted histogram is the truemodel. Using this approach, we estimated the errors on the fittedvalues of η total ; these errors are shown in table 2. We find that thefractional errors on η total are less than 10% in all cases.Using the estimates presented in table 2 and the source po-sitions from table 1, we calculated the two-point correlation func-tions (2PCFs) for the sample. The 2PCFs considered were ξ ± ( θ ) = (cid:80) w a w b [ γ t ( x a ) γ t ( x b ) ± γ x ( x a ) γ x ( x b )] (cid:80) w a w b ,ξ ω ( θ ) = (cid:80) w ω a w ω b ω ( x a ) ω ( x b ) (cid:80) w ω a w ω b ,ξ γω ( θ ) = (cid:80) (w a w ω b + w ω a w b ) ( | γ | ( x a ) ω ( x b ) + ω ( x a ) | γ | ( x b ))2 (cid:80) (w a w ω b + w ω a w b ) , (48)where the weights were taken to be the inverse mean square vari- c (cid:13)000 , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 6.
The recovered shear and rotation estimates for each of the sources shown in Figure 4. In each panel, the recovered estimateis shown as the red dashed line. The green line indicates a zero signal. The black histogram shows the recovered estimates from simulated noise realizations, as discussed in the main text. The black dashed line is the mean estimate from the simulations. Thenumerical results for all of the sources are given in table 2. ance of the estimates. The tangential and cross shear γ t and γ x arethe components of the shear rotated into a frame of reference whichaligns with the separation vector θ = x a − x b . The modulus of theseparation vector, θ , was calculated for each source pair, and thesource pairs were then binned according to their separation intobins of width ∆ θ = 45 ◦ . The summations in equation (48) werecarried out over all source pairs for each bin.The results for the 2PCFs are shown in Figure 9. The blackcurves in the top four panels are theoretical predictions of the shear2PCFs calculated with CosmoSIS using the small angle approxi-mation, and hence the theoretical spectra are only accurate to ∼ ◦ but serve to give an indication of the constraining power of our https://bitbucket.org/joezuntz/cosmosis/wiki/Home method for the small sample of sources used. We assume a Λ CDMcosmology with the Planck best-fit parameters Ω m = 0 . , h = 0 . , Ω b = 0 . , n s = 0 . , A s = 2 . × − ,and σ = 0 . (Planck Collaboration et al. 2016). For the red-shift distribution, we used a cubic spline fit to the distribution ofredshifts for the full sample of 58 sources, shown in Figure 10.From Figure 9, we see that the 95% confidence upper limitson the shear correlation functions are approximately seven ordersof magnitude larger than the predicted correlation functions for thissmall sample of sources. Given that we have only used 30 sourcesfor this analysis, this level of constraining power comes as no sur-prise, and for future surveys where a much larger number of sourceswill be available, such as those performed with the SKA, we expectthat it should be possible to recover a clear detection of the signal;this is discussed in more detail in the following section. One ex- c (cid:13) , 1–19 easuring cosmic shear and birefringence Figure 7.
The recovered rotation estimate and corresponding ∆ α (definedin equation 47) for each of the sources in table 2. The line is the best-fit to the data. The Pearson’s correlation coefficient was calculated to be ρ corr = 0 . ± . , where the errors have been estimated by constructinga distribution of correlation coefficients using random permutations ofthe set of rotation estimates with respect to the set of ∆ α . This indicatesthat there is a positive correlation between ∆ α and the rotation estimate, asexpected. citing result that we see in Figure 9 is that the average rotationsignal has been constrained to ∼ ◦ , which is competetive with cur-rent constraints using CMB observations (see e.g. Kaufman et al.(2016)). Using future radio surveys, we expect that contraints onthe rotation signal may be increased by at least an order of magni-tude. In the previous section, we placed constraints on the shear and ro-tation 2PCFs using only 30 resolved radio sources. Future high res-olution radio surveys will provide a much higher number of wellresolved sources, and in this section we investigate the number ofsources that would be required to make a clear detection of shearpower. To achieve this, we define a figure of merit (FOM) based onthe total signal-to-noise of the power spectrum and assume a toymodel based on a SKA -like survey.For SKA2, the total sky coverage is expected to be ∼ deg , giving a fractional sky coverage of f sky ∼ l min = 2 . Assuming that l max = 2 N side (using HEALPix terminology) and that we haveone source in each pixel, we have N source N pix = N source N f sky = N source l f sky = 1 , (49) http://healpix.sourceforge.net/index.php FOM N source l max σ (cid:104) ω (cid:105) ( × − degs )1 . ×
198 7.163 . ×
337 4.215 . ×
439 3.2310 . ×
651 2.18
Table 3.
The required N source to achieve a detection of shear power fora SKA-like survey with FOM = 1, 3, 5, and 10. In the last column, wealso present the expected error on a global rotation, (cid:104) ω (cid:105) , for this number ofsources. so that l max = (cid:115) N source f sky . (50)The noise power spectrum is given by N l = 4 πσ N source /f sky , (51)where σ rms is the r.m.s. error of the shear estimates for each pixel.As future surveys should provide a high number of high resolutionradio sources, for the following discussion, we set σ rms = 8 . × − , which is the minimum error for the shear estimates given intable 2. The errors on the estimates of the C l s are then σ C l = 2(2 l + 1) f sky ( C l + N l ) . (52)In order to place constraints on the number of galaxies re-quired to make a detection of the shear power spectrum, we binall of the estimated C l s into one bin. Assuming that the errors onthe mean C l in the bin are Gaussian distributed, we can then es-timate how many sources are required to make a detection of thetotal shear power at a given confidence level. To do this, we definea FOM for the C l estimates as FOM = N (cid:80) l max l = l min C l σ < ˆ C l > , (53)where σ < ˆ C l > is the error on the mean estimated C l in the singlebin and l max is a function of the number of sources (equation (50)).Equation (53) can be written in terms of σ C l as FOM = (cid:80) l max l = l min C l (cid:113)(cid:80) l max l = l min σ C l . (54)We calculated the theoretical shear power spectrum for a hy-pothetical SKA2-like radio survey using CosmoSIS . We assumeda Λ CDM model with the Planck best-fit cosmological parame-ters discussed in the previous section. We also assumed the red-shift distribution fitted to the Combined EIS-NVSS Survey Of Ra-dio Sources (CENSORS) (Best et al. 2003; Brookes et al. 2008;Marcos-Caballero et al. 2013), which is given as d n d z = A (cid:18) zz (cid:19) α e − αz/z , (55)where z is the mode of the distribution, α is a shape parameter,and A is a normalization constant. The best-fit parameters to theCENSORS data are z = 0 . +0 . − . and α = 0 . +0 . − . . We thencalculated the FOM, given in equation (53), using equation (52)with f sky = 0 . and σ rms = 8 . × − , as discussed above. c (cid:13) , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 8.
The shear and rotation estimates for all of the sources in the final sample given in table 2. The x-axis of the top and bottom-leftpanels indicates the corresponding source, with the identification numbers defined in the first column of table 2. The bottom-right panelshows the shear estimates in the γ − γ plane. From these plots we see no clear indication of residual systematics in the estimates. The results of this test are shown in Figure (11). We have indi-cated the positions where FOM = 1, 3, 5, and 10, corresponding toa σ , σ , σ , and σ detection of shear power respectively. Thenumerical values are given in table 3. We find that for N source ∼ the shear power is approximately equal to the uncertainty, which isexpected to be dominated by errors on the shear estimates. A σ detection of shear power is expected for N source ∼ × . We alsosee that the maximum multipole which may be probed is l max ∼ for N source ∼ and increases to l max ∼ for N source ∼ . Weexpect that these numbers of well resolved radio sources should beachievable with future surveys.In the last column of table 3, we also show the predicted un-certainty on a global rotation estimate for these source numbers.As with the shear, we have assumed an average error on the rota-tion estimates equal to the minimum error in table 2. We see thatwith these numbers of sources, the errors on (cid:104) ω (cid:105) are ∼ − de- grees. Hence, our estimator has the potential to provide powerfulconstraints on a global rotation signal. We find that the polarization position angles in resolved radiosources are correlated with the position angles of the gradient ofthe total intensity field. We have shown that this correlation is bro-ken in the presence of a lensing signal and/or an overall rotation ofthe plane of polarization, and we have formulated an estimator thatexploits this effect to provide an estimate of the shear and rotationsignals.The estimator has been successfully demonstrated on simu-lations, where we have shown that for very high signal-to-noisesources and assuming zero astrophysical scatter between the polar- c (cid:13) , 1–19 easuring cosmic shear and birefringence Figure 9.
Upper (95% confidence) limits for the shear and rotation 2PCFs calculated using equation (48). The black curves in the top twopanels show the theoretical 2PCFs calculated assuming the Planck best-fit parameters and the spline fit to the source redshift distributionshown in Figure 10. ization and gradient orientations, the bias in the estimator is negli-gible. In the presence of a more realistic signal-to-noise level andastrophysical scatter between the orientations, we find that there isa bias in the estimates. We have developed a method which uses thetotal intensity and polarization maps for a given source to producecalibration simulations in order to mitigate this effect.Using data observed with the VLA and MERLIN array, wehave measured the shear and rotation signals in the directions of 30sources and used these estimates to place constraints on the corre-lation functions. We find that we can constrain the rotation signalto < ◦ with this number of sources. This level of constraint is al-ready comparable with constraints from CMB obsevations. Futureradio surveys, such as those conducted with the SKA, are expectedto deliver a much larger number of higher signal-to-noise sourceswhich should provide a detection of the shear signal, enabling themethod to be used to constrain cosmology. We will also be provided with much tighter constraints on the rotation signal, and hence oncosmic birefringence.The estimate of a cosmological rotation signal was − . ◦ ± . ◦ , where an emphasis should placed on the promising size ofthe error and not upon the magnitude of the estimated signal sincewe do not include a correction for Faraday rotation. In future sur-veys, where a much greater level of constraint on the rotation signalis expected, the effects of Faraday rotation will become more prob-lematic, and a map of the rotation measure across a source will berequired. However, assuming that the effects of Faraday rotation,and other systematics, can be successfully removed from estimatesof the rotation, the level of constraining power on the rotation sig-nal for our approach should be orders of magnitude greater thanthose currently available.A clear detection of the shear signal should be possible for fu-ture high-precision surveys, which will yield a large number of well c (cid:13) , 1–19 Lee Whittaker, Richard A. Battye & Michael L. Brown
Figure 10.
The redshift distribution of the full sample of 58 sources. Thered curve is a cubic spline fit, which was used to calculate the theoreticalshear 2PCFs shown in Figure 9.
Figure 11.
The FOM for an SKA2-like survey as a function of N source .The numerical results are given in table 3. resolved sources. One advantage of our approach is that the shearestimates require no assumptions to be made about the intrinsicsource shapes. We are therefore insensitive to intrinsic alignmentcontamination. Also, the high resolution and depth of future sur-veys should provide many sources with a large number of effectivepixels above the signal-to-noise threshold. This could potentiallyreduce the errors on the shear estimates to well below the level ofirreducible shape noise inherent in traditional approaches to weaklensing. ACKNOWLEDGMENTS
We thank Paddy Leahy and Martin Hardcastle for providing thedata used in this paper. We also thank Ian Browne and Paddy Leahyfor useful comments. LW and MLB are grateful to the ERC for sup-port through the award of an ERC Starting Independent ResearcherGrant (EC FP7 grant number 280127). MLB also thanks the STFCfor the award of Advanced and Halliday fellowships (grant numberST/I005129/1).
REFERENCES
Abbott T., Abdalla F. B., Allam S., Amara A., Annis J., Arm-strong R., Bacon D., Banerji M., Bauer A. H., Baxter E., BeckerM. R., Benoit-L´evy A., Bernstein R. A., Bernstein G. M., BertinE., Blazek J., Bonnett C., Bridle S. L., Brooks D., Bruderer C.,Buckley-Geer E., Burke D. L., Busha M. T., Capozzi D., CarneroRosell A., Carrasco Kind M., Carretero J., Castander F. J., ChangC., Clampitt J., Crocce M., Cunha C. E., D’Andrea C. B., daCosta L. N., Das R., DePoy D. L., Desai S., Diehl H. T., DietrichJ. P., Dodelson S., Doel P., Drlica-Wagner A., Efstathiou G., Ei-fler T. F., Erickson B., Estrada J., Evrard A. E., Fausti Neto A.,Fernandez E., Finley D. A., Flaugher B., Fosalba P., FriedrichO., Frieman J., Gangkofner C., Garcia-Bellido J., Gaztanaga E.,Gerdes D. W., Gruen D., Gruendl R. A., Gutierrez G., HartleyW., Hirsch M., Honscheid K., Huff E. M., Jain B., James D. J.,Jarvis M., Kacprzak T., Kent S., Kirk D., Krause E., KravtsovA., Kuehn K., Kuropatkin N., Kwan J., Lahav O., Leistedt B., LiT. S., Lima M., Lin H., MacCrann N., March M., Marshall J. L.,Martini P., McMahon R. G., Melchior P., Miller C. J., Miquel R.,Mohr J. J., Neilsen E., Nichol R. C., Nicola A., Nord B., OgandoR., Palmese A., Peiris H. V., Plazas A. A., Refregier A., Roe N.,Romer A. K., Roodman A., Rowe B., Rykoff E. S., Sabiu C.,Sadeh I., Sako M., Samuroff S., Sanchez E., S´anchez C., Seo H.,Sevilla-Noarbe I., Sheldon E., Smith R. C., Soares-Santos M.,Sobreira F., Suchyta E., Swanson M. E. C., Tarle G., Thaler J.,Thomas D., Troxel M. A., Vikram V., Walker A. R., WechslerR. H., Weller J., Zhang Y., Zuntz J., Dark Energy Survey Collab-oration, 2016, Phys. Rev. D, 94, 022001Ade P. A. R., Arnold K., Atlas M., Baccigalupi C., Barron D.,Boettger D., Borrill J., Chapman S., Chinone Y., Cukierman A.,Dobbs M., Ducout A., Dunner R., Elleflot T., Errard J., FabbianG., Feeney S., Feng C., Gilbert A., Goeckner-Wald N., Groh J.,Hall G., Halverson N. W., Hasegawa M., Hattori K., Hazumi M.,Hill C., Holzapfel W. L., Hori Y., Howe L., Inoue Y., JaehnigG. C., Jaffe A. H., Jeong O., Katayama N., Kaufman J. P., Keat-ing B., Kermish Z., Keskitalo R., Kisner T., Kusaka A., Le Je-une M., Lee A. T., Leitch E. M., Leon D., Li Y., Linder E.,Lowry L., Matsuda F., Matsumura T., Miller N., MontgomeryJ., Myers M. J., Navaroli M., Nishino H., Okamura T., Paar H.,Peloton J., Pogosian L., Poletti D., Puglisi G., Raum C., RebeizG., Reichardt C. L., Richards P. L., Ross C., Rotermund K. M.,Schenck D. E., Sherwin B. D., Shimon M., Shirley I., SiritanasakP., Smecher G., Stebor N., Steinbach B., Suzuki A., Suzuki J.-i.,Tajima O., Takakura S., Tikhomirov A., Tomaru T., WhitehornN., Wilson B., Yadav A., Zahn A., Zahn O., Polarbear Collabo-ration, 2015, Phys. Rev. D, 92, 123509Best P. N., Arts J. N., R¨ottgering H. J. A., Rengelink R., BrookesM. H., Wall J., 2003, MNRAS, 346, 627 c (cid:13) , 1–19 easuring cosmic shear and birefringence Black A. R. S., Baum S. A., Leahy J. P., Perley R. A., Riley J. M.,Scheuer P. A. G., 1992, MNRAS, 256, 186Brookes M. H., Best P. N., Peacock J. A., R¨ottgering H. J. A.,Dunlop J. S., 2008, MNRAS, 385, 1297Brown M., Bacon D., Camera S., Harrison I., Joachimi B., MetcalfR. B., Pourtsidou A., Takahashi K., Zuntz J., Abdalla F. B., Bri-dle S., Jarvis M., Kitching T., Miller L., Patel P., 2015, Advanc-ing Astrophysics with the Square Kilometre Array (AASKA14),23Brown M. L., Battye R. A., 2011, MNRAS, 410, 2057Burns C. R., Dyer C. C., Kronberg P. P., R¨oser H., 2004, ApJ, 613,672Caldwell R. R., Gluscevic V., Kamionkowski M., 2011,Phys. Rev. D, 84, 043504Carroll S. M., Field G. B., 1997, Physical Review Letters, 79,2394Carroll S. M., Field G. B., Jackiw R., 1990, Phys. Rev. D, 41,1231Catelan P., Kamionkowski M., Blandford R. D., 2001, MNRAS,320, L7Chang T., Refregier A., Helfand D. J., 2004, ApJ, 617, 794Colladay D., Kosteleck´y V. A., 1998, Phys. Rev. D, 58, 116002Crittenden R. G., Natarajan P., Pen U., Theuns T., 2001, ApJ, 559,552Croft R. A. C., Metzler C. A., 2000, ApJ, 545, 561Demetroullas C., Brown M. L., 2016, MNRAS, 456, 3100di Serego Alighieri S., Ni W.-T., Pan W.-P., 2014, ApJ, 792, 35Gruppuso A., Gerbino M., Natoli P., Pagano L., Mandolesi N.,Melchiorri A., Molinari D., 2016, J. Cosmology Astropart.Phys., 6, 001Harari D., Sikivie P., 1992, Physics Letters B, 289, 67Heavens A., Refregier A., Heymans C., 2000, MNRAS, 319, 649Heymans C., Grocutt E., Heavens A., Kilbinger M., KitchingT. D., Simpson F., Benjamin J., Erben T., Hildebrandt H., Hoek-stra H., Mellier Y., Miller L., Van Waerbeke L., Brown M. L.,Coupon J., Fu L., Harnois-D´eraps J., Hudson M. J., Kuijken K.,Rowe B., Schrabback T., Semboloni E., Vafaei S., Velander M.,2013, MNRAS, 432, 2433Hildebrandt H., Viola M., Heymans C., Joudaki S., Kuijken K.,Blake C., Erben T., Joachimi B., Klaes D., Miller L., MorrisonC. B., Nakajima R., Verdoes Kleijn G., Amon A., Choi A., Cov-one G., de Jong J. T. A., Dvornik A., Fenech Conti I., GradoA., Harnois-D´eraps J., Herbonnet R., Hoekstra H., K¨ohlinger F.,McFarland J., Mead A., Merten J., Napolitano N., Peacock J. A.,Radovich M., Schneider P., Simon P., Valentijn E. A., van denBusch J. L., van Uitert E., Van Waerbeke L., 2017, MNRAS,465, 1454Kaufman J. P., Keating B. G., Johnson B. R., 2016, MNRAS, 455,1981Kilbinger M., Fu L., Heymans C., Simpson F., Benjamin J., Er-ben T., Harnois-D´eraps J., Hoekstra H., Hildebrandt H., KitchingT. D., Mellier Y., Miller L., Van Waerbeke L., Benabed K., Bon-nett C., Coupon J., Hudson M. J., Kuijken K., Rowe B., Schrab-back T., Semboloni E., Vafaei S., Velander M., 2013, MNRAS,430, 2200Kosteleck´y V. A., Mewes M., 2009, Phys. Rev. D, 80, 015020Kronberg P. P., Dyer C. C., Burbidge E. M., Junkkarinen V. T.,1991, ApJ, 367, L1Kronberg P. P., Dyer C. C., Roeser H., 1996, ApJ, 472, 115Leahy J. P., Black A. R. S., Dennett-Thorpe J., Hardcastle M. J., Komissarov S., Perley R. A., Riley J. M., Scheuer P. A. G., 1997,MNRAS, 291, 20Lee S., Liu G.-C., Ng K.-W., 2015, Physics Letters B, 746, 406Loredo T. J., Flanagan E. E., Wasserman I. M., 1997,Phys. Rev. D, 56, 7507Marcos-Caballero A., Vielva P., Martinez-Gonzalez E., Finelli F.,Gruppuso A., Schiavon F., 2013, ArXiv e-printsMullin L. M., Riley J. M., Hardcastle M. J., 2008, MNRAS, 390,595Namikawa T., 2017, Phys. Rev. D, 95, 043523Nodland B., Ralston J. P., 1997, Physical Review Letters, 78, 3043Planck Collaboration, Ade P. A. R., Aghanim N., Arnaud M.,Ashdown M., Aumont J., Baccigalupi C., Banday A. J., BarreiroR. B., Bartlett J. G., et al., 2016, A&A, 594, A13Schrabback T., Hartlap J., Joachimi B., Kilbinger M., Simon P.,Benabed K., Bradaˇc M., Eifler T., Erben T., Fassnacht C. D.,High F. W., Hilbert S., Hildebrandt H., Hoekstra H., KuijkenK., Marshall P. J., Mellier Y., Morganson E., Schneider P., Sem-boloni E., van Waerbeke L., Velander M., 2010, A&A, 516, A63Simard-Normandin M., Kronberg P. P., Button S., 1981, ApJS, 45,97Stil J. M., Krause M., Beck R., Taylor A. R., 2009, ApJ, 693, 1392Tabara H., Inoue M., 1980, A&AS, 39, 379Whittaker L., Brown M. L., Battye R. A., 2014, MNRAS, 445,1836—, 2015, MNRAS, 451, 383 c (cid:13)000