Do angular momentum induced ellipticity correlations contaminate
weak lensing measurements?
Priyamvada Natarajan , Robert G. Crittenden,
Ue-Li Pen \& Tom Theuns
, PASA, 18 (2), in press.

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Schematic Outline

We briefly outline the calculation here, details can be found in the following two papers Crittenden et al. (2001a) and Crittenden et al. (2001b). To estimate the strength of intrinsic ellipticity correlations, we approximate the projected shape of a galaxy on the sky by an ellipsoid with semi-axes a, b (a > b). The orientation of the ellipsoid is given by the angle $\psi$ between the major axis and the chosen coordinate system, while its magnitude is given by

$\vert\epsilon \vert = (a^2 - b^2)/(a^2 + b^2)$ . Both the magnitude of the ellipticity and its orientation can be concisely described by the complex quantity

$\epsilon^{(o)}$ ,

$\displaystyle \epsilon^{(o)} = \vert\epsilon^{(o)}\vert e^{2i\psi} = \epsilon_{+}^{(o)} + i \epsilon_{\times}^{(o)}.$

(1)

where the superscript ^(o) denotes the observed shape. In the linear regime and under the assumption of weak lensing, the lensing equation can be written as,

$\displaystyle \epsilon^{(o)}\,=\,\frac{\epsilon\,+\,g}{1\,+\,g^{*}\epsilon},$

(2)

where g is the complex shear and $\epsilon$ the intrinsic shape of the source (Kochanek 1990; Miralda-Escude 1991). Furthermore, in the weak regime, correlations of this distortion field are

$\displaystyle \langle{\epsilon^{(o)}}({\mathbf{x_1}})\,{\epsilon^{(o)*}}({\math... ...f{x_2}})\rangle \,+\,\langle{g({\mathbf{x_1}})}{g^{*}}({\mathbf{x_2}})\rangle\,$

(3)

where the ^* denotes complex conjugation. The first term is the contribution that arises from intrinsic shape correlations. Previous analyses have focused on the third term of this expression, correlations due to weak lensing. We assume in the calculation that shape correlations arise primarily from correlations in the direction of the angular momentum vectors of neighboring galaxies. Spiral galaxies are disk-like with the angular momentum vector perpendicular to the plane of the disk, so that angular momentum couplings will be translated into shape correlations. We will assume that for ellipticals the angular momentum vector also lies along its shortest axis on average, as it does for the spirals. However, since elliptical galaxies are intrinsically rounder, the correlation amplitude will be smaller. We use the observed ellipticity distributions of each morphological type (from the APM survey) in the computation of the shape correlations. For weak lensing, in contrast, the induced shape correlations are independent of the original shapes of the lensed galaxies.

**Figure 1:** The intrinsic correlation signal versus the predictions from weak lensing and current observations. Left panel: predictions for a shallower survey such as SDSS and 2dF with z_m=0.1. The intrinsic signal is plotted for 2 values of a, and the theoretical prediction for weak lensing is the long-dashed line (for z_m=0.1) and dotted-long-dashed (for z_m=0.5). The lensing prediction for z_m=0.1 is extrapolated from the Jain & Seljak fit beyond the stated range of validity. For such low redshifts the intrinsic signal dominates on most scales. Right panel:
$\xi _+(\theta )+\xi _\times (\theta )$ the intrinsic signal for z_m = 1, compared to the measured shear correlation function. At small separations, the intrinsic signal is approximately 1% of the lensing signal. The amplitude depends on the assumed average galaxy thickness ( $\alpha$ ) and the parameter a that describes how well the angular momentum of the galaxy is correlated with the distortion field. We plot a=0.24 (full line) and a=0.55 (short-dashed line) which correspond to the values inferred from the numerical simulations of Lee & Pen (2000) and Heavens et al. (2000). $\alpha =0.73$ corresponds to the value determined from the observed distribution of ellipticities. The data are: van Waerbeke et al. (2000) - solid squares ; Wittman et al. (2000) - filled circles ; Kaiser et al. (2000) - open circles; and Bacon et al. (2000) - filled triangle. The long-dashed line is the extrapolated theoretical prediction from Jain & Seljak (1997) computed for a

$\Omega _\Lambda =0.7$ galaxy cluster normalized flat universe,

$\sim 4.75\times 10^{-4}(\theta /{\rm arc min})^{-0.84}$ .
$\begin{figure} \centerline{\psfig{file=fig_ncpt1.ps,height=2.8in} \psfig{file=fig_ncpt2.ps,height=2.8in}} \end{figure}$

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