# A new observable for cosmic shear

Jérémie Francfort  Ruth Durrer  and Giulia Cusin

###### Abstract

In this paper we introduce a new observable to measure cosmic shear. We show
that if we can measure with good accuracy both, the orientation of a galaxy
and the polarisation direction of its radio emission, the angle between them
is sensitive to the foreground cosmic shear. Even if the signal-to-noise ratio
for a single measurement is expected to be rather small, the fact that all
galaxies in a given pixel are subject to the same shear can be used to
overcome the noise. An additional advantage of this observable is that the
signal is not plagued by intrinsic alignment. We estimate the SNR for the
shear correlation functions $\zeta_{\pm}(\mu,z_{1},z_{2})$ measured in this
way with the future SKA II survey.

##  1 Introduction

Cosmic shear is the coherent deformation of images of background galaxies due
to gravitational field. It gives us precious information about the total
foreground matter density as it is sensitive to both, dark and luminous matter
alike.

However, shear measurements are very difficult. They typically modify the
ellipticity of a galaxy by about 1% or even less [1]. Furthermore, the shear
correlation function is affected by so called intrinsic alignment which can be
of the same order as the shear itself [2, 3]. Nevertheless, in recent years
several observational campains like KiDs (Kilo Degree Survey) and DES (Dark
Energy Survey) and HSC (Hyper Supreme-Cam) have measured the shear correlation
function in different redshift bins, see e.g. [4, 5, 6, 7, 8, 9, 10]. The
shear correlation function is a very important variable to measure
cosmological parameters and, more importantly, to test the consistency of the
cosmological standard model $\Lambda$CDM.

The shear from scalar perturbations is determined by the lensing potential,

$\phi({\boldsymbol{n}},z)=-\int_{0}^{r(z)}\mathrm{d}r\frac{r(z)-r}{r(z)r}\left[\Phi(r{\boldsymbol{n}},t_{0}-r)+\Psi(r{\boldsymbol{n}},t_{0}-r)\right]\,.$
(1.1)

Here $\Phi$ and $\Psi$ are the Bardeen potentials, $r(z)$ is the comoving
distance out to redshift $z$, $t=t_{0}-r$ is conformal time along the light
path and ${\boldsymbol{n}}$ is a direction in the sky. We neglect possible
contributions from tensor perturbations, i.e. gravitational waves, as well as
from vector perturbations since they are generally small [11, 12]. Also, Eq.
(1.1) is the so called ’Born approximation’, i.e. we compute the lensing
potential along the straight, unlensed path, assuming that lensing is a small
perturbation. For non-relativistic matter and a cosmological constant the two
Bardeen potentials are equal and correspond to the Newtonian gravitational
potential. Light from a source at redshift $z$, seen in direction
${\boldsymbol{n}}$ is coming to us from the angular position
${\boldsymbol{n}}+{\boldsymbol{\nabla}}\phi({\boldsymbol{n}},z)$, where
${\boldsymbol{\nabla}}$ denotes the 2D gradient on the unit sphere and
${\boldsymbol{\nabla}}\phi({\boldsymbol{n}},z)$ is the deflection angle.

The shear $\gamma_{ij}$ is given by the traceless part of the second angular
derivatives of $\phi$. The convergence, given by the angular Laplacian can be
measured by galaxy number counts, see e.g. [13, 14, 15] for theoretical
aspects and numerical simulations and [16, 17] for observations.

Usually, the shear is measured via the correlation of the direction of the
ellipticity of galaxies. This assumes that ellipticities are intrinsically
uncorrelated which is evidently not true for galaxies at similar redshifts and
is even relevant for different redshifts, see e.g. [18] for a discussion of
intrinsic alignment. In this paper we derive a new observable which can be
used to measure the shear correlation function and which does not depend on
’intrinsic alignment’ : It is well known that the polarisation of a photon is
parallel transported along its path. However, a small image of finite size is
Lie transported. This is in general described with the Jacobi map [19].
Therefore, if the light from a galaxy is polarised, which is usually the case
for radio galaxies, and if this polarisation is aligned with the ellipticity
of the galaxy, which is also typically the case, this alignment is affected by
foreground shear. Typically, the angle between the polarisation vector and the
axes of the galaxy is of the order of a few degrees, see [20] for more
details. It might also be useful to measure the galaxy shapes with near future
optical telescopes like LSST [21] or the Euclid satellite [22] but the
polarisation has to be measured in the radio since these are the wavelengths
of synchrotron radiation whose polarisation is correlated with the intrinsic
direction of the galaxy.

If the principle axes of the shear tensor and the intrinsic ellipticity of the
galaxy are not aligned, this leads to a slight rotation of the image with
respect to the polarisation, as we have shown in a previous paper [23]. In
that paper we have studied the effect considering galaxies as Schwarzschild
lenses. In this work, we use shear from linear cosmological perturbation
theory and want to outline how one can use the correlation of the orientation
of the image and the polarisation to measure the shear correlation function.
The class of sources we have in mind in this analysis are low frequency radio
galaxies (typically 1-50 GHz as lower frequencies are significantly
depolarised by Faraday rotation [24]), for which the dominant source of linear
polarisation is expected to be synchrotron radiation due to electrons moving
in the magnetic field of the galaxy. For these objects, the magnetic field is
dominantly in the galactic plane (the orthogonal component is very small) and
tends to be aligned with galaxy morphology, i.e. the semi-major axis of the
galaxy (see e.g. [25]). Then polarisation from synchrotron radiation is mainly
orthogonal to the magnetic field component (i.e. it is in the orbital plane).
Hence its projected component (on the observer’s screen) is normal to the
galaxy’s major axis.

Previous authors have exploited the fact that the polarisation position angle
is unaffected by lensing in order to measure gravitational lensing of distant
quasars, see [26, 27, 28]. In [29], the authors proposed to use the
polarisation information in radio galaxies as an indicator of the galaxy
intrinsic shape, with the goal of mitigating shot noise and intrinsic
alignement uncertainties in shear reconstruction. In [30], the same authors
extended this idea to reconstruct maps of the projected dark matter
distribution, or the lensing convergence field. The authors of [31] proposed
to use a proxy for the intrinsic position angle of an observed galaxy, and
propose techniques for cleanly separating weak gravitational lensing signals
from intrinsic alignment contamination in forthcoming radio surveys. Finally,
in [32] it is shown that, thanks to polarisation information, radio weak
lensing surveys will be able to mitigate contamination by intrinsic
alignments, in a way similar but fully complementary to available self-
calibration methods based on position-shear correlations.

Unlike all these works, where the polarisation direction is used to have a
better handle on intrinsic alignment (inferred from the polarisation direction
itself), we propose to measure the offset between the observed polarisation
and galaxy morphology as a new observable on its own. In other words, although
the idea of using polarisation information to access the galaxy intrinsic
orientation is widely explored around in the literature, we believe that this
is the first time where a shear estimator is explicitly written down in terms
of the offset between the (observed) galaxy major axis and polarisation
orientation. A first attempt to do weak lensing with radio surveys is
published in [33]. In this first work, however polarisation is not used.

In Ref. [34] the authors do consider rotation but not the rotation induced by
shear which is considered in the present paper, rather they consider the
rotation from an antisymmetric contribution to the Jacobi map which is much
smaller than shear as it appears only at second order in the perturbations
[35].

This paper is structured as follows. In the next section we develop the
theoretical expressions which determine the shear from a measured angle
$\delta\alpha$ by which the orientation of the galaxy and its polarisation
differ. In Section 3 we present a rough estimate of the error on the
measurement given a typical precision of measured angles. In Section 4 we
discuss our results and in Section 5 we conclude. Some useful properties of
Spin Weighted Spherical Harmonics are presented in Appendix A for
completeness. In Appendix B we derive in detail the error estimates used in
the main text.

Notations and conventions:
We use the signature $(-,+,+,+)$.
The usual spherical angles are $(\theta,\varphi)$, and the corresponding unit
vector is $\boldsymbol{n}$. The surface element of the sphere is denoted
$\mathrm{d}\Omega$. The lensing potential is called $\phi(\boldsymbol{n},z)$.
The Bardeen potentials are $\Phi$ and $\Psi$. The Spin Weighted Spherical
Harmonics are ${}_{s}Y_{\ell,m}$, while the ’usual’ Spherical Harmonics,
${}_{0}Y_{\ell,m}$, are simply denoted $Y_{\ell,m}$.

Figure 1:  The general setup (in $2D$, seen from above): Two pixels are
considered, in directions $\boldsymbol{n}_{j}$ and at redshifts $z_{j}$. The
directions are separated by an angle $\varphi$ (or
$\boldsymbol{n}_{1}\cdot\boldsymbol{n}_{2}=\cos\varphi=\mu$). In each pixel,
one galaxy is chosen (represented by the dots). The computations are made in
the equatorial plane.

##  2 Theoretical development

The lensing potential given in Eq. (1.1) is a stochastic quantity which can be
decomposed into Spherical Harmonics as

$\phi(\boldsymbol{n},z)=\sum_{\ell,m}\phi_{\ell,m}(z)Y_{\ell,m}(\boldsymbol{n})\,,$
(2.1)

where the scalars $\phi_{\ell,m}(z)$ are also random variables. Assuming
statistical isotropy different values of $\ell$ and $m$ are not correlated and
their two-point correlation spectrum is given by

$\langle\phi_{\ell_{1},m_{1}}(z_{1})\phi^{*}_{\ell_{2},m_{2}}(z_{2})\rangle=C_{\ell_{1}}(z_{1},z_{2})\delta_{\ell_{1},\ell_{2}}\delta_{m_{1},m_{2}}\,.$
(2.2)

The $C_{\ell}(z_{1},z_{2})$ are the lensing power spectra for different
resdhifts $z_{1}$ and $z_{2}$. If the fluctuations are Gaussian, these power
spectra encode all the statistical information of the lensing potential. The
lensing potential contains very useful information e.g. about the matter
distribution in the Universe which is not plagued by the biasing problem of
galaxy number counts. Therefore estimating it using different measurements
with different systematics is very important.

In this section we present the main theoretical tools and formulas of the
article. More explanations and details can be found in the Appendix.

We consider radio galaxies which are polarised along their semi-major (or
minor) axis. This polarisation is parallel transported and hence its
components expressed in a parallel transported Sachs basis are constant. The
radio galaxy, represented by an ellipse is sheared and magnified according to
the Jacobi map. If the principle axes of the shear are not aligned with the
principle axes of the galaxy, this leads to a rotation of the galaxies
principle axes expressed in the Sachs basis. In our previous work [23] we have
calculated this rotation which is given by

$\delta\alpha=\frac{\varepsilon^{2}}{2-\varepsilon^{2}}\left(\gamma_{2}\cos
2\alpha-\gamma_{1}\sin 2\alpha\right)\,.$ (2.3)

Here $\varepsilon$ is the eccentricity of the galaxy,
$(\gamma_{1},\gamma_{2})$ are the components of the shear matrix in the Sachs
basis,

$\boldsymbol{\Gamma}=\left(\begin{array}[]{cc}-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}&+\gamma_{1}\end{array}\right)\,,$ (2.4)

and $\alpha$ is the angle between the major-axis of the galaxy shape and the
first basis vector $\boldsymbol{e}_{1}$. We stress that the dependence of the
rotation angle (2.3) on the choice of the Sachs basis is only apparent: under
a rotation of the Sachs basis, the shear transformation compensates the
transformation of the position angle $\alpha$, see [23] for details.

If the semi-major axis of the galaxy is aligned with the shear, $\delta\alpha$
vanishes. For example if we choose $\boldsymbol{e}_{1}$ in the direction of
the semi-major axis of the galaxy such that $\alpha=0$, alignment with the
shear implies $\gamma_{2}=0$ and hence $\delta\alpha=0$. In this case, the
shear just enhances or reduces somewhat the ellipticity of the galaxy. In all
other situation it generates also a rotation by $\delta\alpha$. This rotation
has already been studied long ago as a possible origon of the anisotropy of
galaxy orientations [36].

An additional rotation is in principle also generated by the anti-symmetric
part of the Jacobi matrix. But this part in non-vanishing only at second order
in perturbation theory [35] and we neglect it here.

In addition to $\delta\alpha$, the angle between the polarisation direction
and the semi major axis, also the eccentricity $\varepsilon$ and the direction
of the galaxy’s semi-major axis parametrised by $\alpha$ are observables.
Similar to our previous work [23], we define an observable which we call the
’scaled rotation’111Some versions of the article have a sign mistake in this
definition. The formula given here is correct. by

$\Theta=\frac{2-\varepsilon^{2}}{\varepsilon^{2}}\delta\alpha\,.$ (2.5)

With (2.3) the scaled rotation is related to the shear as

$\Theta=\gamma_{2}\cos 2\alpha-\gamma_{1}\sin 2\alpha\,,$ (2.6)

which is actually simply the shear in the direction $\alpha-\pi/4$, see
Appendix A.4. We want to determine the correlation function
$\langle\Theta({\boldsymbol{n}}_{1},z_{1})\Theta({\boldsymbol{n}}_{2},z_{2})\rangle$
for two directions ${\boldsymbol{n}}_{1}$ and ${\boldsymbol{n}}_{2}$ in the
sky and two redshifts $z_{1}$, $z_{2}$. Our expression for the variable
$\Theta({\boldsymbol{n}},z)$ given in (2.6) in principle depends on our choice
for the Sachs basis via the angle $\alpha$ and via $\gamma_{1}$ and
$\gamma_{2}$. However, as explained in App. A.4, one can circumvent this
problem and define a correlation function that is explicitly coordinate
invariant by choosing $\boldsymbol{e}_{1}$ the direction of the great circle
from ${\boldsymbol{n}}_{1}$ to ${\boldsymbol{n}}_{2}$ which is equivalent to
putting both galaxies on the ’Equator’, with coordinates $(\pi/2,0)$ and
$(\pi/2,\varphi)$ and
$\mu=\cos\varphi={\boldsymbol{n}}_{1}\cdot{\boldsymbol{n}}_{2}$. Note that
there is still a $\mathbb{Z}_{2}$ symmetry where one can swap both galaxies.
However, the correlation function does not depend on this choice. Given two
galaxies and the described setup, the correlation between their scaled
rotation is given by

$\displaystyle\langle{\Theta}(\boldsymbol{n}_{1},\alpha_{1},z_{1}){\Theta}(\boldsymbol{n}_{2},\alpha_{2},z_{2})\rangle$
$\displaystyle=\zeta_{+}(\mu,z_{1},z_{2})\cos(2(\alpha_{1}-\alpha_{2}))+\zeta_{-}(\mu,z_{1},z_{2})\cos(2(\alpha_{1}+\alpha_{2}))\,,$
(2.7)

with $\mu=\boldsymbol{n}_{1}\cdot\boldsymbol{n}_{2}=\cos\varphi$ and
$\zeta_{+}(\mu,z_{1},z_{2})$ and $\zeta_{-}(\mu,z_{1},z_{2})$ the two
coordinate independent shear correlation functions (see App. A for more
details). These correlation functions are related to the power spectrum of the
lensing potential $C_{\ell}(z_{1},z_{2})$ as

$\displaystyle\int_{-1}^{+1}\zeta_{+}(\mu,z_{1},z_{2})\tilde{P}_{\ell}(\mu)\,\mathrm{d}\mu$
$\displaystyle=\frac{1}{4\pi}C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}\,,$ (2.8)
$\displaystyle\int_{-1}^{+1}\zeta_{-}(\mu,z_{1},z_{2})\tilde{Q}_{\ell}(\mu)\,\mathrm{d}\mu$
$\displaystyle=\frac{1}{4\pi}C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}\,,$ (2.9)
$\displaystyle\nu^{2}_{\ell}$ $\displaystyle=\frac{(\ell+2)!}{(\ell-2)!}\,,$
(2.10)

where the polynomials $\tilde{P}_{\ell}(\mu)$ $\tilde{Q}_{\ell}(\mu)$ are
defined by, $\mu=\cos\theta$,

$\displaystyle-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,+2}}(\theta,\pi/2$
$\displaystyle=-\sqrt{\frac{2\ell+1}{4\pi}}\;{{}_{-2}\tensor{Y}{{}_{\ell,-2}}(\theta,\pi/2)}=\frac{2\ell+1}{16\pi}\tilde{Q}_{\ell}(\mu)\,,$
(2.11)
$\displaystyle-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,-2}}(\theta,\pi/2)$
$\displaystyle=-\sqrt{\frac{2\ell+1}{4\pi}}\;{{}_{+2}\tensor{Y}{{}_{\ell,-2}}}(\theta,\pi/2)=\frac{2\ell+1}{16\pi}\tilde{P}_{\ell}(\mu)\,,$
(2.12)

and ${}_{s}Y_{\ell,m}$ are the Spin Weighted Spherical Harmonics. More
details, and the explicit expressions for $\ell=2,\dots,5$ are given in App.
A.

From the observable $\Theta$ we now construct an estimator for the coordinate
independent correlation functions $\zeta_{+}$ and $\zeta_{-}$. Since we want
to estimate two correlation functions, we need two couples of galaxies,
separated by the same angle $\varphi$. Schematically, as
$\Theta\sim\gamma_{1}+\gamma_{2}$, one needs two galaxies to invert this
relation and express $\gamma_{1}$ and $\gamma_{2}$. Moreover, as the
correlation function is given by $\zeta\sim\langle\gamma\gamma\rangle$, we
need the value of $\gamma$ in two different pixels, which can be performed
considering $4$ galaxies in total.

More precisely, the estimator can be computed as follows. We consider two
couples of galaxies both separated by the same angle $\varphi$ and located at
the same redshifts (within the resolution of our survey). The galaxies of the
first couple have the directions and redshifts $(\boldsymbol{n}_{j},z_{j})$
with angles as defined above $\alpha_{j}$ ($j=1,2$), while the second couple
of galaxies are located in different directions $\boldsymbol{n}^{\prime}_{j}$
and with different angles $\alpha^{\prime}_{j}$ but inside the same redshift
bins $z_{j}$. Note that we define the angles $\alpha_{j}$ and
$\alpha^{\prime}_{j}$, with respect to the great circle connecting
${\boldsymbol{n}}_{1}$ and ${\boldsymbol{n}}_{2}$ respectively
${\boldsymbol{n}}^{\prime}_{1}$ and ${\boldsymbol{n}}^{\prime}_{2}$ which can
be different for each couples. The two couples of galaxies, however should be
separated by the same angle $\varphi$ (within our angular resolution), i.e.
$\boldsymbol{n}_{1}\cdot\boldsymbol{n}_{2}=\boldsymbol{n}^{\prime}_{1}\cdot\boldsymbol{n}^{\prime}_{2}=\cos\varphi=\mu$.
The two observables are the product of the scaled rotations, namely

$\displaystyle\Xi$
$\displaystyle={\Theta}(\boldsymbol{n}_{1},\alpha_{1},z_{1})\Theta(\boldsymbol{n}_{2},\alpha_{2},z_{2})\,,$
(2.13) $\displaystyle\Xi^{\prime}$
$\displaystyle={\Theta}(\boldsymbol{n}_{1}^{\prime},\alpha_{1}^{\prime},z_{1})\Theta(\boldsymbol{n}_{2}^{\prime},\alpha_{2}^{\prime},z_{2})\,.$
(2.14)

From these, and using the theoretical expression of the correlation function
of the scaled rotations given by Eq. (2.7), replacing the expectation value by
the observables $\Xi$ and $\Xi^{\prime}$, we can extract the estimators

$\displaystyle\hat{\zeta}_{+}(\mu,z_{1},z_{2})$
$\displaystyle=\Xi\,F_{1}(\alpha_{1}^{\prime},\alpha_{2}^{\prime},\alpha_{1},\alpha_{2})+\Xi^{\prime}\,F_{1}(\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},\alpha_{2}^{\prime})\,,$
(2.15) $\displaystyle\hat{\zeta}_{-}(\mu,z_{1},z_{2})$
$\displaystyle=\Xi\,F_{2}(\alpha_{1}^{\prime},\alpha_{2}^{\prime},\alpha_{1},\alpha_{2})+\Xi^{\prime}\,F_{2}(\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},\alpha_{2}^{\prime})\,,$
(2.16)

with

$F_{1}(\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},\alpha_{2}^{\prime})=\frac{\cos(2(\alpha_{1}+\alpha_{2}))}{\cos(2(\alpha_{1}^{\prime}-\alpha_{2}^{\prime}))\cos(2(\alpha_{1}+\alpha_{2}))-\cos(2(\alpha_{1}^{\prime}+\alpha_{2}^{\prime}))\cos(2(\alpha_{1}-\alpha_{2}))}\,,$
(2.17)

and

$F_{2}(\alpha_{1},\alpha_{2},\alpha_{1}^{\prime},\alpha_{2}^{\prime})=\frac{\cos(2(\alpha_{1}-\alpha_{2}))}{\cos(2(\alpha_{1}^{\prime}+\alpha_{2}^{\prime}))\cos(2(\alpha_{1}-\alpha_{2}))-\cos(2(\alpha_{1}^{\prime}-\alpha_{2}^{\prime}))\cos(2(\alpha_{1}+\alpha_{2}))}\,.$
(2.18)

We observe that in eqs. (2.15) and (2.16) on the left hand side there is no
angle dependence. We used this notation to stress that, observationally, one
chooses a given Sachs frame and for each galaxy quadruplet in pixels
${\boldsymbol{n}}_{1}$ and ${\boldsymbol{n}}_{2}$, one builds the correlations
given in eqs. (2.15) and (2.16). Every single estimator depends on the frame
choice. However, their expectation value obtained by averaging over all
possible quadruplets in the two pixels is independent of the angles
$\alpha_{i}$ and $\alpha^{\prime}_{i}$. In other words, and by construction,

$\langle\hat{\zeta}_{\pm}(\mu,z_{1},z_{2})\rangle=\zeta_{\pm}(\mu,z_{1},z_{2})\,.$
(2.19)

Once an estimator for $\zeta_{\pm}$ is obtained, the estimator for the lensing
potential power spectrum $C_{\ell}(z_{1},z_{2})$ can be given by Eqs. (2.8)
and (2.9).

##  3 Error estimation

In this Section, we estimate the expected error (or signal-to-noise ratio) on
the lensing angular power spectrum extracted via Eqs. (2.8) and (2.9),
starting from our estimator for the correlation functions Eqs. (2.15) and
(2.16).

As explained in the previous section, given two couples of galaxies, each
couple being separated by an angle $\varphi$ (with $\mu=\cos\varphi$), an
estimator for the correlation functions $\zeta_{\pm}$ is given by Eq. (2.15)
and Eq. (2.16). Of course, to obtain a good estimator for
$\zeta_{\pm}(\mu,z_{1},z_{2})$ we need to have many pairs of galaxies at a
given angular separations $\varphi$ (with $\mu=\cos\varphi$) inside the two
redshift bins. Furthermore, we need a good measurement of the scaled rotation
for these pairs and a good measurement of the angles $\alpha_{j}$ and
$\alpha_{j}^{\prime}$. The expressions for $F_{1}$ and $F_{2}$ (see Eqs.
(2.17) and (2.18)) also tell us that for
$\alpha_{1}+\alpha_{2}=\alpha^{\prime}_{1}+\alpha^{\prime}_{2}=\pi/4$ we
cannot determine $\hat{\zeta}_{+}$ while for for
$\alpha_{1}-\alpha_{2}=\alpha^{\prime}_{1}-\alpha^{\prime}_{2}=\pi/4$ we
cannot determine $\hat{\zeta}_{-}$. It follows that to obtain a well-defined
estimator of the correlation functions $\zeta_{\pm}$ we need to select
properly the angles $\alpha_{j}$ and $\alpha_{j}^{\prime}$, excluding galaxy
pairs with
$\alpha_{1}+\alpha_{2}=\alpha^{\prime}_{1}+\alpha^{\prime}_{2}=\pi/4$ or with
$\alpha_{1}-\alpha_{2}=\alpha^{\prime}_{1}-\alpha^{\prime}_{2}=\pi/4$. Note,
however, it does not matter whether the angles
$\alpha_{j},~{}\alpha_{j}^{\prime}$ are correlated, hence intrinsic alignment,
the major concern for traditional shear measurements is not an issue here.
What is important, however, is to have a good measurement of these angles and
of the small and more difficult-to-measure angle $\delta\alpha$ between the
image axis and polarisation.

An optimal estimator can be built as explained in Appendix B, by combining the
information that can be extracted from all possible pairs of couples with the
same angular separation and redshifts. It is optimal to choose the weighting
of each measurement inversely proportional to its error. To determine the
associated signal-to-noise ratio (SNR), we use the results presented in
Appendix B. Let $q$ represent a pair of a couples of galaxies (hence a
quadruplet). For each $q$, we compute an estimator $\hat{\zeta}_{\pm,q}(\mu)$
with its relative error $\tau_{\pm,q}$. The total signal-to-noise ratio for
the measurement of $\zeta_{\pm}(\mu,z_{1},z_{2})$ is given by Eq. (B.6)

${\rm
SNR}_{\pm}(\mu,z_{1},z_{2})=\sqrt{\sum_{q}\frac{1}{\tau^{2}_{\pm,q}}}\,.$
(3.1)

This sum can be computed explicitly if one is given a catalogue of
measurements. Here, we will take a more heuristic approach and admit that the
relative error is roughly equal (or we just consider an average value)

$\tau_{\pm,q}\simeq\tau_{0}\,.$ (3.2)

Then, the signal-to-noise is estimated as

${\rm
SNR}_{\pm}(\mu,z_{1},z_{2})\approx\frac{\sqrt{N_{\mathrm{e}}(\mu,z_{1},z_{2})}}{\tau_{0}}\,,$
(3.3)

where $N_{\mathrm{e}}(\mu,z_{1},z_{2})$ is the number of estimators one can
extract by choosing two couples of galaxies separated by an angle $\varphi$.
The number of quadruplets is computed in Appendix B and is given by

$N_{\mathrm{e}}(\varphi,z_{1},z_{2})=\frac{N_{\mathrm{p}}(\varphi,z_{1},z_{2})(N_{\mathrm{p}}(\varphi,z_{1},z_{2})-1)}{2}\simeq\frac{1}{2}\left(N_{\mathrm{g}}(z_{1})N_{\mathrm{g}}(z_{2})\frac{8f_{\rm
sky}\sin\varphi}{\delta\theta^{3}}\right)^{2}\,,$ (3.4)

where $N_{\mathrm{g}}(z)$ is the number of galaxies in a pixel at redshift $z$
and $\delta\theta$ is the aperture of the angular resolution. Note that the
formula for $N_{\mathrm{e}}$ given here holds for two different redshifts, and
has to be divided by $4$ if the considered redshifts are equal.

The final result for the signal-to-noise ratio given by Eq. (3.3) shows that
even if the erorr on a single estimator is typically rather large so that
$\tau_{0}>1$, the quality of the best estimator can still be good if we have
sufficiently many galaxies at our disposal.

Note that here, we assumed that all the individual estimators are
statistically independent. In reality, this is not the case, as we can assume
that the galaxies in the same pixel are somehow correlated (either their shape
or their orientation). Hence intrinsic alignment enters here in the error
estimate but not in the signal. Furthermore, in the number of estimators given
in (3.4) the same couples of pixels are used multiple times. We therefore
prefer to use a more pessimistic estimation for the number of independent
estimators setting

$N_{\mathrm{e}}(\varphi,z_{1},z_{2})\simeq N_{c}(\varphi)=\frac{8f_{\rm
sky}\sin\varphi}{\delta\theta^{3}}\,,$ (3.5)

where $N_{\mathrm{c}}$ is the number of couples of pixels separated by an
angle $\varphi$. Here we admit just one galaxy from each pixel. More details
can be found in Appendix B.

Finally, and to conclude this Section, another method would be to simply
compute the estimated shear field ${\boldsymbol{\gamma}}(\boldsymbol{n},z)$ in
every pixel using Eq. (2.6). By doing this, the signal-to-noise ratio for
every pixel would be given by $\sqrt{N_{\mathrm{g}}}/\tau_{0}$, where
$N_{\mathrm{g}}$ is the galaxy number in this specific pixel and $\tau_{0}$ is
the mean relative error on one measurement. In this way one could construct a
shear map in the sky for each redshift bin. From this map one can then extract
the power spectrum with its associated error. As we know e.g. from CMB lensing
maps [37], even if the map itself is noise dominated, we can obtain a good
estimator for its power spectrum. Note that to extract the shear in one pixel,
one needs to consider only a pair of galaxy, as the shear has two real
components $\gamma_{1}$ and $\gamma_{1}$. However, to compute the shear
correlation function, one needs to know the shear in two pixels. In other
words, even in this context, it is necessary to have two pairs of galaxies to
build an estimator for the correlation function. The selling argument for the
method we present here is that one could, in principle, construct a map of the
cosmic shear simply considering pairs of galaxies, without taking into account
a potential intrinsic correlation.

##  4 Results and discussion

In Fig. 2, we show an example of the results we can obtain. As discussed in
the previous section, we assume $N_{\mathrm{g}}=1$ to take into account that
the galaxies in the same pixel are not independent from each other, and use
Eq. (3.5). The parameters are taken from SKA2, see [38] for more details. We
choose a sky fraction and a pixel size of

$\displaystyle f_{\rm sky}$ $\displaystyle\approx 0.7\,,$ (4.1)
$\displaystyle\delta\theta$ $\displaystyle=5^{\prime}\approx 1.4\times
10^{-3}\,.$ (4.2)

Moreover, the typical shear signal $\gamma$ will be of order $10^{-3}$. For a
precise estimate of the error per galaxy pair, we would need precise values
for the errors on the various quantities as they are available once a mission
is planned. To get a pessimistic rough estimate, we have realised several
simulation using an error of $\pi/5$ on the angles and $1/2$ on $\varepsilon$.
This leads to a conservative relative error per galaxy pair of the order of
$\tau_{0}\approx 10^{3}$. This estimate is pessimistic, as in real experiments
one can hope to make this error smaller. On the other hand, the assumption
that the polarisation is perfectly aligned with the main axes of the galaxy is
optimistic. The idea is that these two assumptions might roughly compensate
each other,leading to the right order of magnitude for the resulting estimate.
Of course this treatment is simplistic and for a real observational campaign,
detailed simulations will be necessary. Inserting these numbers in (3.5) and
(3.3) we obtain a signal-to-noise ratio of order

${\rm SNR}\approx 45\sqrt{\sin\varphi}\,.$ (4.3)

This is the signal-to-noise ratio for our estimator
$\hat{\zeta}_{\pm}(\varphi,z_{1},z_{2})$ in two redshift bins around $z_{1}$
and $z_{2}$ and within one angular bin. One also needs the estimated value of
$\zeta_{\pm}(\varphi)$, which would be obtained from a catalogue with the
method we describe in this paper. As we do not yet have such a catalogue, we
compute the theoretical value of the correlation function. We compute the
power spectrum of the lensing potential, $C^{\phi}_{\ell}(z_{1},z_{2})$ for
$(z_{1},z_{2})=(1,1),(1,2),(2,2)$ with CLASS [39, 40] using the by default
parameters from the Planck 2018 data [41]
($h=0.6781\,,h^{2}\Omega_{\mathrm{cdm}}=0.0552278\,,h^{2}\Omega_{\mathrm{b}}=0.0102921\,,\log(10^{9}A_{\mathrm{s}})=0.742199\,,n_{\mathrm{s}}=0.9660499$)
To compute the correlation functions, one would need to invert the relations
(2.8) (2.9), i.e. evaluate the sums Eq. (A.82) and Eq. (A.83). However, the
polynomials $\tilde{P}$ and $\tilde{Q}$ are highly oscillating as $\ell$ gets
large and the computation is very badly converging. Instead, we use the flat
sky approximation, see [42] and [43] for more details, to approximate the
correlation functions as

$\displaystyle\zeta_{+}(z_{1},z_{2},\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\;\ell
J_{0}(\ell\varphi)\frac{1}{4}\left(\ell(\ell+1)\right)^{2}C_{\ell}(z_{1},z_{2})\,\mathrm{d}\ell\,$
(4.4) $\displaystyle\zeta_{-}(z_{1},z_{2},\varphi)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{0}^{\infty}\;\ell
J_{4}(\ell\varphi)\frac{1}{4}\left(\ell(\ell+1)\right)^{2}C_{\ell}(z_{1},z_{2})\,\mathrm{d}\ell\,.$
(4.5)

Truncating the integral at $\ell=20^{\prime}000$ seems reasonable, as the
relative error is less than $10^{-3}$ in this case, which is much smaller than
the inverse signal-to-noise ratio.

In Fig. 2 we show the results for the correlation functions
$\zeta_{\pm}(\varphi)$ computed in the flat-sky approximation. The shaded
region around each curve represents the uncertainty computed with ${\rm
SNR}=40\sqrt{\sin\varphi}$. Different panels correspond to different redshift
bins. The result is not very sensitive to the thickness of the redshift bins.
In a true survey this is an advantage as it allows us the enhance the number
of galaxies per bin.

Figure 2:  Correlation functions $\zeta_{\pm}(\varphi)$ computed with the
flat-sky approximation. The $C_{\ell}$ are taken from CLASS, and the sum is
truncated at $\ell=20^{\prime}000$. The error bars were computed with
$\mathrm{SNR}=40\sqrt{\sin\varphi}$. We consider redshift bins
$(z_{1},z_{2})=(1,1),(1,2)$ and $(2,2)$ (from top left to bottom panel).

##  5 Conclusions

In this paper we proposed a new method to extract the shear correlation
function, by measuring the correlation function of the angle between the image
major axis and the polarisation direction of radio galaxies. In particular, we
built an estimator for the shear correlation function given two couples of
galaxies separated by an angle $\varphi$, and estimated the error one gets by
combining all possible pairs separated by this angle.

The advantage of this method with respect to traditional shear measurements is
that we do not rely on the assumption that galaxy eccentricities are
uncorrelated, hence we do not have to deal with a parametrisation of intrinsic
alignment and its uncertainties, which are one of the major source of error in
standard shear measurements in present and planned surveys [2, 3, 6, 7, 8,
10].Even though our signal does not depend on intrinsic alignment, we have
seen that the error does since intrinsic alignment correlates the measurements
from different galaxies which therefore cannot be considered as independent
estimators. In the presented estimation of the signal-to-noise we have taken
this into account in a very conservative way, assuming that we can make only 1
independent measurement per pixel.

We find that even if the signal-to-noise ratio for a single measurements (i.e.
for a given galaxy quadruplet) is expected to be rather small, the fact that
all galaxies in a given pixel are subject to the same shear can be used to
overcome the noise. As a case study, we considered the specifications of SKA2:
the number of independent estimators for a given angular separation $\varphi$
and two redshifts $z_{1}$, $z_{2}$ is expected to scale as $\sim
10^{9}\sin\varphi$. As a consequence, the noise on a single measurement can
exceed the signal by a factor $10^{3}$, and still yield an signal-to-noise of
order 40 which is largely sufficient to detect the signal. Therefore, even if
the maps of $\delta\alpha$ measurements for each redshift bin will be largely
noise dominated, we will be able to obtain a good estimate of for the shear
correlation function when combining all the measurements together.

We stress that the goal of the present paper was to present a new method to
reconstruct the shear correlation functions with a new observable, and to
build an estimator for it. Of course, the limiting factor of our forecasts is
that we had to assume some number for the precision with which the various
angles $\delta\alpha$ and $\alpha$ can be measured. However, as explained
above, our choice of errors is quite conservative, and the crucial factor
setting the signal-to-noise level of our estimator is the high statistics. For
this reason, we do not expect a more refined analysis to drastically change
the conclusions of our study.

Finally we point out that, while in this work we focused on the reconstruction
of the shear correlation function, our new observable can be used also to get
a shear sky map. This is another advantage of our method with respect to
standard shear reconstruction methods, which look at galaxy shapes only (from
the study of galaxy ellipticity it is not possible to get a shear mapping, but
only to extract correlation functions). A natural extension of our work is to
apply this method to simulated (or real) galaxy lensing and polarisation data.
This would provide us with a more realistic estimate of the uncertainties, and
allow us to compare this shear reconstruction method with traditional
LSST/Euclid techniques to measure the shear correlation function.

## Acknowledgements

We thank Richard Battye, Michael Brown, Charles Dalang, Ian Harrison, Alan
Heavens, Azadeh Moradinezhad Dizgah, Serge Parnovskii, Cyril Pitrou and Isaac
Tutusaus for useful discussions and comments. We are very grateful to
Francesca Lepori for her valuable help with class. This work is supported by
the Swiss National Science Foundation.

##  Appendix A Special functions

###  A.1 Spin Weighted Spherical Harmonics

This appendix follows Refs. [44, 45]. Let $(\theta,\varphi)$ be the usual
spherical coordinates on the sphere. We define the Spin Weighted Spherical
Harmonics, ${}_{s}Y_{\ell,m}$, where $s$ represents the weight. The $s=0$
Spherical Harmonics are the usual Spherical Harmonics functions
$\tensor[_{0}]{Y}{}_{\ell,m}\equiv\tensor{Y}{}_{\ell,m}$, with the convention

$Y_{\ell,-m}=(-1)^{m}Y_{\ell,m}^{\star}\,.$ (A.1)

For a generic integer $s$, we define first the spin raising and spin lowering
operations, $\not{\partial}$ and $\not{\partial}^{\star}$, on a function
$\tensor[_{s}]{Y}{}_{\ell,m}$ with spin weight $s$ as

$\displaystyle\not{\partial}_{s}Y_{\ell,m}$
$\displaystyle=\left(s\cot\theta-\partial_{\theta}-\frac{\mathrm{i}}{\sin\theta}\partial_{\varphi}\right)\,_{s}Y_{\ell,m}\,,$
(A.2)

and

$\displaystyle{\not{\partial}^{\star}}_{s}Y_{\ell,m}$
$\displaystyle={\left(-s\cot\theta-\partial_{\theta}+\frac{\mathrm{i}}{\sin\theta}\partial_{\varphi}\right)}\,_{s}Y_{\ell,m}\,.$
(A.3)

The Spin Weighted Spherical Harmonics for generic $s\in\mathbb{Z}$ are
obtained recursively with the spin raising and spin lowering operators given
by Eq. (A.2) and Eq. (A.3) via

$\displaystyle\not{\partial}_{s}Y_{\ell,m}$
$\displaystyle=\sqrt{(\ell-s)(\ell+s+1)}\;_{s+1}Y_{\ell,m}\,,$ (A.4)
$\displaystyle{\not{\partial}^{\star}}_{s}Y_{\ell,m}$
$\displaystyle=-\sqrt{(\ell+s)(\ell-s+1)}\;_{s-1}Y_{\ell,m}\,,$ (A.5)

together with the starting point
${}_{0}\tensor{Y}{}_{\ell,m}\equiv\tensor{Y}{}_{\ell,m}$. Hence, the slashed
derivatives can be interpreted as spin raising/lowering operators. In
particular, for $s=\pm 2$, these definitions yield

$\displaystyle\not{\partial}^{2}{Y}{{}_{\ell,m}}$
$\displaystyle=\nu_{\ell}\;\;{{}_{2}\tensor{Y}{{}_{\ell,m}}}\,,$ (A.6)
$\displaystyle{\not{\partial}^{\star}}^{2}{Y}{{}_{\ell,m}}$
$\displaystyle=\nu_{\ell}\;\;{{}_{-2}\tensor{Y}{{}_{\ell,m}}}\,,$ (A.7)
$\displaystyle\mbox{where}~{}~{}\nu_{\ell}$
$\displaystyle=\sqrt{\frac{(\ell+2)!}{(\ell-2)!}}\,.$ (A.8)

The Spin Weighted Spherical Harmonics satisfy the orthogonality condition (
$d\Omega=\sin\theta d\theta d\varphi$)

$\int\;_{s}Y_{\ell_{1},m_{1}}(\theta,\varphi)\,_{s}Y^{\star}_{\ell_{2},m_{2}}(\theta,\varphi)\,\mathrm{d}\Omega=\delta_{\ell_{1},\ell_{2}}\delta_{m_{1},m_{2}}\,,$
(A.9)

and the conjugation relation

$_{-s}Y_{\ell,-m}=(-1)^{s+m}\,_{s}Y^{\star}_{\ell,m}(\theta,\varphi)\,.$
(A.10)

The Spin Weighted Spherical Harmonics also satisfy the the following addition
theorem

$\sqrt{\frac{4\pi}{2\ell+1}}\sum_{m}\;{{}_{s_{1}}\tensor{Y}{{}_{\ell,m}}}(\theta_{1},\varphi_{1})\;{{}_{-s_{2}}\tensor{Y}{{}^{\star}_{\ell,m}}}(\theta_{2},\varphi_{2})=\;{{}_{s_{1}}\tensor{Y}{{}_{\ell,s_{2}}}}(\beta,\alpha)\mathrm{e}^{-\mathrm{i}s_{1}\gamma}\,,$
(A.11)

where the angles $(\alpha,\beta,\gamma)$ are defined through the implicit
relation

$R_{\mathrm{E}}(\alpha,\beta,\gamma)=R_{\mathrm{E}}(\varphi_{1},\theta_{1},0)^{-1}R_{\mathrm{E}}(\varphi_{2},\theta_{2},0)\,.$
(A.12)

Here $R_{\mathrm{E}}(\alpha,\beta,\gamma)$ is the rotation matrix with the
Euler angles $\alpha$, $\beta$ and $\gamma$. More precisely

$R_{\mathrm{E}}(\alpha,\beta,\gamma)=\begin{pmatrix}\cos\alpha\cos\beta\cos\gamma-\sin\alpha\sin\gamma&\
&-\cos\gamma\sin\alpha-\cos\alpha\cos\beta\sin\gamma&\ &\cos\alpha\sin\beta\\\
\cos\beta\cos\gamma\sin\alpha+\cos\alpha\sin\gamma&\
&\cos\alpha\cos\gamma-\cos\beta\sin\alpha\sin\gamma&\ &\sin\alpha\sin\beta\\\
-\cos\gamma\sin\beta&\ &\sin\beta\sin\gamma&\ &\cos\beta\end{pmatrix}\,.$
(A.13)

Explicit expressions of the Spin Weighted Spherical Harmonics for $s=0,1,2$
and $\ell\leq 2$ are given in Tables 2 and 3. Note that the remaining cases
can be deduced from the conjugation relation given by Eq. (A.10).

We also introduce the auxiliary polynomials $\tilde{P}_{\ell}(\mu)$ and
$\tilde{Q}_{\ell}(\mu)$ which will be useful later. For $\mu=\cos\theta$ they
are defined as

$\displaystyle\frac{2\ell+1}{16\pi}\tilde{Q}_{\ell}(\mu)$
$\displaystyle\equiv$
$\displaystyle-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,+2}}(\theta,\pi/2)~{}=~{}-\sqrt{\frac{2\ell+1}{4\pi}}\;{{}_{-2}\tensor{Y}{{}_{\ell,-2}}(\theta,\pi/2)}\,,$
(A.14) $\displaystyle\frac{2\ell+1}{16\pi}\tilde{P}_{\ell}(\mu)$
$\displaystyle\equiv$
$\displaystyle-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,-2}}(\theta,\pi/2)~{}=~{}-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,-2}}(\theta,\pi/2)\,.$
(A.15)

From the orthonormality condition Eq. (A.9), it is easy to see that

$\int_{-1}^{+1}\;\tilde{P}_{\ell_{1}}(\mu)\tilde{P}_{\ell_{2}}(\mu)\,\mathrm{d}\mu=\int_{-1}^{+1}\;\tilde{Q}_{\ell_{1}}(\mu)\tilde{Q}_{\ell_{2}}(\mu)\,\mathrm{d}\mu=\frac{32}{2\ell_{1}+1}\delta_{\ell_{1},\ell_{2}}\,.$
(A.16)

The explicit expressions for these polynomials for $\ell=2,\dots,5$ are given
in table 1

$\ell$ | $\tilde{P}_{\ell}(\mu)$ | $\tilde{Q}_{\ell}(\mu)$
---|---|---
$2$ | $(\mu+1)^{2}$ | $(\mu-1)^{2}$
$3$ | $(\mu+1)^{2}(3\mu-2)$ | $(\mu-1)^{2}(3\mu+2)$
$4$ | $(\mu+1)^{2}(7\mu^{2}-7\mu+1)$ | $(\mu-1)^{2}(7\mu^{2}+7\mu+1)$
$5$ | $(\mu+1)^{2}(15\mu^{3}-18\mu^{2}+3\mu+1)$ |  $(\mu-1)^{2}(15\mu^{3}+18\mu^{2}+3\mu-1)$
Table 1: The polynomials $\tilde{P}_{\ell}(\mu)$ and $\tilde{Q}_{\ell}(\mu)$.  $m$ | $\tensor{Y}{{}_{1,m}}(\theta,\varphi)$ | ${}_{1}\tensor{Y}{{}_{1,m}}(\theta,\varphi)$
---|---|---
$-1$ | $\frac{1}{2}\sqrt{\frac{3}{2\pi}}\mathrm{e}^{-\mathrm{i}\varphi}\sin\theta$ | $-\frac{1}{4}\sqrt{\frac{3}{\pi}}\mathrm{e}^{-\mathrm{i}\varphi}(1+\cos\theta)$
$0$ | $\frac{1}{2}\sqrt{\frac{3}{\pi}}\cos\theta$ | $\frac{1}{2}\sqrt{\frac{3}{2\pi}}\sin\theta$
$1$ | $-\frac{1}{2}\sqrt{\frac{3}{2\pi}}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta$ | $\frac{1}{4}\sqrt{\frac{3}{\pi}}\mathrm{e}^{\mathrm{i}\varphi}(-1+\cos\theta)$
Table 2: Spherical Harmonics of Spin Weight $s=0,1$ and $\ell=1$ $m$ | $\tensor{Y}{{}_{2,m}}(\theta,\varphi)$ | ${}_{1}\tensor{Y}{{}_{2,m}}(\theta,\varphi)$ | ${}_{2}\tensor{Y}{{}_{2,m}}(\theta,\varphi)$
---|---|---|---
$-2$ | $\frac{1}{4}\sqrt{\frac{15}{2\pi}}\mathrm{e}^{-2\mathrm{i}\varphi}\sin^{2}\theta$ | $-\frac{1}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{-2\mathrm{i}\varphi}(1+\cos\theta)\sin\theta$ | $\frac{1}{8}\sqrt{\frac{5}{\pi}}\mathrm{e}^{-2\mathrm{i}\varphi}(1+\cos\theta)^{2}$
$-1$ | $\frac{1}{2}\sqrt{\frac{15}{2\pi}}\mathrm{e}^{-\mathrm{i}\varphi}\sin\theta\cos\theta$ | $-\frac{1}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{-\mathrm{i}\varphi}(2\cos^{2}\theta+\cos\theta-1)$ | $-\frac{1}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{-\mathrm{i}\varphi}\sin\theta(1+\cos\theta)$
$0$ | $\frac{1}{8}\sqrt{\frac{5}{\pi}}(1+3\cos(2\theta))$ | $\frac{1}{2}\sqrt{\frac{15}{2\pi}}\sin\theta\cos\theta$ | $\frac{1}{4}\sqrt{\frac{15}{2\pi}}\sin^{2}\theta$
$1$ | $-\frac{1}{2}\sqrt{\frac{15}{2\pi}}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta\cos\theta$ | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{\mathrm{i}\varphi}(2\cos^{2}\theta-\cos\theta-1)$ | $\frac{1}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta(-1+\cos(\theta))$
$2$ | $\frac{1}{4}\sqrt{\frac{15}{2\pi}}\mathrm{e}^{2\mathrm{i}\varphi}\sin^{2}\theta$ | $\frac{2}{4}\sqrt{\frac{5}{\pi}}\mathrm{e}^{2\mathrm{i}\varphi}\sin\theta(-1+\cos\theta)$ | $\frac{1}{8}\sqrt{\frac{5}{\pi}}\mathrm{e}^{2\mathrm{i}\varphi}(1-\cos\theta)^{2}$
Table 3: Spherical Harmonics of Spin Weight $s=0,1,2$ and $\ell=2$

###  A.2 Expression of the shear

In this Appendix, we present useful relations involving spin Spherical
Harmonics. More details can be found in [44, 45]. This second reference is a
very useful PhD thesis covering the topic in depth. The interested reader is
referred to it for further details.

Let $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$ be an orthonormal basis on the
sphere associated with the usual spherical coordinates $(\theta,\varphi)$. We
define the $(+,-)$ basis

$\boldsymbol{e}_{\pm}=\frac{1}{\sqrt{2}}\left(\boldsymbol{e}_{1}\mp\mathrm{i}\boldsymbol{e}_{2}\right)\,.$
(A.53)

The spin raising and lowering operators are simply related to the covariant
derivatives in directions $\boldsymbol{e}_{\pm}$,

$\boldsymbol{\nabla}_{\boldsymbol{e}_{-}}=-\frac{1}{\sqrt{2}}\not{\partial}\,,\quad\boldsymbol{\nabla}_{\boldsymbol{e}_{+}}=-\frac{1}{\sqrt{2}}{\not{\partial}^{\star}}\,.$
(A.54)

With these identities, the relevant operators to compute the shear from the
lensing potential are

$\boldsymbol{\nabla}_{1}^{2}-\boldsymbol{\nabla}_{2}^{2}=\frac{1}{2}(\not{\partial}^{2}+{{\not{\partial}^{\star}}}^{2})\,,$
(A.55)

and

$\boldsymbol{\nabla}_{1}\boldsymbol{\nabla}_{2}=-\frac{\mathrm{i}}{4}(\not{\partial}^{2}-{{\not{\partial}^{\star}}}^{2})\,,$
(A.56)

where it is assumed
$\not{\partial}{\not{\partial}^{\star}}={\not{\partial}^{\star}}\not{\partial}$,
as in this context it acts on the scalar lensing potential $\phi$. The
definition of the shear in the $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$ basis
is

$\displaystyle\gamma_{1}$
$\displaystyle=-\frac{1}{2}(\boldsymbol{\nabla}_{1}^{2}-\boldsymbol{\nabla}_{2}^{2})\phi\,,$
(A.57) $\displaystyle\gamma_{2}$
$\displaystyle=-\boldsymbol{\nabla}_{1}\boldsymbol{\nabla}_{2}\phi\,,$ (A.58)

where $\phi$ is the lensing potential. This shows that the shear is a spin $2$
object. Using $\gamma^{\pm}=\gamma_{1}\pm\mathrm{i}\gamma_{2}$ and the
relations given above, the shear in the $(+,-)$ basis is given by the slashed
derivatives of the lensing potential as

$\displaystyle\gamma^{+}$
$\displaystyle=-\frac{1}{2}{\not{\partial}}^{2}\phi\,,$ (A.59)
$\displaystyle\gamma^{-}$
$\displaystyle=-\frac{1}{2}{{\not{\partial}^{\star}}}^{2}\phi\,.$ (A.60)

Hence $\gamma^{+}$ has helicity $+2$ while $\gamma^{-}$ has helicity $-2$.
Using the standard decomposition for the lensing potential

$\phi(\boldsymbol{n},z)=\sum_{\ell,m}\phi_{\ell,m}(z)Y_{\ell,m}(\boldsymbol{n})\,,$
(A.61)

and the squared raising/lowering operators given in Eq (A.6), one obtains the
decomposition of the shear in the $(+,-)$ basis as

$\displaystyle\gamma^{+}(\boldsymbol{n},z)$
$\displaystyle=-\frac{1}{2}\sum_{\ell=2,m}\phi_{\ell,m}(z)\nu_{\ell}\;{}_{+2}{Y}{{}_{\ell,m}}(\boldsymbol{n})\,,$
(A.62) $\displaystyle\gamma^{-}(\boldsymbol{n},z)$
$\displaystyle=-\frac{1}{2}\sum_{\ell=2,m}\phi_{\ell,m}(z)\nu_{\ell}\;{}_{-2}{Y}{{}_{\ell,m}}(\boldsymbol{n})\,.$
(A.63)

The complex numbers $\phi_{\ell,m}(z)$ are random variables whose expectation
values define the angular power spectrum of the lensing potential,

$\langle\phi_{\ell_{1},m_{1}}(z_{1})\phi^{\star}_{\ell_{2},m_{2}}(z_{2})\rangle=C_{\ell}(z_{1},z_{2})\delta_{\ell_{1},\ell_{2}}\delta_{m_{1},m_{2}}\,.$
(A.64)

As the lensing potential $\phi(\boldsymbol{n},z)$ is real. They satisfy

$\phi_{\ell,m}^{\star}=(-1)^{m}\phi_{\ell,-m}\,.$ (A.65)

###  A.3 Correlation functions on the equator

In this Section, we compute the correlation functions of the shear in the
$(+,-)$ and $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$ basis. Using the
decomposition given by Eq. (A.62) and Eq. (A.63) we find for the correlation
function of $\gamma^{+}$ and $\gamma^{-}$

$\displaystyle\langle\gamma^{+}(\boldsymbol{n}_{1},z_{1})\gamma^{-}(\boldsymbol{n}_{2},z_{2})\rangle$
$\displaystyle=\frac{1}{4}\sum\langle\phi_{\ell_{1},m_{1}}(z_{1})\phi_{\ell_{2},m_{2}}(z_{2})\rangle\nu_{\ell_{1}}\nu_{\ell_{2}}\;{}_{+2}{Y}{{}_{\ell_{1},m_{1}}}(\boldsymbol{n}_{1})\;_{-2}{Y}{{}_{\ell_{2},m_{2}}}(\boldsymbol{n}_{2})$
(A.66) $\displaystyle=\frac{1}{4}\sum
C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}(-1)^{m}\;_{+2}{Y}{{}_{\ell,m}}(\boldsymbol{n}_{1})\;_{-2}{Y}{{}_{\ell,-m}}(\boldsymbol{n}_{2})$
(A.67) $\displaystyle=\frac{1}{4}\sum
C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}\;{}_{+2}{Y}{{}_{\ell,m}}(\boldsymbol{n}_{1})\;_{+2}{Y}{{}^{\star}_{\ell,m}}(\boldsymbol{n}_{2})\,,$
(A.68)

where we used the conjugation properties Eq. (A.65) and Eq. (A.10). Using the
addition theorem Eq. (A.11) (with $s_{1}=+2$ and $s_{2}=-2$), the sum over $m$
reads

$\displaystyle\sum_{m}{{}_{+2}{Y}{{}_{\ell,m}}(\boldsymbol{n}_{1})}\;_{+2}Y^{\star}_{\ell,m}(\boldsymbol{n}_{2})$
$\displaystyle=-\sqrt{\frac{2\ell+1}{4\pi}}\;_{+2}\tensor{Y}{{}_{\ell,-2}}(\varphi,\pi/2)$
(A.69) $\displaystyle\equiv\frac{2\ell+1}{16\pi}\tilde{P}_{\ell}(\mu)\,,$
(A.70)

Finally, the correlation function is given by

$\langle\gamma^{+}(\boldsymbol{n}_{1},z_{1})\gamma^{-}(\boldsymbol{n}_{2},z_{2})\rangle=\sum_{\ell}\frac{2\ell+1}{64\pi}\nu_{\ell}^{2}C_{\ell}(z_{1},z_{2})\tilde{P}_{\ell}(\mu)\,.$
(A.71)

The two other correlations can be obtained following exactly the same steps
(the values of the Euler angles are the same), yielding

$\displaystyle\langle\gamma^{+}(\boldsymbol{n}_{1},z_{1})\gamma^{+}(\boldsymbol{n}_{2},z_{2})\rangle=\langle\gamma^{-}(\boldsymbol{n}_{1},z_{1})\gamma^{-}(\boldsymbol{n}_{2},z_{2})\rangle=\sum_{\ell}\frac{2\ell+1}{64\pi}\nu_{\ell}^{2}C_{\ell}(z_{1},z_{2})\tilde{Q}_{\ell}(\mu)\,.$
(A.72)

Inverting the relations $\gamma^{\pm}=\gamma_{1}\pm\mathrm{i}\gamma_{2}$
yields

$\displaystyle\gamma_{1}$
$\displaystyle=\frac{1}{2}(\gamma^{+}+\gamma^{-})\,,$ (A.73)
$\displaystyle\gamma_{2}$
$\displaystyle=\frac{\mathrm{i}}{2}(\gamma^{-}-\gamma^{+})\,.$ (A.74)

Using the correlations given above by Eq. (A.71) and Eq. (A.72) yield

$\displaystyle\langle\gamma_{1}(\boldsymbol{n}_{1},z_{1})\gamma_{1}(\boldsymbol{n}_{2},z_{2})\rangle$
$\displaystyle=\sum_{\ell}\frac{2\ell+1}{128\pi}\nu_{\ell}^{2}C_{\ell}(z_{1},z_{2})(\tilde{P}_{\ell}(\mu)+\tilde{Q}_{\ell}(\mu))\,,$
(A.75)
$\displaystyle\langle\gamma_{2}(\boldsymbol{n}_{1},z_{1})\gamma_{2}(\boldsymbol{n}_{2},z_{2})\rangle$
$\displaystyle=\sum_{\ell}\frac{2\ell+1}{128\pi}\nu_{\ell}^{2}C_{\ell}(z_{1},z_{2})(\tilde{P}_{\ell}(\mu)-\tilde{Q}_{\ell}(\mu))\,,$
(A.76)
$\displaystyle\langle\gamma_{1}(\boldsymbol{n}_{1},z_{1})\gamma_{2}(\boldsymbol{n}_{2},z_{2})\rangle$
$\displaystyle=0\,,$ (A.77)

where the points $\boldsymbol{n}_{1}$ and $\boldsymbol{n}_{1}$ lie on the
equator and subtend and angle $\varphi$ with $\mu=\cos\varphi$.

###  A.4 Invariant correlation functions

Here, we compute the shear and its correlation functions in a coordinate
invariant way, see for example [46]. Let $(\theta,\varphi)$ be the spherical
coordinates and $(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$ the associated
orthonormal frame. With such a basis, the shear is a 2-tensor of the form

$\boldsymbol{\Gamma}=\begin{pmatrix}-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}&\gamma_{1}\end{pmatrix}\,.$ (A.78)

For a generic tangent vector $\boldsymbol{e}=(\cos\alpha,\sin\alpha)$ in the
$(\boldsymbol{e}_{1},\boldsymbol{e}_{2})$, the shear in direction
$\boldsymbol{e}$ is defined as

$\gamma_{\alpha}\equiv\gamma_{\boldsymbol{e}}\equiv
e^{a}e^{b}\Gamma_{ab}=-\gamma_{1}\cos(2\alpha)-\gamma_{2}\sin(2\alpha)\,.$
(A.79)

It is clear from the definition that $\gamma_{1,2}$ and the angle $\alpha$ do
depend on the coordinate system. However, for a fixed (physically defined)
vector $\boldsymbol{e}$, the shear in direction $\boldsymbol{e}$,
$\gamma_{\boldsymbol{e}}$ does not depend on the coordinates, which makes this
quantity a good candidate to study correlation functions. For two galaxies
located at $(\boldsymbol{n}_{1},z_{1})$ and $(\boldsymbol{n}_{2},z_{2})$, we
can define the geodesic joining them to be the equator of our system of
coordinates. As this process does not depend on the coordinates and is well-
defined for every pair of galaxies, the result that follows is also coordinate
independent. From this construction, we define the two invariant correlation
functions

$\displaystyle\zeta_{\mathrm{p}}(\mu,z_{1},z_{2})$
$\displaystyle=\langle\gamma_{0}(\boldsymbol{n}_{1},z_{1})\gamma_{\pi}(\boldsymbol{n}_{2},z_{2})\rangle=\langle\gamma_{1}(\boldsymbol{n}_{1},z_{1})\gamma_{1}(\boldsymbol{n}_{2},z_{2})\rangle\,,$
(A.80) $\displaystyle\zeta_{\mathrm{c}}(\mu,z_{1},z_{2})$
$\displaystyle=\langle\gamma_{-\pi/4}(\boldsymbol{n}_{1},z_{1})\gamma_{3\pi/4}(\boldsymbol{n}_{2},z_{2})\rangle=\langle\gamma_{2}(\boldsymbol{n}_{1},z_{1})\gamma_{2}(\boldsymbol{n}_{1},z_{1})\rangle\,,$
(A.81)

with $\mu=\boldsymbol{n}_{1}\cdot\boldsymbol{n}_{2}=\cos\varphi$. The last
equality is valid in the preferred system of coordinates, where both galaxies
lie on the equator. An illustration of this definition is shown in Fig. 3.
Using the results of Sec. A.3 yields

$\displaystyle\zeta_{\mathrm{p}}(\mu,z,z^{\prime})$
$\displaystyle=\sum_{\ell}\frac{2\ell+1}{128\pi}C_{\ell}(z,z^{\prime})\nu_{\ell}^{2}(\tilde{P}_{\ell}(\mu)+\tilde{Q}_{\ell}(\mu))\,,$
(A.82) $\displaystyle\zeta_{\mathrm{c}}(\mu,z,z^{\prime})$
$\displaystyle=\sum_{\ell}\frac{2\ell+1}{128\pi}C_{\ell}(z,z^{\prime})\nu_{\ell}^{2}(\tilde{P}_{\ell}(\mu)-\tilde{Q}_{\ell}(\mu))\,.$
(A.83)

Note that the sums start at $\ell=2$. Defining,

$\displaystyle\zeta_{\pm}$
$\displaystyle=\frac{1}{2}(\zeta_{\mathrm{p}}\pm\zeta_{\mathrm{c}})\,,$ (A.84)

and using the orthogonality properties of the polynomials $\tilde{P}_{\ell}$
and $\tilde{Q}_{\ell}$ given in Eq.(A.16), we have

$\displaystyle\int_{-1}^{+1}\zeta_{+}(\mu,z_{1},z_{2})\tilde{P}_{\ell}(\mu)\,\mathrm{d}\mu$
$\displaystyle=\frac{1}{4\pi}C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}\,,$ (A.85)
$\displaystyle\int_{-1}^{+1}\zeta_{-}(\mu,z_{1},z_{2})\tilde{Q}_{\ell}(\mu)\,\mathrm{d}\mu$
$\displaystyle=\frac{1}{4\pi}C_{\ell}(z_{1},z_{2})\nu_{\ell}^{2}\,.$ (A.86)

Note that the relations Eq. (A.85) and Eq. (A.86) relate only coordinate
independent observables. The correlation functions $\zeta_{\pm}$ can be
estimated by observations as explained in the main text. Via (A.85) we can
then use them to estimate the lensing power spectrum $C_{\ell}$.

Figure 3:  The correlation functions $\zeta_{\mathrm{p}}$ and
$\zeta_{\mathrm{c}}$. Both galaxies are located on the ’Equator’ - represented
by the dotted lines - which defines a preferred system of coordinates. The
angles between the direction of the shear and the connecting line are
indicated. For a fixed separation angle, these correlations are intrinsically
given and do not depend on the coordinate system.

##  Appendix B Error estimation

###  B.1 Best estimator

Given $X_{j}$ measurements of an observable $X$, each of them with error
$\delta X_{j}=\tau_{j}X_{j}$ ($\tau$ is the relative error). We want to
construct an estimator for $X$. We define

$\hat{X}=\sum w_{j}X_{j}\,,\qquad\sum w_{j}=1n\,.$ (B.1)

In order to obtain the best possible estimator for $X$ We want to choose the
weights $w_{j}$ which yield the highest signal-to-noise ratio (SNR). We claim

$w_{j}=\frac{1}{Z}\frac{1}{X_{j}\tau_{j}^{2}}\qquad\mbox{where}\quad
Z=\sum\frac{1}{X_{j}\tau_{j}^{2}}\,.$ (B.2)

To see that this is the best choice, we note that the error on the estimator
is given by

$N^{2}=\sum w_{j}^{2}\delta X_{j}^{2}=\sum w_{j}^{2}\tau_{j}^{2}X_{j}^{2}\,.$
(B.3)

The square of the SNR which we want to maximise is the quantity

$A=\frac{\hat{X}^{2}}{N^{2}}.$ (B.4)

Using the Ansatz (B.2), one can verify directly that this is choice of the
weights gives

$\frac{\partial A}{\partial w_{i}}=0\,,$ (B.5)

and it is the only zero of the gradient of $A$ (with positive weights which
sum up to 1) and it is a minimum. Hence the $w_{i}$ given above are the best
choice if one wants to maximize the SNR of an observable. The constant $Z$ is
determined by the requirement that

$\sum w_{j}=1\,.$

Computing $A$ explicitly one finds the well known result

${\rm SNR}=\sqrt{A}=\sqrt{\sum\frac{1}{\tau_{j}^{2}}}\,.$ (B.6)

###  B.2 Specific example

If we consider our estimator of the correlation function,
$\zeta_{\pm}(\mu,z_{1},z_{2})$ and denote the value obtained from two pairs of
galaxies by $\hat{\zeta}_{j}$ and the error by $\delta\hat{\zeta}_{j}$, then
we find

$A=\sum\left(\frac{\hat{\zeta}_{j}}{\delta\hat{\zeta}_{j}}\right)^{2}\,,$
(B.7)

and the optimal estimator for $\zeta_{\pm}(\mu,z_{1},z_{2})$ is

$\hat{\zeta}(\mu,z_{1},z_{2})=\sum\frac{1}{Z}\left(\frac{\hat{\zeta}_{j}}{\delta\hat{\zeta}_{j}}\right)^{2}\,,$
(B.8)

with

$Z=\sum\frac{\hat{\zeta}_{j}}{(\delta\hat{\zeta}_{j})^{2}}\,.$ (B.9)

###  B.3 Counting the pairs of galaxies

Here we want to count the number of pairs of galaxies which can be used to
estimate $\xi_{\pm}(\mu,z_{1},z_{2})$. For this we need to estimate the number
of galaxies with fixed opening angle $\varphi$, $\mu=\cos\varphi$. We suppose
that we have pixels of angular aperture $\delta\theta$. The solid angle of a
cone with this opening angle is at lowest order

$\delta\Omega=\delta\theta^{2}\pi\,.$ (B.10)

Let us set the first pixel at the North Pole. We want to count the number of
pixels whose center is at an angle $\varphi\pm\delta\theta/2$ from this first
pixel. The solid volume of these pixels is

$\delta\Omega_{\varphi}=2\pi\int_{\varphi-\delta\theta/2}^{\varphi+\delta\theta/2}\;\sin\theta\,\mathrm{d}\theta=2\pi\delta\theta\sin\varphi\,.$
(B.11)

Here we assume that the full ring with angle $\varphi$ around the first pixel
is observed. For incomplete sky coverage this is not true for all values of
$\varphi$, but we neglect this in our treatment and take the sky coverage into
account as an overall factor $f_{\rm sky}$ which denotes the fraction of the
sky covered by the survey. Hence, the number of pixels forming such an angle
with the original pixel is given by

$N(\varphi)=\frac{\delta\Omega_{\varphi}}{\delta\Omega}=\frac{2\sin(\varphi)}{\delta\theta}\,.$
(B.12)

We also need the total number of pixels which we can choose as our first
pixel, given by

$N_{\rm tot}=f_{\rm sky}\frac{4\pi}{\delta\Omega}=\frac{4f_{\rm
sky}}{\delta\theta^{2}}\,.$ (B.13)

Here we have introduced $f_{\rm sky}$, the observed sky fraction. The total
number of couples separated by an angle $\varphi$ is

$N_{\mathrm{c}}(\varphi)=N_{\rm tot}\times N(\varphi)=\frac{8f_{\rm
sky}\sin\varphi}{\delta\theta^{3}}\,.$ (B.14)

If we consider auto-correlations, $z_{1}=z_{2}$, this number has to be be
divided by $2$ due to symmetry. Let us now denote the number of galaxies in a
pixel at redshift $z$ by $N_{\mathrm{g}}(z)$. For a given pair of pixels at
$z_{1}$ and $z_{2}$, one can choose
$N_{\mathrm{g}}(z_{1})N_{\mathrm{g}}(z_{2})$ pairs of galaxies. Hence, the
total number of pairs of galaxies which we can consider for the estimator
$\hat{\zeta}_{\pm}(\varphi,z_{1},z_{2})$ is

$N_{\mathrm{p}}(\varphi,z_{1},z_{2})=N_{\mathrm{g}}(z_{1})N_{\mathrm{g}}(z_{2})N_{\mathrm{c}}(\varphi)\,,$
(B.15)

To compute an estimator $\hat{\zeta}_{\pm}$, we need $4$ galaxies, or $2$
different pairs. The number of estimators we can form is therefore

$N_{\mathrm{e}}(\varphi,z_{1},z_{2})=\frac{N_{\mathrm{p}}(\varphi,z_{1},z_{2})(N_{\mathrm{p}}(\varphi,z_{1},z_{2})-1)}{2}\simeq\frac{1}{2}\left(N_{\mathrm{g}}(z_{1})N_{\mathrm{g}}(z_{2})\frac{8f_{\rm
sky}\sin\varphi}{\delta\theta^{3}}\right)^{2}\,.$ (B.16)

The division by $2$ of $N_{\mathrm{c}}$ becomes a division by $4$ of
$N_{\mathrm{e}}$ if we consider auto-correlations, $z_{1}=z_{2}$.

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