A new observable for cosmic shear

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 measured in this way with the future SKA II survey.


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 ΛCDM.
The shear from scalar perturbations is determined by the lensing potential, 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 δα 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 (θ, ϕ), and the corresponding unit vector is n. The surface element of the sphere is denoted dΩ. The lensing potential is called φ(n, z). The Bardeen potentials are Φ and Ψ. The Spin Weighted Spherical Harmonics are s Y ,m , while the 'usual' Spherical Harmonics, 0 Y ,m , are simply denoted Y ,m . Figure 1. The general setup (in 2D, seen from above): Two pixels are considered, in directions n j and at redshifts z j . The directions are separated by an angle ϕ (or n 1 · n 2 = cos ϕ = µ). In each pixel, one galaxy is chosen (represented by the dots). The computations are made in the equatorial plane.

Theoretical development
The lensing potential given in Eq. (1.1) is a stochastic quantity which can be decomposed into Spherical Harmonics as where the scalars φ ,m (z) are also random variables. Assuming statistical isotropy different values of and m are not correlated and their two-point correlation spectrum is given by 2) The C (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 Here ε is the eccentricity of the galaxy, (γ 1 , γ 2 ) are the components of the shear matrix in the Sachs basis, and α is the angle between the major-axis of the galaxy shape and the first basis vector 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 α, see [23] for details. If the semi-major axis of the galaxy is aligned with the shear, δα vanishes. For example if we choose e 1 in the direction of the semi-major axis of the galaxy such that α = 0, alignment with the shear implies γ 2 = 0 and hence δα = 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 δα. 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 δα, the angle between the polarisation direction and the semi major axis, also the eccentricity ε and the direction of the galaxy's semi-major axis parametrised by α are observables. Similar to our previous work [23], we define an observable which we call the 'scaled rotation' 1 by 3) the scaled rotation is related to the shear as which is actually simply the shear in the direction α−π/4, see Appendix A.4. We want to determine the correlation function Θ(n 1 , z 1 )Θ(n 2 , z 2 ) for two directions n 1 and n 2 in the sky and two redshifts z 1 , z 2 . Our expression for the variable Θ(n, z) given in (2.6) in principle depends on our choice for the Sachs basis via the angle α and via γ 1 and γ 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 e 1 the direction of the great circle from n 1 to n 2 which is equivalent to putting both galaxies on the 'Equator', with coordinates (π/2, 0) and (π/2, ϕ) and µ = cos ϕ = n 1 ·n 2 . Note that there is still a 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 with µ = n 1 · n 2 = cos ϕ and ζ + (µ, z 1 , z 2 ) and ζ − (µ, 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 (z 1 , z 2 ) as where the polynomialsP (µ)Q (µ) are defined by, µ = cos θ, and s Y ,m are the Spin Weighted Spherical Harmonics. More details, and the explicit expressions for = 2, . . . , 5 are given in App. A. From the observable Θ we now construct an estimator for the coordinate independent correlation functions ζ + and ζ − . Since we want to estimate two correlation functions, we need two couples of galaxies, separated by the same angle ϕ. Schematically, as Θ ∼ γ 1 + γ 2 , one needs two galaxies to invert this relation and express γ 1 and γ 2 . Moreover, as the correlation function is given by ζ ∼ γγ , we need the value of γ 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 ϕ and located at the same redshifts (within the resolution of our survey). The galaxies of the first couple have the directions and redshifts (n j , z j ) with angles as defined above α j (j = 1, 2), while the second couple of galaxies are located in different directions n j and with different angles α j but inside the same redshift bins z j . Note that we define the angles α j and α j , with respect to the great circle connecting n 1 and n 2 respectively n 1 and n 2 which can be different for each couples. The two couples of galaxies, however should be separated by the same angle ϕ (within our angular resolution), i.e. n 1 · n 2 = n 1 · n 2 = cos ϕ = µ. The two observables are the product of the scaled rotations, namely (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 Ξ and Ξ , we can extract the estimatorsζ , (2.17) and . (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 n 1 and 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 α i and α i . In other words, and by construction, Once an estimator for ζ ± is obtained, the estimator for the lensing potential power spectrum C (z 1 , z 2 ) can be given by Eqs. (2.8) and (2.9).

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 ϕ (with µ = cos ϕ), an estimator for the correlation functions ζ ± is given by Eq. (2.15) and Eq. (2.16). Of course, to obtain a good estimator for ζ ± (µ, z 1 , z 2 ) we need to have many pairs of galaxies at a given angular separations ϕ (with µ = cos ϕ) 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 α j and α j . The expressions for F 1 and F 2 (see Eqs. (2.17) and (2.18)) also tell us that for α 1 + α 2 = α 1 + α 2 = π/4 we cannot determineζ + while for for α 1 − α 2 = α 1 − α 2 = π/4 we cannot determineζ − . It follows that to obtain a well-defined estimator of the correlation functions ζ ± we need to select properly the angles α j and α j , excluding galaxy pairs with α 1 + α 2 = α 1 + α 2 = π/4 or with α 1 − α 2 = α 1 − α 2 = π/4. Note, however, it does not matter whether the angles α j , α j 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 δα 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 estimatorζ ±,q (µ) with its relative error τ ±,q . The total signal-to-noise ratio for the measurement of ζ ± (µ, z 1 , z 2 ) is given by Eq. (B.6) 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) τ ±,q τ 0 .
Then, the signal-to-noise is estimated as where N e (µ, z 1 , z 2 ) is the number of estimators one can extract by choosing two couples of galaxies separated by an angle ϕ. The number of quadruplets is computed in Appendix B and is given by where N g (z) is the number of galaxies in a pixel at redshift z and δθ is the aperture of the angular resolution. Note that the formula for N 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 τ 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 where N c is the number of couples of pixels separated by an angle ϕ. 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 γ(n, z) in every pixel using Eq. (2.6). By doing this, the signal-to-noise ratio for every pixel would be given by N g /τ 0 , where N g is the galaxy number in this specific pixel and τ 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 γ 1 and γ 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.

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 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 f sky ≈ 0.7 , (4.1) Moreover, the typical shear signal γ 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 π/5 on the angles and 1/2 on ε. This leads to a conservative relative error per galaxy pair of the order of τ 0 ≈ 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 SNR ≈ 45 sin ϕ . This is the signal-to-noise ratio for our estimatorζ ± (ϕ, 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 ζ ± (ϕ), 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 φ (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 Ω cdm = 0.0552278 , h 2 Ω b = 0.0102921 , log(10 9 A s ) = 0.742199 , n 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.46) and Eq. (A.47). However, the polynomialsP andQ are highly oscillating as 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 Truncating the integral at = 20 000 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 ζ ± (ϕ) computed in the flat-sky approximation. The shaded region around each curve represents the uncertainty computed with SNR = 40 √ sin ϕ. 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.

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 ϕ, 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, 6-8, 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 ϕ and two redshifts z 1 , z 2 is expected to scale as ∼ 10 9 sin ϕ. 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 δα 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 δα and α 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.
The Spin Weighted Spherical Harmonics for generic s ∈ Z are obtained recursively with the spin raising and spin lowering operators given by Eq. (A.2) and Eq. ( together with the starting point 0 Y ,m ≡ Y ,m . Hence, the slashed derivatives can be interpreted as spin raising/lowering operators. In particular, for s = ±2, these definitions yield The Spin Weighted Spherical Harmonics satisfy the orthogonality condition ( dΩ = sin θdθdϕ) The Spin Weighted Spherical Harmonics also satisfy the the following addition theorem where the angles (α, β, γ) are defined through the implicit relation Here R E (α, β, γ) is the rotation matrix with the Euler angles α, β and γ. More precisely cos α cos β cos γ − sin α sin γ − cos γ sin α − cos α cos β sin γ cos α sin β cos β cos γ sin α + cos α sin γ cos α cos γ − cos β sin α sin γ sin α sin β − cos γ sin β sin β sin γ cos β   .
(A.13) Explicit expressions of the Spin Weighted Spherical Harmonics for s = 0, 1, 2 and ≤ 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 polynomialsP (µ) andQ (µ) which will be useful later. For µ = cos θ they are defined as From the orthonormality condition Eq. (A.9), it is easy to see that The explicit expressions for these polynomials for = 2, . . . , 5 are given in table 1

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 (e 1 , e 2 ) be an orthonormal basis on the sphere associated with the usual spherical coordinates (θ, ϕ). We define the (+, −) basis The spin raising and lowering operators are simply related to the covariant derivatives in directions e ± , With these identities, the relevant operators to compute the shear from the lensing potential are and  where it is assumed / ∂ / ∂ = / ∂ / ∂, as in this context it acts on the scalar lensing potential φ. The definition of the shear in the (e 1 , e 2 ) basis is where φ is the lensing potential. This shows that the shear is a spin 2 object. Using γ ± = γ 1 ± iγ 2 and the relations given above, the shear in the (+, −) basis is given by the slashed derivatives of the lensing potential as Hence γ + has helicity +2 while γ − has helicity −2. Using the standard decomposition for the lensing potential 25) and the squared raising/lowering operators given in Eq (A.6), one obtains the decomposition of the shear in the (+, −) basis as The complex numbers φ ,m (z) are random variables whose expectation values define the angular power spectrum of the lensing potential, As the lensing potential φ(n, z) is real. They satisfy

A.4 Invariant correlation functions
Here, we compute the shear and its correlation functions in a coordinate invariant way, see for example [46]. Let (θ, ϕ) be the spherical coordinates and (e 1 , e 2 ) the associated orthonormal frame.
With such a basis, the shear is a 2-tensor of the form For a generic tangent vector e = (cos α, sin α) in the (e 1 , e 2 ), the shear in direction e is defined as It is clear from the definition that γ 1,2 and the angle α do depend on the coordinate system. However, for a fixed (physically defined) vector e, the shear in direction e, γ e does not depend on the coordinates, which makes this quantity a good candidate to study correlation functions. For two galaxies located at (n 1 , z 1 ) and (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 ζ c (µ, z 1 , z 2 ) = γ −π/4 (n 1 , z 1 )γ 3π/4 (n 2 , z 2 ) = γ 2 (n 1 , z 1 )γ 2 (n 1 , z 1 ) , (A. 45) with µ = n 1 · n 2 = cos ϕ. 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 ζ p (µ, z, z ) = 2 + 1 128π C (z, z )ν 2 (P (µ) +Q (µ)) , (A. 46) ζ c (µ, z, z ) = 2 + 1 128π C (z, z )ν 2 (P (µ) −Q (µ)) . (A.47) Note that the sums start at = 2. Defining, and using the orthogonality properties of the polynomialsP andQ given in Eq.(A.16), we have  Given X j measurements of an observable X, each of them with error δX j = τ j X j (τ is the relative error). We want to construct an estimator for X. We definê 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 To see that this is the best choice, we note that the error on the estimator is given by 3) The square of the SNR which we want to maximise is the quantity A =X 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 w j = 1 .
Computing A explicitly one finds the well known result (B.6)

B.2 Specific example
If we consider our estimator of the correlation function, ζ ± (µ, z 1 , z 2 ) and denote the value obtained from two pairs of galaxies byζ j and the error by δζ j , then we find and the optimal estimator for ζ ± (µ, z 1 , z 2 ) iŝ ζ(µ, z 1 , z 2 ) = 1 Z

B.3 Counting the pairs of galaxies
Here we want to count the number of pairs of galaxies which can be used to estimate ξ ± (µ, z 1 , z 2 ). For this we need to estimate the number of galaxies with fixed opening angle ϕ, µ = cos ϕ. We suppose that we have pixels of angular aperture δθ. The solid angle of a cone with this opening angle is at lowest order δΩ = δθ 2 π . (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 ϕ ± δθ/2 from this first pixel. The solid volume of these pixels is Here we assume that the full ring with angle ϕ around the first pixel is observed. For incomplete sky coverage this is not true for all values of ϕ, but we neglect this in our treatment and take the sky coverage into account as an overall factor f 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 We also need the total number of pixels which we can choose as our first pixel, given by Here we have introduced f sky , the observed sky fraction. The total number of couples separated by an angle ϕ is N c (ϕ) = N tot × N (ϕ) = 8f sky sin ϕ δθ 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 g (z). For a given pair of pixels at z 1 and z 2 , one can choose N g (z 1 )N g (z 2 ) pairs of galaxies. Hence, the total number of pairs of galaxies which we can consider for the estimatorζ ± (ϕ, z 1 , z 2 ) is N p (ϕ, z 1 , z 2 ) = N g (z 1 )N g (z 2 )N c (ϕ) , (B.15) To compute an estimatorζ ± , we need 4 galaxies, or 2 different pairs. The number of estimators we can form is therefore N e (ϕ, z 1 , z 2 ) = N p (ϕ, z 1 , z 2 )(N p (ϕ, z 1 , z 2 ) − 1) 2 1 2 N g (z 1 )N g (z 2 ) 8f sky sin ϕ δθ 3 2 . (B.16) The division by 2 of N c becomes a division by 4 of N e if we consider auto-correlations, z 1 = z 2 .