Weak lensing reconstruction through cosmic magnification I: a minimal variance map reconstruction

We present a concept study on weak lensing map reconstruction through the cosmic magnification effect in galaxy number density distribution. We propose a minimal variance linear estimator to minimize both the dominant systematical and statistical errors in the map reconstruction. It utilizes the distinctively different flux dependences to separate the cosmic magnification signal from the overwhelming galaxy intrinsic clustering noise. It also minimizes the shot noise error by an optimal weighting scheme on the galaxy number density in each flux bin. Our method is in principle applicable to all galaxy surveys with reasonable redshift information. We demonstrate its applicability against the planned Square Kilometer Array survey, under simplified conditions. Weak lensing maps reconstructed through our method are complementary to that from cosmic shear and CMB and 21cm lensing. They are useful for cross checking over systematical errors in weak lensing reconstruction and for improving cosmological constraints.


INTRODUCTION
Weak gravitational lensing has been established as one of the most powerful probes of the dark universe (Refregier 2003;Albrecht et al. 2006;Munshi et al. 2008;Hoekstra & Jain 2008). It is rich in physics and contains tremendous information on dark matter, dark energy and the nature of gravity at cosmological scales. Its modeling is relatively clean. At the multipole ℓ < 2000, gravity is the dominant force shaping the weak lensing power spectrum while complicated gas physics only affects the lensing power spectrum at less than 1% level (White 2004;Zhan & Knox 2004;Jing et al. 2006;Rudd et al. 2008). This makes the weak lensing precision modeling feasible, through high precision simulations.
Precision weak lensing measurement is also promising. The most sophisticated and successful method so far is to measure the cosmic shear, lensing induced galaxy shape distortion. After the first detections in the year 2000 (Bacon et al. 2000;Kaiser et al. 2000;Van Waerbeke et al. 2000;Wittman et al. 2000), data quality has been improved dramatically (e.g. Fu et al. 2008). Ongoing and planed surveys, such as DES 1 , LSST 2 , JDEM 3 , and Pan-STARRS 4 , have great promise for further significant improvement. However, weak lensing reconstruction through cosmic shear still suffers from practical difficulties associated with galaxy shape. These include shape measurement errors (additive and multiplicative) (Heymans et al. 2006;Massey et al. 2007) and the galaxy intrinsic alignment (Croft & Metzler 2000;Heavens et al. 2000;Jing 2002;Hirata & Seljak 2004;Mandelbaum et al. 2006;Hirata et al. 2007;).
An alternative method for weak lensing reconstruction is through cosmic magnification, the lensing induced fluctuation in galaxy (or quasar and any other celestial objects) number density (Menard 2002 and references therein). Since it does not involve galaxy shape, it automatically avoids all problems associated with galaxy shape.
However, the amplitude of cosmic magnification in galaxy number density fluctuation is usually one or more orders of magnitude overwhelmed by the intrinsic galaxy number fluctuation associated with the large scale structure of the universe. Existing cosmic magnification measurements (Scranton et al. 2005;Hildebrandt et al. 2009;Menard et al. 2010;Van Waerbeke 2010) circumvent this problem by cross-correlating two galaxy (quasar) samples widely separated in redshift. Unfortunately, the measured galaxy-galaxy cross correlation is often dominated by the foreground galaxy density-background cosmic magnification correlation and is hence proportional to a unknown galaxy bias of foreground galaxies. This severely limits its cosmology application.
The intrinsic galaxy clustering and cosmic magnification have different redshift and flux dependence, which can be utilized to extract cosmic magnification. Zhang & Pen (2006) demonstrated that, by choosing (foreground and background) galaxy samples sufficiently bright and sufficiently far away, the measured cross correlation signal can be dominated by the cosmic magnificationcosmic magnification correlation, which is free of the unknown galaxy bias. However, even for those galaxy samples, the foreground galaxy density-background cosmic magnification correlation is still non-negligible (Zhang & Pen 2006). This again limits its cosmology application due to the galaxy bias problem. Zhang & Pen (2005) further showed that, since the galaxy intrinsic clustering and cosmic magnification have distinctive dependence on galaxy flux, cosmic magnification can be extracted by appropriate weighting over the observed galaxy number density in each flux bins. Weak lensing reconstructed from spectroscopic redshift surveys such as the square kilometer array (SKA) in this way can achieve accuracy comparable to that of cosmic shear of stage IV projects. These works demonstrate the great potential of cosmic magnification as a tool of precision weak lensing reconstruction. Furthermore, since cosmic shear and cosmic magnification are independent methods, they would provide valuable cross-check of systematical errors in weak lensing measurement and useful information on galaxy physics such as galaxy intrinsic alignment. Zhang & Pen (2005) only discussed two limiting cases, completely deterministic and completely stochastic biases with known flux dependence. In reality, galaxy bias could be partly stochastic. Furthermore, the flux dependence of galaxy bias is not given a priori. In this paper, we aim to investigate a key question. Are we able to simultaneously measure both cosmic magnification and the intrinsic galaxy clustering, given the galaxy number density measurements in flux and redshift bins?
The paper is organized as follows. In §2, we present the basics of cosmic magnification and derive a minimal variance estimator for weak lensing reconstruction through cosmic magnification. In a companion paper we will discuss an alternative method to measure the lensing power spectrum through cosmic magnification. In §3, we discuss various statistical and systematical errors associated with this reconstruction. In §4, we target at SKA to demonstrate the performance of the proposed estimator. We discuss and summarize in §5. We SKA specifications are specified in the appendix A. Throughout the paper, we adopt the WMAP five year data (Komatsu et al. 2009) with Ωm = 0.26, ΩΛ = 0.74, Ω b = 0.044, h = 0.72, ns = 0.96 and σ8 = 0.80.

A MINIMAL VARIANCE LINEAR ESTIMATOR
Weak lensing changes the number density of background objects, which is called cosmic magnification. It involves two competing effects. A magnification of solid angle of source objects, leading to dilution of the source number density, and a magnification of the flux, making objects brighter. Let n(s, zs) be the unlensed number density at flux s and redshift zs, and the corresponding lensed quantity be n L (s L , zs). Throughout the paper the superscript "L" denotes the lensed quantity. The galaxy number conservation then relates the two by where s L = sµ. µ(θ, zs) is the lensing magnification at corresponding direction θ and redshift zs, The last expression has adopted the weak lensing approximation such that the lensing convergence κ and the lensing shear γ are both much smaller than unity (|κ|, |γ| ≪ 1). κ for a source at redshift zs is related to the matter overdensity δm along the line of sight by (3) Here, χ and χs are the radial comoving distances to the lens at redshift z and the source at redshift zs, respectively. D(χ) denotes the comoving angular diameter distance, which is equaling to χ for a flat universe.
Taylor expanding Eq. 1 around the flux s L , we obtain Where the parameter α is related to the logarithmic slope of the luminosity function, Notice that n(s L ) is related to the cumulative luminosity function N (> s) by N (> s) = ∞ s n(s L )ds L . Weak lensing then modifies the galaxy number over-density to Here, δg is the intrinsic (unlensed) galaxy number over-density (galaxy intrinsic clustering). For brevity, we have defined g ≡ 2(α−1) and will use this notation throughout the paper. Obviously, to the first order of gravitational lensing, the cosmic magnification effect can be totally described by the cosmic convergence and the slope of galaxy number count. The luminosity function averaged over sufficiently large sky area is essentially unchanged by lensing, The last expression is accurate to 0.1% since κ 2 ∼ 10 −3 . The above approximation is important in cosmic magnification. It allows us to calculate α by replacing the unlensed (and hence unknown) luminosity function n with n L , the directly observed one. This means that α is also an observable and its flux dependence is directly given by observations. We will see that this is the key to extract cosmic magnification from galaxy number density distribution. The biggest challenge in weak lensing reconstruction through cosmic magnification is to remove the galaxy intrinsic clustering δg. This kind of noise in cosmic magnification measurement is analogous to the galaxy intrinsic alignment in cosmic shear measurement. But the situation here is much more severe, since δg is known to be strongly correlated over wide range of angular scales and making it overwhelming the lensing signal at virtually all angular scales ( Fig. 1 & 2).
To a good approximation, δg = bgδm, where δm is defined as matter surface over-density and bg is the galaxy bias. This is the limiting case of deterministic bias. In general, for statistics no higher than second order, δg can be described with an extra parameter, the galaxy-matter cross correlation coefficient r. bg and r are Figure 1. Contaminations of the galaxy intrinsic clustering to the weak lensing reconstruction through cosmic magnification. The dominant contamination comes from the galaxy auto correlation, which is often several orders of magnitude larger than the lensing correlation. The galaxy intrinsic clustering also induces a galaxy-lensing cross correlation. This contamination can be comparable to the lensing signal. For a redshift bin 1.0 < z b < 1.2, we plot the resulting matter power spectrum Cm b m b , matter-lensing cross power spectrum Cm b κ b and the lensing power spectrum Cκ b κ b . For galaxies with bias bg = 1, a factor b 2 g shall be applied to Cm b m b and a factor bg shall be applied to Cm b κ b . defined through b 2 g (l, z) ≡ Cg(l, z) Cm(l, z) , r(l, z) ≡ Cgm(l, z) Cg(l, z)Cm(l, z) .

The estimator
The data we have are measurements of δ L g (θ) at each angular pixel of each redshift and flux bin. Throughout the paper we use the subscript "i" and "j" to denote the flux bins. For convenience, we will work in Fourier space. Then for a given redshift bin and a given multipole ℓ, we have the Fourier transform of the galaxy overdensity of the i-th flux bin, δ L g,i (ℓ). For brevity, we simply denote it as δ L i hereafter. The g factor of the i-th flux bin is denoted as gi. As explained earlier, it is a measurable quantity. bi is the galaxy bias of the i-th flux bin.
We want to find an unbiased linear estimator of the form Such that the expectation value ofκ is equal to the true κ of this pixel. Thus the weighting function w must satisfy the following conditions, Since the measured δ L i is contaminated by shot noise δ shot i , we shall minimize the shot noise. This corresponds to minimize Here,ni is the average galaxy surface number density of the i-th flux bin, a directly measurable quantity. The three sets of requirement (Eq. 10, 11 & 12) uniquely fix the solution. Using the Lagrangian multiplier method, we find the solution to be Here, the two Lagrangian multipliers λ1,2 are given by wi is invariant under a flux independent scaling in bi. For this reason, we only need to figure out the relative flux dependence in the galaxy bias, instead of its absolute value. Despite the neat mathematical solution above, in reality we do not know the galaxy bias a priori. We adopt a recursive procedure to simultaneously solve bi, wi and henceκ.
• The first step. In general, the power spectrum is dominated by the galaxy intrinsic clustering, namely Here, Cm b m b is the matter density angular power spectrum, Cκ b κ b is the lensing power spectrum and Cm b κ b is the cross-power spectrum, for the background redshift bin. Hence a natural initial guess for bi is given by the following equation, Plug b (1) i into in Eq. 14 and Eq. 13, we obtain the weighting w (1) i , and then the first guess of lensing convergence, is cosmology dependent, so one may think the bias reconstruction and hence the proposed lensing reconstruction are cosmology dependent. However, this is not the case. As explained earlier, the weighting function wi does not depend on the absolute value of bi. So in the exercise, Cm b m b can be fixed to any value of convenience in determining the bias since it is independent with flux. In this sense, the bias reconstruction is cosmology independent and is hence free of uncertainties from cosmological parameters.
• The second step. We subtract the lensing contribution from the measurement δ L i , using κ (1) constructed above. Our second guess for bi is then We then obtain w (2) i and κ (2) . However, this solution is still not exact. The expectation value of the r.h.s is j bj. • The iteration. We repeat the above step to obtain b (p) i and κ (p) (p = 3, · · · ) until the iteration converges. Since we know that the lensing contribution is sub-dominant and we start our iteration by neglecting the lensing contribution, and since the flux dependence of the intrinsic clustering and cosmic magnification (bi and gi) are different , such iteration should be stable and converge. We numerically check it to be the case.
The information of galaxy bias is not only encoded in the observed power spectrum of the same flux bin, but also encoded in the cross power spectra between different flux bins. In our exercise, we do not take this extra information into account. So there are possibilities for further improvement.
In the above description, we have neglected shot noise (or equivalently assumed that it can be subtracted completely). In reality, we are only able to subtract shot noise up to the limit of cosmic variance. Residual shot noise introduces systematical error in the lensing reconstruction, which we will quantify in next section.
Finally, we obtain the optimal estimator of cosmic convergence Our estimator explicitly satisfies i w can deviate from its real value bi, our estimator can be biased. This is yet another source of systematical errors in our weak lensing reconstruction through cosmic magnification. However, later we will show that such systematical error is under control.

The reconstructed weak lensing power spectrum
From the reconstructed κ, we can reconstruct the lensing power spectrum. For the same redshift bin, we have Throughout the paper we use the superscript or subscript "b" for quantities of background redshift bin and "f" for quantities of foreground redshift bin, and we neglect the superscript "(n)" of weighting w (n) i from this section. For the cross correlation between foreground and background populations, we have Here, Cκ f κ b is the lensing power spectrum between foreground and background redshift bins, and Cm f κ b is the cross power spectrum between foreground dark matter and background lensing convergence. We have neglected the correlation Cm f m b between foreground and background matter distributions. It is natural for nonadjacent redshift bins with separation ∆z ≥ 0.1, because of physical irrelevance. For two adjacent redshift bins, there is indeed a non-vanishing matter correlation. However, this correlation is also safely neglected since both the foreground and background intrinsic clustering are sharply suppressed by factors

STATISTICAL AND SYSTEMATICAL ERRORS
Our reconstruction method does not rely on priors on galaxy bias, in this sense, it is robust. However, there is still a number of statistical and systematical errors in the κ reconstruction and weak lensing power spectrum reconstruction. In this section, we outline and quantify associated errors in the lensing power spectrum.

The statistical error
Our estimator minimizes the shot noise in the measurement. For auto power spectrum C bb , the associated statistic measurement error is For the cross-power spectrum C bf , the statistical error is Throughout the paper we apply ∆l = 0.2l, and f sky is the sky coverage.
The above errors are purely statistical errors causing by shot noise resulting from sparse galaxy distribution. For this reason, we call them the weighted shot noise. We reconstruct the κ from the measurements of lensed galaxy number over-density δ L i . We solve the galaxy bias and weighting function from the observed galaxy power spectra |δ L i | 2 , and this whole process is free of any given cosmological model. So our proposed method to reconstruct the weak lensing map is the one at the given sky coverage, with the right cosmic variance. Only when we compare our measured lensing power spectra with their ensemble average predicted by a given theory, we must add cosmic variance. Since statistical errors arising . The HI galaxy bias b(s, z) and the magnification bias g(s, z) = 2(α − 1) as a function of flux s by fixing the redshift to be the central value of each redshift bin. The plotted curves are started from the flux limit at the fixed redshift. The cosmic magnification bias strongly depends on the flux, while the galaxy bias weakly changes with it. Such difference in flux dependence ensures us to find a optimal estimator to reconstruct the weak lensing from the cosmic magnification. from cosmic variance are independent to shot noise, it can be taken into account straightforwardly.

Systematical errors
We use the symbol δC to denote systematical errors in the lensing power spectrum measurement. We have identified three types of major systematical errors. Throughout the paper we use the superscript "(o)" (o = 1, 2, 3, · · · ) to denote them.

The systematical error from deterministic bias
The first set of systematical errors come from errors in determining the galaxy bias bi through Eq. 15 , even if we neglect the shot noise contribution to it. This bias arises due to a degeneracy that is among the power spectra Cm b m b , Cm b κ b , Cκ b κ b and galaxy bias bi in Eq. 15, which causes a slight deviation from its true flux dependence. In a companion paper we will address and clarify this issue in more detail (Yang & Zhang, in preparation). We will also show that this systematical error is correctable.
The consequence is that i w (n) i bi = 0 in Eq. 19. It causes a systematical error in the auto correlation of the same redshift bin It also biases the cross correlation measurement between two different redshift bins,

The systematical error from stochastic bias
Our reconstruction has approximated the galaxy bias to be deterministic, namely rij = 1. rij is the correlation coefficients be-tween galaxies with different flux. It is known that galaxy bias exists a stochastic component and hence rij < 1, especially at nonlinear scales (Wang et al. 2007;Swanson et al. 2008;Gil-Marin et al. 2010). This does not affect the determination of the galaxy bias bi, since we only use the auto correlation between the same flux and redshift bin to determine the bias. However, stochasticity does bias the power spectrum measurement, since now the condition (Eq. 11) no longer guarantees a complete removal of the galaxy intrinsic clustering. The systematical error induced to the auto correlation measurement is Here, ∆rij ≡ 1 − rij . Given present poor understanding of galaxy stochasticity, we demonstrate this bias by adopting a very simple toy model, with This model is by no way realistic. The particular reason to choose this toy model is that readers can conveniently scale the resulting δC (2) to their favorite models of galaxy stochasticity by multiplying a factor 100∆rij (ℓ, z). As we will show later, this systematical error could become the dominant error source. However, measuring the lensing power spectrum between two redshift bins can avoid this problem. Clearly, stochasticity in galaxy distribution does not bias such cross power spectrum measurement.

The systematical error from shot noise
By far we have neglected the influence of galaxy shot noise in determining the galaxy bias (Eq.15). The induced error is denoted as δbj = b r j −bj . Here, b r j is the true bias andbj is the final obtained bias b (n) j from the iteration. It is reasonable to consider the case that shot noise is subdominant to the galaxy intrinsic clustering. Under this limit, Where C shot j is the shot noise power spectrum of j-th flux bin in the observed power spectrum (Eq. 15). We are then able to Taylor expand flux weighting wi aroundbj to estimate the induced bias in it, Where wi(bj) is the final obtained flux weighting. After a lengthy but straightforward derivation, we derived the induced bias in the . The contaminations before and after using the estimator to reconstruct the weak lensing auto power spectrum. We choose two redshift bins 0.4 < z b < 0.6 and 1.0 < z b < 1.2, for which, the upper panel shows us the suppression of matter auto power spectra, and the lower panel presents the suppression of matter-lensing cross power spectra, respectively. Clearly, for the same redshift bin our reconstruction method can sharply reduce the two correlations both induced by the intrinsic clustering. The power spectrum Cm b m b can be suppressed by an order of ∼ 10 4 and the power spectrum Cm b κ b can be reduced by one or more orders of magnitude. The suppression is more stronger for the lower redshift bin, since the value of i w b i b b i rises with redshift. Roughly speaking, this is caused by the increasing error in the final galaxy bias. With the increasing redshift, the contribution from the weak lensing power spectrum to the observed power spectrum increases, so the error in the initial galaxy bias increases and then leads to the rising error in the final galaxy bias because of the existing degeneracy in the process of solving galaxy bias. auto correlation measurement, The induced bias in cross correlation measurement is

Other sources of error
There are other sources of error that we will neglect in the simplified treatment presented here. First of all, we only deal with idealized surveys with uniform survey depth without any masks. Complexities in real surveys will not only impact the estimation of errors listed in previous sections, but may also induce new sources of error. These errors can be investigated with mock catalog mimicking real observations. We will postpone such investigation elsewhere.
Another source of error is the determination of α (or equivalently g). For SKA that we will target at, errors in α are negligible since the galaxy luminosity function can be determined to high accuracy given billions of SKA galaxies with spectroscopic redshifts. However, this may not be the case for other surveys, due to at least two reasons. (1) Some surveys may not have sufficient galaxies at bright end. Large Poisson fluctuations then forbidden precision measurement of α there. (2) α is defined with respect to the galaxy luminosity function in a given redshift bin. However, a large fraction of galaxy surveys may only have photometric redshift measurement. Errors in redshift, especially the catastrophic photo-z error, affect the measurement of α.
Dust extinction is also a problem for optical surveys, but not for radio surveys like SKA. Dust extinction also induces fluctuations in galaxy number density, with a characteristic flux dependence α. This flux dependence differs from the α − 1 dependence in cosmic magnification and bg(s) dependence in galaxy bias. The minimal variance estimator can be modified such that i wiαi = 0 to eliminate this potential source of error, when necessary.
In next section, we will quantify statistical and systematical errors listed in §3.1 & 3.2 for the planned 21cm survey SKA. Although the estimation is done under very simplified conditions, it nevertheless demonstrates that these errors are likely under control.

THE PERFORMANCE OF THE MINIMAL VARIANCE ESTIMATOR
We target at SKA to investigate the feasibility of our proposal. SKA is able to detect billions of galaxies through their neutral hydrogen 21cm emission. The survey specifications are adopted as field of view FOV = 10deg 2 , total survey period t all = 5yr and total sky coverage 10 4 deg 2 (Dewdney et al. 2009;Abdalla et al. 2010;Faulkner et al. 2010). More details of the survey are given in the appendix. Fig. 1 and Fig. 2 demonstrate contaminations of galaxy intrinsic clustering to the cosmic magnification measurement from one same redshift bin and one couple of foreground and background redshift bins. For a typical redshift bin 1.0 < z b < 1.2, the auto matter angular power spectrum Cm b m b is larger than the lensing power spectrum by two or more orders of magnitude. Fig. 1 also shows Cm b κ b is comparable to Cκ b κ b . Since typical bias of 21cm galaxies is ∼ 1 (Fig. 3), this means that the galaxy intrinsic clustering overwhelms the lensing signal by orders of magnitude. Similarly, in Fig. 2 the cross power spectrum Cm f κ b induced by the foreground intrinsic clustering overwhelms the weak lensing power spectrum Cκ f κ b by one or more orders of magnitude. These big contaminations related galaxy intrinsic clustering make the weak lensing measurement difficult from the directly cosmic magnification measurement, unless for sufficiently bright foreground and background galaxies at sufficiently high redshifts (Zhang & Pen 2006).
As we explained in earlier sections, the key to extract the lensing signal from the overwhelming noise is the different dependences of signal and noise on the galaxy flux. Fig. 3 shows the lensing signal and the intrinsic clustering indeed have very different dependences on the galaxy flux. For the 21cm emitting galaxies, the flux dependence in g ≡ 2(α − 1) is much stronger than that in the galaxy bias. Furthermore, g changes sign from faint end to bright end. Such behavior can not be mimicked by bias, which keeps positive.
From such difference in the flux dependences, we expect that our estimator to significantly reduce contaminations from the galaxy intrinsic clustering. As explained earlier, in the galaxy correlation between the same redshift bin, the intrinsic clustering induces a systematical error proportional to Cm b m b (Eq. 24) and an error proportional to Cm b κ b (Eq. 24). In the ideal case that both the stochasticity and shot noise in galaxy distribution can be neglected, the systematical error proportional to Cm b m b will be suppressed by a factor 1/ Fig. 4 shows that this suppression factor is of the order ∼ 10 4 at interesting scales 10 < ∼ ℓ < ∼ 10 4 . For the same reason, the systematical error proportional to Cm b κ b will be suppressed by a factor 1/ i w b i b b i ∼ 10 2 over the same angular scales. The systematical error induced by foreground intrinsic clustering in the weak lensing reconstruction between two redshift bins is proportional to Cm f κ b (Eq. 25). Our estimator also suppresses it by a factor 1/ i w f i b f i ∼ 10 2 , as shown in Fig. 5.

Same redshift bins
The lensing power spectrum can then be directly measured through the reconstructed lensing maps. This can be done for maps of the same redshift bin. Fig. 6 compares the residual systematical errors to the lensing power spectrum signal. Overall, our minimal variance estimator significantly suppresses the galaxy intrinsic clustering and makes the weak lensing reconstruction feasible. For source redshift z b ∼ 1 (e.g. 1.0 < z b < 1.4), all investigated systematical errors and statistical errors are suppressed to be subdominant to the signal. We expect the lensing power spectrum to be measured with an accuracy of ∼ 10%-20%. However, at lower and higher redshifts, the reconstruction is not successful. We observe the systematical error δC (1) bb increases with increasing redshift and overwhelms the lensing signal at z b > ∼ 2. In the current formalism it is difficult to explain this behavior straightforwardly. But roughly speaking, this is caused by worse initial guess on the flux dependence of galaxy bias coupled with the degeneracy explained earlier. The initial guess is accurate to the level of Cκ b κ b /Cm b m b . So the associated error increases with redshift.
This systematical error looks rather frustrating. However in the companion paper (Yang & Zhang, in preparation) we will show that the systematical error δC (1) is correctable. Where we can separate the degeneracy, and reconstruct a quantity through a direct multiparameter fitting against the measured power spectra, which perfectly mimics the flux dependence of galaxy bias b b i , because the power spectrum Cm b m b is independent with the flux and the correction term Cm b κ b /Cm b m b is small especially for high redshift. Although it is bad to find that the final convergence depends on the initial guess of galaxy bias, y b i as a guess of galaxy bias does re- bb from the stochastic bias, and δC bb from the shot noise are presented by the solid, dotted and dash-dot-dot-dotted lines, respectively. While statistical error is plotted by the dashed line. Here the statistical error is called weighted shot noise only from the sparse galaxy distribution, since we aim to reconstruct the weak lensing at each angular pixel with the corresponding cosmic variance. For the intermediate redshift bins 1.0 < z b < 1.2 and 1.2 < z b < 1.4, the signal overwhelms all these errors which can be controlled to ∼ 10%-20% level. duce the systematical error δC (1) and then works far better than the obtained galaxy bias in this paper.
The systematical error δC (3) arising from shot noise becomes non-negligible at z b ∼ 2, due to sharply decreasing galaxy density and increasing shot noise at these redshifts (see Fig. A2 ). We find that this error is always subdominant to either δC bb . This term is also ∝ (δb b bb and ∆C bb are the sums of 1/nj weighted in different ways and hence have similar shapes. (2) The galaxy bias in our fiducial model is scale independent. This results in scale independent weighting function wj, as long the bias can be determined to high accuracy. For these reasons, δC bb , ∆C bb ∝ l 0 . This is the case at low redshift. (3) However, the accuracy in determining galaxy bias is significantly degraded by contamination proportional to Cκ b κ b /Cm b m b , which is scale dependent and increases with redshift. This is the reason both δC (3) bb and ∆C bb show complicated angular dependence at z b ≃ 2 (Fig. 6).
At low redshift (e.g. 0.4 < z b < 0.6), the systematical error δC (2) bb arising from galaxy bias stochasticity becomes dominant or even exceeds the lensing signal. This is what expected. The galaxy intrinsic clustering is stronger at lower redshift. This amplifies the impact of galaxy stochasticity. Weaker lensing signal at lower redshift further amplifies its relative impact. As expected, Fig. 6 shows that δC (2) bb /Cκ b κ b decreases with increasing redshift and becomes negligible at z b ∼ 2. Since δC (2) bb ∝ ∆rij, the importance of δC (2) bb is sensitive to the true nature of galaxy stochasticity. Its value should be multiplied by a factor 100∆rij for fiducial value of ∆rij = 1%. We hence conclude that galaxy stochasticity is likely the dominant source of error in weak lensing reconstruction through cosmic magnification.
bf from the deterministic galaxy bias is dominant and the systematical error from the stochastic bias can be avoided in such cross lensing power spectrum, namely δC (2) bf = 0. For every couple of foreground and background redshift bins, the cross reconstructed weak lensing signal dominates at scale range 10 < ∼ ℓ < ∼ 10 4 and it can be measured to reach ∼ 10%-20% accuracy.

Different redshift bins
Fortunately this stochasticity issue can be safely overcome in the lensing power spectrum measurement through lensing maps reconstructed in two different redshift bins (foreground and background bins). The results are shown in Fig. 7. In this case, the systematical error is dominated by δC (1) bf . The stochasticity does not induce systematical error so that δC (2) bf = 0. Overall, the lensing power spectrum measurement through cross correlating reconstructed maps of different redshift bins is more robust than the one based on the same redshift bin. The reconstruction accuracy can be controlled to 10%-20% over a wide range of foreground and background redshifts.
At last, for a consistency test, we check whether our results depend on the division of the flux bin. As expected, various systematical errors and statistical error change little with respect to different choices of flux bins, as long as these bins are sufficiently fine.

Uncertainties in the forecast
There are a number of uncertainties in the forecast, besides the ones discussed in previous sections. (1) In the fiducial galaxy intrinsic clustering model, we have neglected the scale dependence in galaxy bias.
(2) We have neglected cosmic variance in the lensing signal so the fiducial power spectrum is the ensemble average. But we do not expect it can significantly impact our result, since the cosmic variance at most relevant scale is small due to l∆lf sky ≫ 1 with a given large sky coverage of SKA. (3) The toy model of galaxy stochasticity is too simplified. In reality, the cross correlation coefficient r should be a function of redshift, angular scale and galaxy type and flux.
For these reasons, the numerical results presented in this paper should only be trusted as rough estimation. Robust evaluation of the weak lensing reconstruction performance through cosmic magnification requires much more comprehensive investigation. Nevertheless, the concept study shown in this paper demonstrate that weak lensing reconstruction through cosmic magnification is indeed promising.

CONCLUSIONS AND DISCUSSIONS
We propose a minimal variance estimator to reconstruct the weak lensing convergence κ field through the cosmic magnification effect in the observed galaxy number density distribution. This estimator separates the galaxy intrinsic clustering from the lensing signal due to their distinctive dependences on the galaxy flux. Using SKA as an example, we demonstrate the applicability of our method, under highly simplified conditions. It is indeed able to significantly reduce systematical errors. We have identified and quantified residual systematical errors and found them in general under control. Extensive efforts shall be made to test our reconstruction method more robustly and to improve this method.
Comparing to previous works, our method has several key features/advantages.
• Unlike weak lensing reconstruction through cosmic shear, it does not involve galaxy shape measurement and reconstruction and hence avoids all potential problems associated with galaxy shapes. Hence the reconstructed lensing maps provide useful independent check against cosmic shear measurement.
• Unlike existing cosmic magnification measurements which actually measure the galaxy-lensing cross correlation, our estimator allows directly reconstruction of the weak lensing κ field. From the reconstructed κ, we are able to directly measure the lensing power spectra of the same source redshift bin and between two redshift bins. These statistics do not involve galaxy bias, making them more robust cosmological probes. The usual lensing tomography is also directly applicable.
• Unlike our previous works (Zhang & Pen 2005, the new method does not require priors on the galaxy bias, especially its flux dependence. Our methods is able to simultaneously measure the galaxy bias (scaled with a flux independent factor) and the lensing signal. Hence we do not adopt any priors on the galaxy bias (other than that it is deterministic) and treat the galaxy bias as a free function of scale and flux. The price to pay is degradation in constraining galaxy bias and in lensing reconstruction. Adding priors on the galaxy bias can further improve the reconstruction precision, although the reconstruction accuracy will be affected by uncertainties/biases in the galaxy bias prior. For this reason, we do not attempt to add priors on galaxy bias in the reconstruction.
• Our method is complementary to a recent proposal by Heavens & Joachimi (2011), which proposes a nulling technique to reduce the galaxy intrinsic clustering by proper weighting in redshift. Comparing to this method, our method only utilizes extra information encoded in the flux dependence to reduce/remove the galaxy intrinsic clustering. It keeps the cosmological information encoded in the redshift dependence disentangled from the process of removing the intrinsic clustering.
The proposed approach is not the only way for weak lensing reconstruction through cosmic magnification. The current paper focuses on direct reconstruction of the lensing convergence κ map. In a companion paper, we will focus on direct reconstruction of the lensing power spectrum (Yang & Zhang, in preparation). We will show that combining two-point correlation measurements between all flux bins, the lensing power spectrum can be reconstructed free of assumptions on the galaxy intrinsic clustering. We will see that this approach is more straightforward, more consistent and easier to carry out. However the method presented in this paper does have advantages. Since it reconstructs the lensing κ map, higher order lensing statistics such as the bispectrum can be measured straightforwardly. Furthermore, the reconstructed κ map can be straightforwardly correlated with other tracers of the large scale structure. For example, it can be correlated with the lensing map reconstructed from CMB lensing (Seljak & Zaldarriaga 1999;Hu & Okamoto 2002) or 21cm lensing. Furthermore, through this approach we can have better understanding on the origin of various systematical errors, which can be entangled in the alternative approach.
Our reconstruction method is versatile to include other components of fluctuation in the galaxy number density. The extinction induced fluctuation discussed earlier is one. High order corrections to the cosmic magnification is another. Taylor expanding Eq. 1 to the second order, we obtain δ L g = δg + 2(α − 1)κ (32) +2(α − 1) (κδg − κδg ) + 1 2 (γ 2 − γ 2 ) Here g2 ≡ (s 2 /n)d 2 n/ds 2 is related with the second derivative of luminosity function. The above result shows that the κ reconstruction through cosmic magnification is biased by terms proportional to κδg and γ 2 (second line in the above equation). Similar biases also exist in cosmic shear measurement. We recognize κδg as the source-lens coupling. The γ 2 term is analogous to the κγ term caused by reduced shear γ/(1 − κ). Precision lensing cosmology has to model these corrections appropriately.
The high order corrections ∝ 1 − 5α + 2g2 can in principle be separated due to its unique flux dependence. However, it is unclear whether the reconstruction is doable, even for a survey as advanced as SKA.

APPENDIX A: SKA SURVEY
SKA is a future radio survey with aiming to construct the world's largest radio telescope. Through the neutral hydrogen emitting 21 cm hyperfine transition line, it can observe large sample of 21 cm galaxies. Even at high-redshift, the observed 21 cm galaxies is extremely excess than the QSOs or LBGs that is usually used as background object (Scranton et al. 2005;Hildebrandt et al. 2009), so it is easily to overcome the shot noise in the measurement of cosmic magnification effect, of which precise measurement is sensitive to the shot noise. In addition, compared with photometric survey, the spectroscopic survey can determine the redshift more precise by using the redshifted wavelength of 21 cm line (λ = λ0(1 + z)). Furthermore, radio survey is free of galactic dust extinction, which is correlated with foreground galaxies and then induces a correction to the galaxy-galaxy cross correlation. As a consequence, it is expected a good performance of our proposed method to measure the cosmic magnification for SKA. In this paper, we adopt the survey specifications as follows: the telescope field-of-view (FOV) is 10deg 2 without evolution, the overall survey time is 5yr and the sky coverage is 10000deg 2 .

A1 HI mass limit
By assuming a flat profile for the emission line with frequency, we obtain the following equation linking MHI with the observed flux density s by atomic Physics (detail in Abdalla & Rawlings (2005)), MHI = 16π 3 mH A21hc χ 2 (z)sV (z)(1 + z) .
Where χ(z) is the comoving angular distance, A21 is spontaneous transition rate, mH is the atomic mass of hydrogen, h is Planck's constant and V (z) is the line-of-sight velocity spread. We assume a scaling with redshift for the typical rest-frame velocity width of the line V (z) = V0/ √ 1 + z, where V0 = 300kms −1 , and ignore the effects of inclination. The r.m.s. sensitivity for a dual-polarization radio receiver at system temperature Tsys for an integration of duration t on a telescope of effective collecting area A eff is given by Here, the correlation efficiency ηc is adopted as ηc = 0.9, the duration time t = 40h for each area of the sky and the width of the HI emission line determines the relevant frequency bandwidth ∆ν, which is related to a line-of-sight velocity width V (z) at redshift z The flux detection limit s lim for galaxy is defined by the threshold parameter nσ = s lim /srms. Here, we apply nσ = 5. From Eq .
(A1), we can obtain the HI mass limit.