CALIBRATING PHOTOMETRIC REDSHIFT DISTRIBUTIONS WITH CROSS-CORRELATIONS

We use the halo model to populate N-body cold dark matter simulations with mock galaxies to test the cross-correlation method. The simulations compute the evolution of large-scale structure in a periodic, cubical box of side 1 Gpc/h using a Tree-PM code White (2002). There are 1024³ dark matter particles of mass 6 × 10¹⁰ M_☉/h. The randomly generated Gaussian initial conditions are evolved from a starting redshift of z = 75. The Plummer softening is 35 kpc/h (comoving). The simulated cosmology has the parameters: Ω_m = 0.25, Ω_Λ = 0.75, Ω_bh² = 0.0224, h = 0.72, n = 0.97, and σ₈ = 0.8.

Halo catalogs are constructed from this simulation using a friends-of-friends algorithm (Davis et al. 1985) with a linking length of b = 0.168 in units of the mean inter-particle spacing. There are approximately 7.5 million halos of mass greater than 5 × 10¹¹ M_☉/h resolved with ∼8 or more dark matter particles. The galaxy catalogs are constructed by populating these halos according to the halo-model prescription with a center-satellite split (Zehavi et al. 2005). It is assumed that very small halos host no galaxies. Halos that cross a mass threshold M_min are assumed to host one central galaxy. Halos of much higher mass are assumed to have formed through mergers of smaller halos and will host both central and satellite objects. The mean number of galaxies in a halo of mass M (i.e., the Halo Occupation Distribution or HOD; Cooray & Sheth 2002) is given by

$$\begin{eqnarray} &&\langle N_{\rm gal}(M)\rangle =\Theta (M-M_{\rm min}) \left(1+\frac{M-M_{\rm min}}{10\,M_{\rm min}}\right). \end{eqnarray} \tag{ 11 }$$

The position of the central galaxy is assumed to be at the halo center defined by the position of the most gravitationally bound dark matter particle. The number of satellite galaxies in any particular halo is drawn from a Poisson distribution, and the satellites are assumed to trace the dark matter. The mock catalogs are constructed from a single simulation snapshot at redshift z = 0.1, rather than from a true light cone. The HOD also has no redshift dependence, so there is no clustering evolution of the galaxy samples in the mock catalogs unless it is intentionally introduced through the selection function, as described below.

We use this prescription to populate the simulation with several different galaxy samples. One population, used as the mock photometric catalog, has a relatively low value of M_min = 5 × 10¹¹ M_☉/h (this choice is driven by the smallest identifiable halos in our catalog). These are fairly common galaxies with large-scale bias b_p ≈ 1.6 with respect to the dark matter. Although we know the true value of the redshifts of these objects from the simulations, we do not use any redshift information for the photometric sample when testing the reconstruction algorithm developed here. The other populations we have created are mock spectroscopic samples, with values of M_min = 1 × 10¹² M_☉/h and 7 × 10¹² M_☉/h that generate much rarer mock galaxies with mean number densities of 1.5 × 10⁻³ and 1.2 × 10⁻⁴ h Mpc⁻¹ (a factor of roughly two and 30 fewer galaxies than the photometric sample, before the selection function is applied). The two mock spectroscopic samples have higher biases b_s ≈ 1.8 and b_s ≈ 2.2 with respect to the underlying dark matter. For some tests of the cross-correlation method, we also use a fair subsample of 50% or 80% of the mock photometric population to define the spectroscopic sample. These mock galaxy samples are all somewhat sparser than those available in real galaxy surveys (our choices are driven by the resolution of our simulations). Thus, trends identified in this study, particularly those that render the cross-correlation method less effective due to noisy correlation function measurements of very sparse samples, may be less severe in real observations.

To test the cross-correlation method, we also need to impose both a selection function and an observation window on the photometric sample. We do not impose any selection function on the spectroscopic sample. For convenience, we choose to perform the reconstruction in bins of comoving distance χ_i rather than redshift bins z_i, but the method can be used with either choice of bins. We place the observer in the center of one face of our simulation cube, and assume that (s)he observes a conical volume with a 12 deg opening angle from the center line (24 deg in total). The cone stretches the length of the box (1 Gpc/h) along the line of sight and has a diameter of roughly half of the box at the far end. We adopt a toy model for the selection function, the sum of two Gaussians, which defines the fraction of galaxies "detected" N_keep(χ)/N(χ) at each of 70 slices in comoving distance χ through the box.

$$\begin{eqnarray} \frac{N_{\rm keep}(\chi)}{N(\chi)}& \propto & \frac{1}{\sqrt{2\pi \sigma _1^2}} \, e^{-(\chi -\chi _1)^2/2\sigma _1^2}\nonumber\\ && + \frac{1}{\sqrt{2\pi \sigma _2^2}} \, e^{-(\chi -\chi _2)^2/2\sigma _2^2} \end{eqnarray} \tag{ 12 }$$

We have normalized this quantity so that the maximum value is 1, and we refer to it as the "detection fraction" later in the paper. We present two models for comparison in this paper, [χ₁, σ₁, χ₂, σ₂] = [0, 0.15, 0.8, 0.16] and [0.3, 0.07, 0.7, 0.10]. In the limit of small galaxy numbers in the slice, it is significant that we apply the selection function before the geometry for reasons of cosmic variance. The shape of the final photometric redshift distribution is affected both by the selection function and the geometry of the mock observation. This is illustrated graphically in Figure 1, which shows the first of the two selection functions. Once we have mock photometric and spectroscopic samples in place, we compute in each bin χ_i the autocorrelation function of the spectroscopic sample ξ_ss(r, χ_i), and the angular cross-correlation between the spectro-z objects in the ith bin and the entire photometric sample w_ps(θ, χ_i). We have used the algorithm of Landy & Szalay (1993) to compute the correlation functions. To compute w_ps(θ, χ_i), we opt to measure the 3D correlation function ξ_ps(r, χ_i) and perform the integral of Equation (6) numerically, because the mock observation volume is small which renders direct calculation of w_ps(θ, χ_i) noisy. This will not be a problem for surveys that cover a large fraction of the sky.

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We are eager to test how sensitive the reconstruction of the photometric distribution function ϕ_p(χ) may be to redshift dependence in the bias b_p that is not accounted for. We addressed this question analytically in Section 2, but it is useful to examine the issue with simulations to see if the systematic biases that occur are significant compared to the error bars on the reconstructed distribution. Redshift dependence in the bias can occur because of evolution in the underlying large-scale structure, the clustering properties of galaxies of a given type, and it can occur because the population of photometric objects detected by an instrument changes as a function of redshift, as expected in a magnitude limited sample. Our simulation volume is made of a single time-slice, so we cannot examine the effects of the first two mechanisms at present, but we expect the effects of the second mechanism to dominate the redshift dependence of the bias. We have devised a simple method inspired by the results of Vale & Ostriker (2006) and Conroy & Wechsler (2009) to introduce this type of bias evolution into our mock galaxy sample. Whereas before we applied the selection function in Equation (12) randomly to the galaxies in each slice, now we choose to keep galaxies preferentially that live in the largest halos. We assume that the brightest galaxies live in the biggest halos and that central galaxies are brighter than satellite galaxies at a given redshift. First, we rank order the halos in the slice. We determine the number of galaxies to be kept from Equation (12) and place one in the center of each halo beginning with the halo of highest mass. If the number of galaxies exceeds the number of halos in the slice, we begin placing satellite galaxies. We again start with the largest halo and continue populating each halo with satellites until we have placed all the galaxies. This procedure generates a mock catalog of photometric galaxies whose clustering strength will depend strongly on the selection function. The effect of this procedure on the large-scale bias is somewhat complicated. In the regime where only satellites in the lowest mass halos are being eliminated, if the detection fraction from Equation (12) is high the bias will be close to the bias of the whole population. However as the detection fraction gets lower, the sample will be more highly biased with respect to the dark matter because only satellites in the largest halos are being kept, thus weighting the largest halos more heavily than the smaller halos in the clustering measurement. However, if the detection fraction is so low that all of the satellites are eliminated and the population consists only of centrals, the large-scale bias will be lower than for the whole population, because more weight from the largest halos has been discarded than from the smaller halos. In this analysis we are principally in the second regime, which means the large-scale bias will tend to follow the shape of the selection function.

This is demonstrated in Figure 2, which plots the correlation function measurement of the mock photometric sample in each conical slice. The top panel shows the result if the galaxies are eliminated randomly. In this case the variation among the curves is due to cosmic variance (with more noise in the closest bins because of smaller volume), and the ratio b_p/b_s is independent of redshift. The bottom panel shows the results from the procedure we just described, and the ratio b_p/b_s will vary with redshift by construction. The inset shows the variation in the value of ξ_pp(r) along the line of sight (where we have picked a particular scale, indicated with a vertical line). On the top we show that compared to the last slice (marked with blue squares), the value of ξ_pp(r) varies by some tens of percent, and is a relatively flat function of the line-of-sight distance. The bottom inset on the other hand clearly reflects the shape of the redshift distribution function used to create the sample (plotted later in the bottom panel of Figure 4) and shows variations in the normalization of ξ_pp(r) of up to 150%.

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We emphasize that we do not expect this method to quantitatively capture the redshift dependence of the bias in a real galaxy survey, but we expect it is qualitatively similar, and as long as the bias changes significantly with redshift, it will be useful in testing the magnitude of the effect on the reconstruction. We will refer to this method of applying the selection function as the ordered selection function (OSF) and the simple random method as the random selection function (RSF). By comparing the reconstruction in the two cases, we determine how important it is to account for redshift dependence in the bias.

Once the cross-correlation w_ps(θ, χ_i) and autocorrelation function ξ_ss(r, χ_i) have been measured, a procedure is needed to perform the inversion of the integral of Equation (6) to obtain the line-of-sight distribution of the photometric sample in some bins χ_j and to obtain error bars on those measurements. For a single angular scale θ and bin i in spectroscopic redshift, we approximate the integral in Equation (6) as a discreet sum over N redshift bins (ignoring bias evolution for the moment):

$$\begin{eqnarray} w_{{\rm ps}}(\theta,\chi _i) &\propto& \int _0^\infty \xi _{{\rm ss}}(r(\chi,\chi _i,\theta))\,\phi _p(\chi) \, d\chi \nonumber \\ &\approx& \sum _{j=0}^{N-1} X_{ij} \phi _{p,j} \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} &X_{ij}=\frac{b_p}{b_s}\xi _{{\rm ss}}\big(\chi _i,r=\sqrt{(\theta \chi _i)^2+(\chi _j-\chi _i)^2}\big). \end{eqnarray} \tag{ 14 }$$

The observation will be performed in multiple angular bins θ. The values of r in X_ij will not be at the exact positions where we have made the measurement of ξ_ss (in our case in 60 bins between 5 and 50 comoving Mpc/h). If the r falls within the domain of our data, we interpolate ξ_ss with a spline, if it is larger than the biggest scale we measure, the value of ξ_ss is extrapolated using a fit of the form r⁻² to a subset of data points (larger than 15 Mpc/h) measured in the volume. If the value of r is less than the minimum value of r measured, we reject the entire θ bin, since we do not trust the method for ξ_ss measured on very small scales. At a distance of 250 Mpc/h (for example), this results in rejecting θ values less than ∼1 deg.

Collecting all the observed w_ps values in a single vector w (one element for each unique combination of χ_i and θ), the solution for ϕ_p(χ_j) will be obtained by inverting relation:

$$\begin{eqnarray} &&{\bm w} \propto {\bf X} \cdot {\bm \phi }. \end{eqnarray} \tag{ 15 }$$

Here, X is a matrix whose elements are given by Equation (14) and whose dimension is

$$\begin{eqnarray} &&{\rm (\# spec \, bins\,} i * {\rm \# theta\,bins)} \times N. \nonumber \end{eqnarray}$$

ϕ will be a vector of length N, and w will be a vector of length (# spec bins i* # theta bins). It is significant that measurements at different values of θ can mix in the inversion; since they are correlated it is important that they do so. We do not know the pre-factor b_p/b_s, but as long as it is assumed to be independent of redshift (as in the special case where the spectroscopic sample is a fair subsample of the photometric population) this is not relevant. The reconstructed ϕ_p(χ_j) will not be correctly normalized after the matrix inversion because there is no information in Equation (13) about the total number of galaxies in the photometric sample. The normalization of ϕ_p(χ_j) will be set by requiring that it integrate to 1. Later, we will discuss the consequences of violating the assumption that b_p/b_s is independent of redshift.

In principle, solving Equation (15) for ϕ_p(χ_j) is a simple matter of inverting a matrix, but in practice doing so is numerically unstable and furthermore errors in the observables w and X must be accounted for in the solution. Since w is a projection through the same dark matter structures traced by X, there will be non-negligible correlations between the errors in the two observables. To extract ϕ_p(χ), we write down the expression for the χ² statistic (the first term of Equation (16)), and minimize it. Typically, we are attempting to reconstruct ϕ_p(χ_j) in as many bins N as possible. There is often insufficient information in the correlation functions to completely constrain ϕ_p in the chosen number of bins, thus it becomes necessary to stabilize against solutions that have highly anti-correlated adjacent bins. Since it is reasonable to assume that the true distribution will not be highly oscillatory, we adopt a smoothness prior to regularize the solution and add it to our expression for χ² below:

$$\begin{eqnarray} &&\chi ^2=({\bm w}-{\bf X}{\bm \phi })^T {\bf C}^{-1}_{\bm \phi }({\bm w}-{\bf X}{\bm \phi }) + \lambda \,{\bm \phi }^T {\bm B}^T {\bm B} {\bm \phi }. \end{eqnarray} \tag{ 16 }$$

Here, C_ϕ is the covariance matrix incorporating errors in both w and X, and the subscript denotes that it is an explicit function of ϕ. For purposes of the present analysis, however, we will assume that the spectroscopic correlation function is perfectly determined, and we will propagate errors from w_ps only, specifically C⁻¹_ϕ = C⁻¹ = inverse covariance matrix of w_ps(θ, χ_i). This is not a good approximation as we will soon show (in fact it is unlikely that ξ_ss will ever be better determined than w_ps for realistic scenario), and we refer the reader to the Appendix for a description of the full error propagation from C⁻¹_ϕ into errors on ϕ_p(χ_j). The reader should consider all errors reported on ϕ_p(χ_j) as lower limits on the total error.

The regularization scheme we have adopted is described in detail in Press (2002, Section 18.5). The intention is to add a term to χ² that gets large when neighboring points have widely different values. Minimization of χ² will then tend to solutions that do not have anti-correlated neighboring points. The matrix B is given by

$$\begin{eqnarray} {\bm B}=\left({\begin{array}{@{}ccccccccc@{}} -1 & 1 & 0 & 0 & 0 & 0 & 0 & \cdots & 0 \\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & \cdots & 0 \\ \vdots & & & & \ddots & & & & \vdots \\ 0 &\cdots & 0 & 0 & 0 & 0 & -1 & 1 & 0 \\ 0 & \cdots & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\ \end{array} } \right).\qquad \end{eqnarray} \tag{ 17 }$$

Note that B has one fewer row than column. The factor λ should be chosen such that the first and second terms in χ² contribute roughly equal weight. This can be approximately arranged if λ is taken to be

$$\begin{eqnarray} &&\lambda =\frac{{\rm Tr}({\bm X}^T {\bm C}^{-1}{\bm X})}{{\rm Tr}({\bm B}^T{\bm B})}, \end{eqnarray} \tag{ 18 }$$

but in practice the weight in χ² will also depend on the solution and the values of w_ps. We find that the quality of the reconstruction depends on how well the two terms in Equation (16) are balanced. Since this depends on the answer, we opt to refine the value of lambda and recompute the solution iteratively as follows.

• 1.  
  Compute a tolerance parameter defined from the two terms in χ² as tol = 1–(term 1/term 2).
• 2.  
  If tol >0, λ is increased by a factor of 10. If tol <0, we decrease it by 10.
• 3.  
  We recompute the solution and the tol.
• 4.  
  If the absolute value of tol has decreased, we repeat the refinement of lambda, otherwise we exit and keep the previous value of the solution.

In practice, the procedure usually requires only one refinement of λ. We observe that changing the algorithm to have smaller steps in lambda (e.g., a factor of two rather than 10) does not improve the solution and occasionally oversmooths it. We emphasize that while we have identified an algorithm that works, we have not optimized the application of a smoothness prior. Since the solution is moderately sensitive to the smoothing, care should be taken to understand the properties of the smoothing before applying this technique to real data. Imposing a smoothness prior may be reasonable for a magnitude limited photometric sample, but may be problematic when attempting to calibrate a color-selected sample, which can exhibit jumps in the redshift distribution as features of the galaxy spectra pass in and out of the photometric bands. We have not studied the effects of different smoothing algorithms because it is likely that the need for a smoothness prior will be mitigated in future analyses by using the photometric probability distributions as an additional prior. Smooth portions of the photo-z prior will similarly penalize oscillatory solutions as the smoothness prior we have used in this work, while discontinuous behavior in the photometric priors reflects actual photo-z degeneracies and will guide the reconstruction toward solutions that resolve multi-modal behavior. Since we have no mock photometric redshift probabilities, examining that technique is beyond the scope of this paper, but will be the subject of further research.

To minimize χ² we take the derivative with respect to ϕ and set it equal to zero. Taking C⁻¹ to be independent of ϕ and noting that it is symmetric, we find

$$\begin{eqnarray} &&-\!2{\bm w}^T {\bm C}^{-1} {\bm X} + 2 {\bm \phi }^T {\bm X}^T {\bm C}^{-1} {\bm X} +2\lambda {\bm \phi }^T{\bm B}^T{\bm B} = 0 \hspace{0.5cm} \end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray} &&{\bm X}^T {\bm C}^{-1}{\bm w} = [{\bm X}^T {\bm C}^{-1} {\bm X} +\lambda {\bm B}^T{\bm B}]{\bm \phi } \end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray} &&{\bm \phi }= [{\bm X}^T {\bm C}^{-1} {\bm X} +\lambda {\bm B}^T{\bm B}]^{-1} {\bm X}^T {\bm C}^{-1}{\bm w}. \end{eqnarray} \tag{ 21 }$$

The covariance matrix of the recovered ϕ is given by

$$\begin{eqnarray} &&{\rm Cov}[\bm \phi ] \propto [{\bm X}^T {\bm C}^{-1} {\bm X}+\lambda {\bm B}^T{\bm B}]^{-1}. \end{eqnarray} \tag{ 22 }$$

To recover the constant of proportionality, we will need to rescale the elements of the covariance matrix by the square of the factor used to rescale ϕ (since we renormalize such that ϕ(χ) integrates to 1). We caution that all matrix multiplications above are finite sum approximations to integral quantities, so care must be taken to ensure that phi and its error bars come out to scale. For clarity, let us refer to w_ps(z_i) as a function of a continuous variable z_i (which will label the spectroscopic bins). ξ(z_i, χ_i) will be a function of both z_i and another continuous variable χ_i (which will label the bins in which ϕ is reconstructed). We are considering a single value of θ, and z and χ can represent either redshifts or comoving distances, we simply use both letters to distinguish them. Ignoring regularization, the continuous expression for χ² is then

$$\begin{eqnarray} &&\chi ^2=\int dz_1 dz_2\left[ \left(w_{{\rm ps}}(z_1) - \int d\chi _1 \phi (\chi _1) \xi (z_1,\chi _1) \right) \right. \nonumber \\ &&\left. C^{-1}(z_1,z_2) \left(w_{{\rm ps}}(z_2) - \int d\chi _2 \phi (\chi _2) \xi (z_2,\chi _2) \right) \right]. \hspace{0.2cm} \end{eqnarray} \tag{ 23 }$$

To minimize χ² we must set the functional derivative δχ²/δϕ(χ) = 0. We leave the details to the reader, but note that in Equation (19), the factors of Δz that correspond to integrals over z cancel but the factor of Δχ does not.

In this section, we present a series of reconstructions that examine various aspects of the problem and identify the potential difficulties in applying this method. We begin in Figure 3 with the simplest case and gradually add complexity. The heavy solid line shows the theoretical redshift distribution (selection function + geometry) that has been applied mock photometric catalog of galaxies with M_min = 5 × 10¹¹ in Equation (11). This redshift distribution corresponds to the selection function shown on the left of Figure 1. We have chosen galaxies randomly (the RSF method described in Section 3.1) in applying the selection function. The resulting catalog has 54,000 galaxies. We have measured the detection fraction in each slice for this particular realization and plot it with a thin wavy line in Figure 3. The spectroscopic calibrating sample is taken to be a different population of galaxies with M_min = 7 × 10¹² yielding around 5600 galaxies, i.e., a highly biased tracer of the density field compared to the photometric sample. We assume that both the observables are measured with perfect accuracy. Since we have assumed perfect knowledge of the spectroscopic correlation function, we opt to measure it in the entire light cone and use the result in each of the i spectroscopic bins ξ_ss(r, χ_i) = ξ_{whole volume}(r). This is only justified because there is no evolution in the light cone, and the calibrators are not affected by the selection function which changes along the line of sight. We perform the reconstruction by computing the matrices X, w, C⁻¹, and B and using them in Equation (21). The result is plotted in Figure 3 with points. Notice that the reconstruction follows the values of this realization (the actual redshift distribution in this cone) rather than the theoretical redshift distribution used to create the sample.

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Figure 4 reveals a more realistic picture. The lines and points in Figure 4 are all the same as in Figure 3. We now measure the autocorrelation of the spectroscopic sample ξ_ss(r, χ_i) in each of the conic sections and use those measurements to perform the reconstruction. The top and bottom panels show two different choices of selection function, whose parameters are marked in the caption. The top is difficult to reconstruct because it peaks at χ = 0 where there is no volume in the mock observation, and the bottom is a challenge because the feature coincides with the bin spacing, which will be a problem to reconstruct because of the smoothness prior. We have drastically decreased the number of bins, so that there are a reasonable number of calibrators in most of the bins (a few hundred to ${\cal O}(1000)$). The normalization criterion, which comes from the condition that ϕ_p integrate to 1, becomes more sensitive to noise in the reconstruction when the number of bins is decreased. In this plot and the plots that follow, we set the normalization by hand so that we can illustrate other points; we let the maximum value of the reconstruction (points) equal the maximum value of the theoretical input ϕ (thick solid line). While the reconstructions in Figure 4 show the correct trends, they are not sufficiently high quality to recover the bimodal behavior in the reconstructed selection function.

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We now briefly discuss the error bars in Figure 4 obtained from Equation (22). The Cov[ϕ] matrix is not diagonal; we conservatively report $1/\sqrt{\vphantom{A^A}\smash{\hbox{${{\rm Cov}^{-1}[\phi ]_{i,i}}$}}}$ as the error on the ith bin. These error bars represent a lower bound on the error because we have only propagated error from w_ps and not from ξ_ss. The matrix C⁻¹ in Equation (22) is the inverse covariance matrix of the cross-correlation measurement w, and with sufficient volume can be estimated by dividing the observation into a number of bins and bootstrapping. Here, however, we opt to follow Peebles (1980) and assume that on these scales the errors are dominated by Poisson noise. In this case, C⁻¹ is diagonal and its elements are given by

$$\begin{eqnarray} &&{\bm C}_{ii}=\frac{1}{\sqrt{n_{\rm pairs}}}=\bar{n}_s \bar{\Sigma }_p \chi _s^2 \Delta \chi _s \, \Omega \sin (\theta)\Delta \theta. \end{eqnarray} \tag{ 24 }$$

We use survey parameters appropriate to large upcoming missions. We take $\bar{n}_s=1\times 10^5$ galaxies per (Gpc/h)³, $\bar{\Sigma }_p=100$ photometric galaxies per square arcmin, and Ω = 20, 000 deg² on the sky. Δχ_s is the width of the bin of spectroscopic data. The resulting error bars in Figure 4 and those that follow display a surprising trend. Even though the number of pairs is dramatically larger for bins at farther comoving distance, the reconstruction of the photometric distribution is less accurate in the most distant bins. The key to understanding this trend is that for a fixed angular scale θ, the physical scale being probed in distant bins is much larger than at nearby distances. Since the correlation function is a steeply dropping function of separation, the impact of shrinking values in X of Equation (22) dominates over the growing number of pairs in C⁻¹. By measuring w_ps on smaller angular scales the errors in the farthest bins can be improved, but there is a limit to how far this can be pushed because measurements at small angular scales will send the near field into the trans-linear and one-halo regime, for which there are corrections to the underlying assumption that ξ_ps ∝ ξ_ss.

In Figure 5 we switch to a larger calibration sample with M_min = 1 × 10¹², which allows a much more accurate measurement of the spectroscopic correlation function. We see that the reconstruction now captures the bimodal behavior, but both the resolution (bin spacing) and errors are modest and the smoothing is still evident in the last bin. Notice that the error bars have increased; this is because this calibration sample has lower bias, so the elements in X in the covariance matrix of ϕ(χ) (Equation (22)) are all smaller. This is, however, an illusion. Had we propagated the error from ξ_ss, it would contribute larger errors to Figure 4 than to Figure 5 because the measurement is noisier for the smaller sample. In moving to the larger sample, the number of spectra required has increased by an order of magnitude and is now the same size as the mock photometric catalog (after the selection function has been applied). In summary, comparison of Figures 3, 4, and 5 demonstrates that ξ_ss(χ) must be reasonably well determined for the shape of the reconstructed photometric distribution to be well captured. It is worth noting that we have used no priors in the determination of ξ_ss(χ), improving and applying theoretical priors may yield significantly better results with fewer spectra.

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The fact that the more numerous calibrators generated such improved results suggested that the noisiness of the spectroscopic correlation function is very likely driving the poorness of the reconstruction. One alternative way to avoid noisiness in the calibrating sample might be to assume a template for the spectroscopic correlation function and fit it to the noisy data. We attempted to improve the reconstruction with the rarer calibrators by simply fitting the spectroscopic observations with a two-parameter power law, and performing the reconstruction using the fin instead of interpolating the measured data. This alternate procedure appears to discard too much information contained in ξ_ss(χ). The result was visually similar to Figure 4, namely, the bimodal behavior in the distribution is lost.

We add another layer of complexity in Figure 6. We apply our selection function in such a way that the bias of the remaining objects will exhibit redshift dependence (the OSF method described in Section 3.1). We continue using the less rare calibrating sample with halo model M_min = 1 × 10¹² for this illustrative proof of concept. Although it is not very statistically significant, the eye can pick out a trend in the reconstruction; the reconstructed distribution is suppressed in areas of low detection fraction. Since the bias is tracking the detection fraction, the areas of low detection are enhanced less than the areas of high detection, thus they appear suppressed when we normalize to the highest point. At this level of accuracy, it is unlikely that this systematic will dominate the error in tomographic analyses, however for much larger surveys it could be a significant concern.

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In Figure 7, we show how the situation is altered if a fair subsample of the photometric population is used instead of a rarer biased tracer population. This is a special case because the bias of the calibrators will have identical redshift dependence as the photometric sample (the ratio b_p/b_s = 1). We show two different subsamples in this plot, 50% and 80% of the photometric catalog, and test the reconstruction for the distribution on the right in Figures 4, 5, and 6. We see that for a survey of this volume, 50% is too small to capture the bimodal behavior in the reconstruction, but with 80%, the reconstruction works well. Indeed the systematic offset in Figure 6 is remedied and the reconstruction follows the realization to much better accuracy. Of course spectroscopic follow-up of such a large fraction of the sample is more than sufficient to calibrate the redshift distribution without using the cross-correlation technique (black stars). For larger surveys the fraction needed will be smaller than indicated here, because they will be able to measure the spectroscopic correlation functions with greater accuracy. For surveys large enough not to be limited by noisy spectroscopic correlation functions, it may be necessary to calibrate with a fair subsample of galaxies to avoid systematic bias in the redshift distribution that comes from redshift dependence in the bias of the photometric sample.

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