MAPPING THE GALAXY COLOR–REDSHIFT RELATION: OPTIMAL PHOTOMETRIC REDSHIFT CALIBRATION STRATEGIES FOR COSMOLOGY SURVEYS

Upcoming large-scale surveys such as LSST, Euclid, and WFIRST will measure the three-dimensional cosmological weak lensing shear field from broadband imaging of billions of galaxies. Weak lensing is widely considered to be one of the most promising probes of the growth of dark matter structure, as it is sensitive to gravitation alone and requires minimal assumptions about the coupling of dark matter and baryons (Bartelmann & Schneider 2001; Weinberg et al. 2013). Moreover, weak lensing tomography is sensitive to the dark energy equation of state through its impact on the growth of structure with time (Hu & Tegmark 1999). However, it is observationally demanding: in addition to requiring accurately measured shapes for the weak lensing sample, robust redshift estimates to the galaxies are needed in order to reconstruct the three-dimensional matter distribution. Because it is infeasible to obtain spectroscopic redshifts (spec-z's) for the huge numbers of faint galaxies these studies will detect, photometric redshift (photo-z) estimates derived from imaging in some number of broad filters will be required for nearly all galaxies in the weak lensing samples.

Photo-z estimation has become an indispensable tool in extragalactic astronomy, as the pace of galaxy detection in imaging surveys far outstrips the rate at which follow-up spectroscopy can be performed. While photo-z techniques have grown in sophistication in recent years, the requirements for cosmology present novel challenges. In particular, cosmological parameters derived from weak lensing are sensitive to small, systematic errors in the photo-z estimates (Huterer et al. 2006; Ma et al. 2006). Such biases are generally much smaller than the random scatter in photo-z estimates (Dahlen et al. 2013), and are of little consequence for galaxy evolution studies; however, they can easily dominate all other uncertainties in weak lensing experiments (Newman et al. 2015). In addition to weak lensing cosmology, accurate and well-characterized photo-z's will be crucial to other cosmological experiments. For example, baryon acoustic oscillation (BAO) experiments that rely on redshifts measured from faint near-infrared grism spectra will often have to resort to photo-z's in order to determine the correct redshift assignment for galaxies with only a single detected line. Well-characterized photo-z estimates will be needed to correctly account for any errors thus introduced.

There are two key requirements placed on the photo-z estimates for weak lensing cosmology. First, redshift estimates for individual objects must have sufficient precision to correct for intrinsic galaxy shape alignments as well as other potential systematics arising from physically associated galaxies that may affect the interpretation of the shear signal. While not trivial, meeting the requirement on the precision of individual photo-z estimates (${\sigma }_{z}\lt 0.05(1+z)$ for Euclid, Laureijs et al. 2011) should be achievable (Hildebrandt et al. 2010). The second, more difficult, requirement is that the overall redshift distributions N(z) of galaxies in ∼10–20 tomographic bins used for the shear analysis must be known with high accuracy. Specifically, the mean redshift $\langle z\rangle$ of the N(z) distribution must be constrained to better than 2 × 10⁻³(1 + z) in order to interpret the amplitude of the lensing signal and achieve acceptable error levels on the cosmological parameter estimates (Huterer et al. 2006; Amara & Réfrégier 2007; Laureijs et al. 2011). Small biases in the photo-z estimates, or a relatively small number of objects with catastrophically incorrect photo-z's, can cause unacceptably large errors in the estimated N(z) distribution. Photo-z estimates alone are not sufficient to meet this requirement, and spectroscopic calibration samples will be needed to ensure low bias in the N(z) estimates. The significant difficulties associated with this requirement are summarized by Newman et al. (2015).

The most straightforward approach to constrain N(z) is to measure it directly by random spectroscopic sampling of galaxies in each tomographic redshift bin (Abdalla et al. 2008). The total number of spectra needed to meet the requirement is then set by the central limit theorem. For upcoming "Stage IV" cosmology surveys (LSST, Euclid,¹⁷ and WFIRST) it is estimated that direct measurement of N(z) for the tomographic bins would require total spectroscopic samples of ∼30,000–100,000 galaxies, fully representative in flux, color, and spatial distribution of the galaxies used to measure the weak lensing shear field (e.g., Ma & Bernstein 2008, Hearin et al. 2012). Moreover, the spectroscopic redshifts would need to have a very high success rate (≳99.5%), with no subpopulation of galaxies systematically missed in the redshift survey. Newman et al. (2015) note that current deep redshift surveys fail to obtain secure redshifts for ∼30%–60% of the targeted galaxies; given the depths of the planned dark energy surveys, this "direct" method of calibrating the redshifts seems to be unfeasible.

Because of the difficulty of direct spectroscopic calibration, Newman et al. (2015) argue that the most realistic method of meeting the requirements on N(z) for the dark energy experiments may be some form of spatial cross-correlation of photometric samples with a reference spectroscopic sample, with the idea that the power in the cross-correlation will be highest when the samples match in redshift (Newman 2008; Schmidt et al. 2013; Rahman et al. 2015). This approach shows significant promise, but is not without uncertainties and potential systematics. For example, it requires assumptions regarding the growth of structure and galaxy bias with redshift, which may be covariant with the cosmological inferences drawn from the weak lensing analysis itself. Further work may clarify these issues and show that the technique is indeed viable for upcoming cosmological surveys. However, it seems safe to say that this method cannot solely be relied on for the weak lensing missions, particularly as at least two approaches will be needed: one to calibrate N(z) for the tomographic bins, and another to test and validate the calibration.

In light of these arguments, it is clear that targeted spectroscopic training and calibration samples will have to be obtained to achieve the accuracy in the $\langle z\rangle$ estimates of tomographic bins required by the weak lensing missions. Moreover, careful optimization of these efforts will be required to make the problem tractable. Here we present a technique, based on the simple but powerful self-organizing map (SOM) (Kohonen 1982, 1990), to map the empirical distribution of galaxies in the multidimensional color space defined by a photometric survey. Importantly, this technique provides us with a completely data-driven understanding of what constitutes a representative photometric galaxy sample. We can thereby evaluate whether a spectroscopic sample used for training and calibration spans the full photometric parameter space; if it does not, there will be regions where the photo-z results are untested and untrained. Machine learning—based photo-z algorithms, in particular, depend critically on representative spectroscopic training sets, and their performance will be degraded in regions of color space without spectroscopic coverage¹⁸ (Collister & Lahav 2004; Hoyle et al. 2015).

We show that the empirical color mapping described here can be used to optimize the training and calibration effort by focusing spectroscopic effort on regions of galaxy parameter space that are currently poorly explored, as well as regions with a less certain mapping to redshift. Alternatively, we can use the technique to identify and discard specific regions of color space for which spectroscopy will prove to be too expensive, or for which the redshift uncertainty is too large. In effect, the method lets us systematize our understanding of the mapping from color to redshift. By doing so, the number of spectroscopic redshifts needed to calibrate N(z) for the weak lensing tomographic bins can be minimized. This approach will also naturally produce a "gold standard" training sample for machine learning algorithms.

The technique we adopt also provides insight into the nature of catastrophic photo-z failures by illustrating regions of color space in which the mapping between color and redshift becomes degenerate. This is possible because the self-organized map is topological, with nearby regions representing similar objects, and widely separated regions representing dissimilar ones. In addition, template-fitting photo-z codes can potentially be refined with the map, particularly through the development of data-based priors and by using the empirical color mapping to test and refine the galaxy template sets used for fitting.

Here our focus is on the Euclid survey, one of the three Stage IV dark energy surveys planned for the next decade, the other two being LSST and WFIRST. Euclid will consist of a 1.2 meter space telescope operating at L2, which will be used to measure accurate shapes of galaxies out to z ∼ 2 over ∼15,000 deg² with a single, broad (riz) filter. These observations will reach an AB magnitude of ≃24.5 (10σ). In addition to these observations, a near-infrared camera on Euclid will obtain Y, J, and H band photometry to an AB magnitude ≃24 (5σ), which, together with complementary ground-based optical data, will be used for photo-z determination. The mission will also constrain cosmological parameters using BAO and redshift space distortions (RSD), using redshifts obtained with a low-resolution grism on the near-infrared camera. A more detailed description of the survey can be found in Laureijs et al. (2011).

For this work, we assume that Euclid will obtain ugrizYJH photometry for photo-z estimation. We select galaxies from the COSMOS survey (Scoville et al. 2007) that closely approximate the Euclid weak lensing sample, with photometry in similar bands and at similar depths as the planned Euclid survey. While our focus is on Euclid, the method we present is general and directly applicable to other weak lensing surveys facing the same calibration problem.

This paper is organized as follows. In Section 2 we give an overview of the methodology used to map the galaxy multicolor space. In Section 3 we discuss the galaxy sample from the COSMOS survey used to approximate the anticipated Euclid weak lensing sample. In Section 4 we describe the self-organizing map algorithm and its implementation for this application. In Section 5 we discuss the map in detail, including what it reveals about the current extent of spectroscopic coverage in galaxy multicolor space. In Section 6 we address the problem of determining the spectroscopic sample needed to meet the weak lensing requirement, and in Section 7 we conclude with a discussion.

Galaxies with imaging in a set of N filters will follow some distribution in the multidimensional space (of dimension N − 1) defined by the unique colors measured by the filters. These colors together determine the shape of the low-resolution spectral energy distribution (SED) measured by the filters. Henceforth, we will call the position a galaxy occupies in color space simply its color, or ${\boldsymbol{C}}.$ For example, the Euclid survey is expected to have eight bands of photometry (ugrizYJH),¹⁹ and therefore a galaxy's position in color space is uniquely determined by seven colors: $u-g,g-r,\;...,J-H.$ Galaxy color is the primary driver of photometric redshift estimates: template-based methods predict ${\boldsymbol{C}}$ for different template/redshift/reddening combinations and assign redshifts to galaxies based on where the models best fit the observed photometry, while machine learning methods assume the existence of a mapping from ${\boldsymbol{C}}$ to redshift, and attempt to discover it using spectroscopic training samples.

Our goal here is to empirically map the distribution of galaxies in the color space defined by the anticipated Euclid broadband filters. We refer to this distribution as $\rho ({\boldsymbol{C}}).$ Once we understand how galaxies are distributed in color space, optimal methods of sampling the distribution with spectroscopy can be developed to make an informed calibration of the color–redshift relation.

The general problem of mapping a high-dimensional data distribution arises in many fields. Because the volume of the data space grows exponentially with the number of dimensions, data rapidly become sparse as the dimensionality increases. This effect—the so-called "curse of dimensionality" (Bellman 1957)—makes normal data sorting strategies impractical. A number of algorithms, collectively referred to as nonlinear dimensionality reduction (NLDR), have been developed to address this problem by projecting high-dimensional data onto a lower-dimensional representation, thus facilitating visualization and analysis of relationships that exist in the data.

We adopt the SOM algorithm, described in more detail in Section 4. As emphasized by Geach (2012), self-organized mapping is a powerful, empirical method to understand the multidimensional distributions common in modern astronomical surveys. Two primary motivations for choosing this technique over others are the relative simplicity of the algorithm and the highly visual nature of the resulting map, which facilitates human understanding of the data.

We use multiwaveband data from the COSMOS survey (Capak et al. 2007) to provide a close approximation to the expected Euclid weak lensing data. Photo-z estimates for the Euclid sample will rely on three near-infrared filters on the telescope (YJH), reaching an AB depth of 24 mag (5σ) for point sources, as well as complementary ground-based imaging in the optical, which we assume will consist of ugriz imaging with LSST (in the northern sky the ground-based imaging data may be restricted to griz, affecting the analysis somewhat but not changing the overall conclusions).

To provide a close analog to the expected Euclid data, we use COSMOS u band imaging from CFHT, griz imaging from Subaru Suprime Cam, and YJH imaging from the UltraVista survey (McCracken et al. 2012), spanning a 1.44 deg² patch of COSMOS with highly uniform depth. We apply a flux cut to the average flux measured across the Subaru r, i, and z bands to match the expected depth limit of the single, broad visible filter Euclid will use for the weak lensing shear measurement. The resulting "Euclid analog" sample consists of 131,609 objects from COSMOS.

The SOM (Kohonen 1982, 1990) is a neural network model widely used to map and identify correlations in high-dimensional data. Its use for some astronomical applications has been explored previously (see, e.g., Naim et al. 1997; Brett et al. 2004; Way & Klose 2012; Fustes et al. 2013; Carrasco Kind & Brunner 2014; Greisel et al. 2015). The algorithm uses unsupervised, competitive learning of "neurons" to project high-dimensional data onto a lower-dimensional grid. The SOM algorithm can be thought of as a type of nonlinear principal component analysis, and is also similar in some respects to the k-means clustering algorithm (MacQueen 1967). In contrast to these and other methods, the SOM preserves the topology of the high-dimensional data in the low-dimension representation. Similar objects are thus grouped together on the self-organized map, and clusters that exist in the high-dimensional data space are reflected in the lower-dimensional representation. This feature makes the maps visually understandable and thus useful for identifying correlations that exist in high-dimensional data. More detailed descriptions of the algorithm and its variants can be found in a number of references (see, e.g., Vesanto 2002; Carrasco Kind & Brunner 2014).

The SOM consists of a fixed number of cells arranged on a grid. The grid can be of arbitrary dimension, although two-dimensional grids are most common as they are the easiest to visualize. Each cell in the grid is assigned a weight vector ${\boldsymbol{w}}$ having the same number of dimensions as the training data. This vector can be thought of as pointing to a particular region of the multidimensional parameter space occupied by the data. The weight vectors are initialized prior to training, either randomly or by sampling from the input data. The training of the map is unsupervised, in the sense that the output variable of interest (here, redshift) is not considered. Only the input attributes (galaxy photometry) drive the training. We note that any measured galaxy property (size, magnitude, shape, environment, surface brightness, etc.) could be used in the training. We consider only colors here, as these are the primary drivers of the photo-z estimates, and the quantities most physically tied to redshift. The other properties mentioned can still be used after the map has been created to identify and help break redshift degeneracies within particular regions of galaxy color space.

Training proceeds by presenting the map with a random galaxy from the training sample, which the cells "compete" for. The cell whose weight vector most closely resembles the training galaxy is considered the winner, and is called the Best Matching Unit, or BMU. The BMU as well as cells in its neighborhood on the map are then modified to more closely resemble the training galaxy. This pattern is repeated for many training iterations, over which the responsiveness of the map to new data gradually decreases, through what is known as the learning rate function. Additionally, the extent of the neighborhood around the BMU affected by new training data shrinks with iteration number as well, through what is known as the neighborhood function. These effects cause the map to settle to a stable solution by the end of the training iterations.

To compute the winning cell for a given training object, a distance metric must be chosen. Most often, the Euclidean distance between the training object ${\boldsymbol{x}}$ and the cell weight vector ${{\boldsymbol{w}}}_{k}$ is used. With data of dimension m, this distance is given by:

$$\begin{eqnarray}&&{d}_{k}^{2}={d}_{k}^{2}({\boldsymbol{x}},{{\boldsymbol{w}}}_{k})=\displaystyle \sum _{i=1}^{m}{({x}_{i}-{w}_{k,i})}^{2}.\end{eqnarray} \tag{ 1 }$$

However, dimensions with intrinsically larger error than others will be overweighted in this distance metric. To account for this, we instead use the reduced χ² distance between the training object and the cell weight vector. With ${\sigma }_{{x}_{i}}$ representing the uncertainty in the ith component of ${\boldsymbol{x}},$ this becomes

$$\begin{eqnarray}&&{d}_{k}^{2}={d}_{k}^{2}({\boldsymbol{x}},{{\boldsymbol{w}}}_{k})=\displaystyle \frac{1}{m}\displaystyle \sum _{i=1}^{m}\displaystyle \frac{{({x}_{i}-{w}_{k,i})}^{2}}{{\sigma }_{{x}_{i}}^{2}}.\end{eqnarray} \tag{ 2 }$$

The BMU is the cell minimizing the χ² distance. Once the BMU has been identified, the weight vectors of cells in the map are updated with the relation

$$\begin{eqnarray}&&{{\boldsymbol{w}}}_{k}(t+1)={{\boldsymbol{w}}}_{k}(t)+a(t){H}_{b,k}(t)[{\boldsymbol{x}}(t)-{{\boldsymbol{w}}}_{k}(t)].\end{eqnarray} \tag{ 3 }$$

Here t represents the current timestep in the training. The learning rate function a(t) is a monotonically decreasing function of the timestep (with a(t) ≤ 1), such that the SOM becomes progressively less responsive to new training data. With N_iter representing the total number of training iterations, we adopt the following functional form for a(t):

$$\begin{eqnarray}&&a(t)={0.5}^{(t/{N}_{\mathrm{iter}})}.\end{eqnarray} \tag{ 4 }$$

The term H_b,k(t) is the value of the neighborhood function at the current timestep for cell k, given that the current BMU is cell b. This function is encoded as a normalized Gaussian kernel centered on the BMU:

$$\begin{eqnarray}&&{H}_{b,k}(t)={e}^{-{D}_{b,k}^{2}/{\sigma }^{2}(t)}.\end{eqnarray} \tag{ 5 }$$

Here D_b,k is the Euclidean distance on the map separating the kth cell and the current BMU. The width of the Gaussian neighborhood function is set by σ(t) and is given by

$$\begin{eqnarray}&&\sigma (t)={\sigma }_{s}{(1/{\sigma }_{s})}^{(t/{N}_{\mathrm{iter}})}.\end{eqnarray} \tag{ 6 }$$

The starting value, σ_s, is large enough that the neighborhood function initially encompasses most of the map. In practice, we set σ_s equal to the the size (in pixels) of the smaller dimension of the rectangular map. The width of the neighborhood function shrinks by the end of training such that only the BMU and cells directly adjacent to it are significantly affected by new data.

4.1. Optimizing the Map for the Photo-z Problem

4.2. Algorithm Implementation

4.3. Assessing Map Quality

Ideally, the SOM should be highly representative of the data, in the sense that the SEDs of most galaxies in the sample are well-approximated by some cell in the map. To assess the representativeness of the map we calculate what is known as the average quantization error over the entire training sample of N objects:

$$\begin{eqnarray}&&{\epsilon }_{q}=\displaystyle \frac{1}{N}\displaystyle \sum _{i=1}^{N}| | {{\boldsymbol{x}}}_{i}-{{\boldsymbol{b}}}_{i}| | .\end{eqnarray} \tag{ 7 }$$

Here ${{\boldsymbol{b}}}_{i}$ is the best matching cell for the ith training object. We find that the average quantization error is 0.2 for the sample. The quantization error is the average vector distance between an object and its best-matching cell in the map.²⁰ Therefore, with seven colors used to generate the map, the average offset of a particular color (e.g., g − r) of a given galaxy from its corresponding cell in the map is $0.2/\sqrt{7}=0.08\;\mathrm{mag}$. Note that the map provides a straightforward way of identifying unusual or anomalous sources. Such objects will be poorly represented by the map due to their rarity—in effect, they are unable to train their properties into the SOM. Simply checking whether an object is well represented by some cell in the map is therefore a way of testing whether it is "normal," and may be useful for flagging, for example, blended objects, contaminated photometry, or truly rare sources.

Figure 1 provides an overview of the SOM generated from COSMOS galaxies, which encodes the 8-band SEDs that appear in the data with non-negligible frequency. Note that the final structure of the map is to some extent random and depends on the initial conditions combined with the order in which training objects are presented, but the overall topological structure will be similar from run to run; this was verified by generating and comparing a number of maps.²¹ Figure 2 illustrates the variation of two colors (u − g and g − r) across the map, demonstrating how these features help drive the overall structure. In the following analysis we probe the map by analyzing the characteristics of the galaxies that associate best with each cell in color space.

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5.1. The Distribution of Galaxies in Color Space, $\rho ({\boldsymbol{C}})$

In Figure 3 we show the self-organized map colored by the number of galaxies associating best with each cell. This coloration is effectively our estimate of $\rho ({\boldsymbol{C}}),$ the density of galaxies as a function of position in color space. An important caveat is that the density estimate derived from the COSMOS survey data is likely to be affected to some degree by cosmic variance (and perhaps, to a lesser extent, by shot noise). The true $\rho ({\boldsymbol{C}})$ can ultimately be constrained firmly with the wide-area survey data from LSST, Euclid, and WFIRST. However, the COSMOS-based $\rho ({\boldsymbol{C}})$ should be a close approximation of what the full surveys will find.

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5.2. Photometric Redshift Estimates across the Map

Because the cells in the SOM represent galaxy SEDs that appear in the data, we can compute photometric redshifts for them to see how they are distributed in redshift. We used the Le Phare template fitting code (Arnouts et al. 1999; Ilbert et al. 2006) to compute cell photo-z's. We used the cell weight vectors (converting the colors to photometric magnitudes normalized in i-band) as inputs for Le Phare, assigning realistic error bars to these model SEDs based on the scatter in the photometry of galaxies associated with each cell. The result of the photo-z fitting is shown on the left side of Figure 4.

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We also estimate redshifts on the map by computing the median photo-z of the galaxies associated with each cell, using the 30-band photo-z estimates provided by the COSMOS survey (Ilbert et al. 2009). These photo-z estimates take advantage of more photometric information than is contained in the eight Euclid-like filters used to generate the map. Nevertheless, as can be seen on the right side of Figure 4, the resulting map is quite smooth, indicating that the eight Euclid bands capture much of the relevant information for photo-z estimation contained in the 30-band data.

Redshift probability density functions (PDFs) generated by the Le Phare template fitting can be used to estimate redshift uncertainty across the map, letting us identify cells that have high redshift variance or multiple redshift solutions, as well as cells with a well-defined mapping to redshift. In Figure 5 we show the photo-z dispersion results from the Le Phare code. The dispersion is the modeled uncertainty in the redshift assigned to each cell, based on the spread in the cell's redshift PDF. Figure 5 shows that there are well-defined regions in which the modeled uncertainties are much higher, and that these regions tend to cluster around sharp boundaries between low- and high-redshift galaxies. Note that these boundaries are inherent to the data and indicate regions of significant redshift degeneracy. A possible improvement in this analysis is to more rigorously estimate the photometric uncertainty for each cell using a metric for the volume of color space it represents; we defer this more detailed analysis to future work.

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5.3. Current Spectroscopic Coverage in COSMOS

One of the most important results of the mapping is that it lets us directly test the representativeness of existing spectroscopic coverage. To do so, we used the master spectroscopic catalog from the COSMOS collaboration (M. Salvato et al. 2016, in preparation). The catalog includes redshifts from VLT VIMOS (zCOSMOS, Lilly et al. 2007; VUDS, Le Fèvre et al. 2015), Keck Multi-object Spectrograph for Infrared Exploration (MOSFIRE) (N. Scoville et al., 2015 in preparation; MOSDEF, Kriek et al. 2014), Keck Deep Extragalactic Imaging Multi-object Spectrograph (DEIMOS) (Kartaltepe et al. 2010, G. Hasinger et al. 2015, in preparation), Magellan IMACS (Trump et al. 2007), Gemini-S (Balogh et al. 2014), Subaru FMOS (Silverman et al. 2014), as well as a non-negligible fraction of sources provided by a number of smaller programs. It is important to note that the spectroscopic coverage of the COSMOS field is not representative of the typical coverage for surveys. Multiple instruments with different wavelength coverages and resolutions were employed. Moreover, the spectroscopic programs targeted different types of sources: from AGN to flux-limited samples, from group and cluster members to high-redshift candidates, etc., providing an exceptional coverage in parameter space.

In the left panel of Figure 6, we show the map colored by the median spectroscopic redshift of galaxies associated with each cell, using only galaxies with the highest confidence redshift assignments (corresponding to ∼100% certainty). The gray regions on the map correspond to cells of color space for which no galaxies have such high confidence spectrosopic redshifts; 64% of cells fall in this category. In the right panel of Figure 6 we show the same plot, but using all confidence ≳95% redshifts in the master catalog. Significantly more of the galaxy color space is covered with spectroscopy when the requirement on the quality of the redshifts is relaxed, with only 51% of color cells remaining gray. However, for calibration purposes very high confidence redshifts will be needed, so that the right-hand panel may be overly optimistic. As can be seen in both panels, large and often continuous regions of galaxy color space remain unexplored with spectroscopy.

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It should be noted that Figure 6 is entirely data-driven, demonstrating the direct association of observed SED with observed redshift. An interesting possibility suggested by this figure is that the color–redshift relation may be smoother than expected from photo-z variance estimates from template fitting (e.g., Figure 5). High intrinsic variance in the color–redshift mapping should result in large cell-to-cell variation in median spec-z, whereas the actual distribution appears to be rather smooth overall.

5.4. Magnitude Variation across Color Space

Not surprisingly, the median galaxy magnitude varies strongly with location in color space, as illustrated in Figure 7. This variation largely determines the regions of color space that have been explored with spectroscopy, with intrinsically fainter galaxies less likely to have been observed. In fact, as we will discuss further in Section 6.6, the majority of galaxies in unexplored regions of color space are faint, star-forming galaxies at z ∼ 0.2–1.5, which are simply too "uninteresting" (from a galaxy evolution standpoint) to have been targeted in current spectroscopic surveys. Such sources will, however, be critically important for weak lensing cosmology.

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We have demonstrated that the SOM, when applied to a large photometric dataset, efficiently characterizes the distribution of galaxies in the parameter space relevant for photo-z estimation. We now consider the problem of determining the spectroscopic sample needed to calibrate the $\langle z\rangle$ of the tomographic redshift bins to the required level for weak lensing cosmology. We show that allocating spectroscopic efforts using the color space mapping can minimize the spectroscopy needed to reach the requirement on the calibration of N(z).

6.1. Estimating the Spectroscopic Sample Needed for Calibration

Obtaining spectroscopic redshifts over the full color space of galaxies is obviously beneficial, but the question arises: precisely how many spectra are needed in different regions of color space in order to meet the dark energy requirement? Here we provide a framework for understanding this question in terms of the color space mapping.

First we note that each color cell has some subset of galaxies that best associate with it; let the total number of galaxies associating with the ith cell be n_i. We refer to the true redshift probability distribution of these galaxies as P_i(z). For the sake of this argument we assume that a tomographic redshift bin for weak lensing will be constructed by selecting all galaxies associating with some subset of the cells in the SOM. Let the total number of cells used in that tomographic bin be c. Then the true N(z) distribution for galaxies in the resulting tomographic redshift bin is

$$\begin{eqnarray}&&N(z)=\displaystyle \sum _{i=1}^{c}{n}_{i}{P}_{i}(z).\end{eqnarray} \tag{ 8 }$$

The mean of the N(z) distribution is given by

$$\begin{eqnarray}&&\langle z\rangle =\displaystyle \frac{\int {zN}(z){dz}}{{N}_{T}},\end{eqnarray} \tag{ 9 }$$

where the integral is taken over all redshifts and N_T is the total number of galaxies in the redshift bin. Inserting Equation (8) into (9), we find that the mean redshift of the bin can be expressed as

$$\begin{eqnarray}\langle z\rangle & = & \displaystyle \frac{1}{{N}_{T}}\displaystyle \int z[{n}_{1}{P}_{1}(z)+\cdots +{n}_{c}{P}_{c}(z)]{dz}\\ & = & \displaystyle \frac{1}{{N}_{T}}[{n}_{1}\langle {z}_{1}\rangle +\cdots +{n}_{c}\langle {z}_{c}\rangle ].\end{eqnarray} \tag{ 10 }$$

Equation (10) is the straightforward result that the mean redshift of the full N(z) distribution is proportional to the sum of the mean redshifts of each color cell, weighted by the number of galaxies per cell. The uncertainty in $\langle z\rangle$ depends on the uncertainty of the mean redshift of each cell, and is expressed as

$$\begin{eqnarray}&&{\rm{\Delta }}\langle z\rangle =\displaystyle \frac{1}{{N}_{T}}\sqrt{\displaystyle \sum _{i=1}^{c}{n}_{i}^{2}{\sigma }_{\langle {z}_{i}\rangle }^{2}}.\end{eqnarray} \tag{ 11 }$$

Equation (11) shows quantitatively what is intuitively clear, namely that the uncertainty in $\langle z\rangle$ is influenced more strongly by cells with both high uncertainty in their mean redshift and a significant number of galaxies associating with them. This indicates that the largest gain can be realized by sampling more heavily in denser regions of galaxy color space, as well as those regions with higher redshift uncertainty. Conversely, cells with very high redshift dispersion could simply be excluded from the weak lensing sample (although caution would be needed to ensure that no systematic errors are introduced by doing so).

If we assume that the c color cells have roughly equal numbers of galaxies and that ${\sigma }_{\langle {z}_{i}\rangle }$ is roughly constant across cells, then Equation (11) becomes

$$\begin{eqnarray}&&{\rm{\Delta }}\langle z\rangle ={\sigma }_{\langle {z}_{i}\rangle }/\sqrt{c}.\end{eqnarray} \tag{ 12 }$$

With ${\sigma }_{\langle {z}_{i}\rangle }\sim 0.05(1+\langle z\rangle ),$ we find ∼600 color cells with this level of uncertainty would be needed to reach the Euclid calibration requirement for the redshift bin. With one spectrum per cell required to reach this level of uncertainty in ${\sigma }_{\langle {z}_{i}\rangle },$ this estimate of the number of spectra needed is in rough agreement with that of Bordoloi et al. (2010), and much lower than estimates for direct calibration through random sampling. Note that the mean redshifts $\langle {z}_{i}\rangle$ for each color cell used in Equation (10) should be based on spectroscopic redshifts, to ensure that the estimates are not systematically biased. The error in a cell's mean redshift estimate, ${\sigma }_{\langle {z}_{i}\rangle },$ will depend on the dispersion in the P_i(z) distribution for the cell, and will scale inversely with the square root of the number of spectra obtained to estimate it.

The preceding analysis treats the photo-z calibration as a stratified sampling problem, in which the overall statistics of a population are inferred through targeted sampling from relatively homogeneous subpopulations. The gain in statistical precision from using Equation (10) to estimate $\langle z\rangle$ can be attributed to the systematic way in which the full color space is sampled, relative to blind direct sampling. However, stratified sampling will only outperform random sampling in the case that the subpopulations being sampled do in fact have lower dispersion than the overall distribution—i.e., in the case that the P_i(z) distributions for the color cells have lower redshift dispersion than the N(z) distribution of all the galaxies in a tomographic bin.

6.2. Simulating Different Sampling Strategies

Now we attempt to more realistically estimate the spectroscopic coverage needed to achieve the requirement in our knowledge of $\langle z\rangle .$ To begin, we assume that the cell redshift PDFs from Le Phare are reasonably accurate, and can be taken to represent the true P_i(z) distributions for galaxies in each color cell. (This assumption is, of course, far from certain, and simply serves as a first approximation.) With the known occupation density of cells of the map (Figure 3), we can then use Equation (8) to generate realistic N(z) distributions for different tomographic bins. For this illustration, we break the map up into photo-z-derived tomographic bins of width Δ z = 0.2 over 0 < z < 2 (although Euclid will most likely use somewhat different bins in practice). An example of one of the N(z) distributions modeled in this way is shown in Figure 8.

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The uncertainty in the estimated $\langle z\rangle$ of these N(z) distributions can then be tested for different spectroscopic sampling strategies through Monte Carlo simulations, in which spectroscopy is simulated by randomly drawing from the P_i(z) distributions. (Alternatively, given our knowledge of the individual ${\sigma }_{\langle {z}_{i}\rangle }$ uncertainties, Equation (11) can be used directly. In fact, the results were checked in both ways and found to be in agreement).

The results of three possible sampling strategies are given in Table 1. The simplest strategy tested ("Strategy 1") is to obtain one spectrum per color cell in order to estimate the cell mean redshifts. Equation (10) is then used to compute the overall mean of the tomographic bin. We expect to meet the Euclid requirement, ${\rm{\Delta }}\langle z\rangle \leqslant 0.002(1+\langle z\rangle ),$ for 3/10 bins (and come close in the others) with this approach, which would require ∼11k spectra in total.

Table 1.  Simulated Uncertainty in $\langle z\rangle$ for Representative Redshift Bins for Different Sampling Strategies

	Strategy 1a		Strategy 2b			Strategy 3c
Redshift Bin	#Spectra	${\sigma }_{\langle z\rangle }/(1+\langle z\rangle )$	#Spectra	% Sample Lostd	${\sigma }_{\langle z\rangle }/(1+\langle z\rangle )$	#Spectra	${\sigma }_{\langle z\rangle }/(1+\langle z\rangle )$
0.0–0.2	659	0.0034	627	4.2	0.0024	723	0.0028
0.2–0.4	1383	0.0028	1314	4.6	0.0015	1521	0.0020
0.4–0.6	2226	0.0014	2115	3.9	0.0007	2448	0.0010
0.6–0.8	2027	0.0018	1926	4.3	0.0005	2229	0.0012
0.8–1.0	1357	0.0021	1290	4.4	0.0009	1491	0.0013
1.0–1.2	1705	0.0011	1620	4.6	0.0005	1875	0.0008
1.2–1.4	559	0.0029	532	4.4	0.0015	613	0.0021
1.4–1.6	391	0.0044	372	3.3	0.0021	429	0.0031
1.6–1.8	268	0.0064	255	2.7	0.0050	294	0.0055
1.8–2.0	164	0.0093	156	2.1	0.0085	180	0.0088
Total #spectra:	10739		10207			11793

Notes.

^aObtaining one spectrum per color cell to estimate $\langle {z}_{i}\rangle ,$ with $\langle z\rangle$ computed using Equation (10). ^bAgain obtaining one spectrum per color cell to estimate $\langle {z}_{i}\rangle ,$ but rejecting the 5% of cells with the highest redshift uncertainty. ^cObtaining three spectra per color cell for the 5% of cells with the highest redshift uncertainty, one spectrum per cell for the other 95%. ^dThe fraction of galaxies lost from the weak lensing sample for that tomographic bin due to excluding 5% of the most uncertain color cells.

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The second strategy tested is similar to the first, in that one spectrum per cell is obtained. However, galaxies associated with the 5% of the cells in each bin with the highest redshift uncertainty are rejected from the weak lensing sample, and these cells are ignored in the sampling. This significantly reduces the uncertainty in the $\langle z\rangle$ estimates, with 6/10 bins meeting the requirement; moreover, it reduces the total number of spectra needed by 5%. However, it comes at the cost of reducing the number of galaxies in the weak lensing sample.

The third strategy is to sample the 5% of the cells with the highest redshift uncertainty with three spectra each in order to estimate their mean redshifts with greater accuracy, again obtaining one spectrum for the other 95% of the cells. This strategy again lowers the uncertainty in the $\langle z\rangle$ estimates substantially, but at the cost of increased spectroscopic effort, requiring ∼12k spectra in total. The additional spectra needed may also prove to be the more difficult ones to obtain, so the effort needed cannot be assumed to scale linearly with the number of spectra.

These examples are simply meant to be illustrative of the possible strategies that can be adopted for the spectroscopic calibration. More refined strategies are possible—for example, an optimal allocation of spectroscopic effort could be devised that scales the number of spectra in a given region of color space proportionately to the redshift uncertainty in that region, while rejecting limited regions of color space that are both highly uncertain and difficult for spectroscopy. Additional spectroscopy may need to be allocated to the higher redshift bins, for which there tend to be fewer cells overall as well as higher dispersion within cells. Tomographic bins could also be intentionally generated to minimize the uncertainty in $\langle z\rangle .$ The simpler examples shown here do illustrate that, if we believe the cell P_i(z) estimates from template fitting, the Euclid calibration requirement ${\rm{\Delta }}\langle z\rangle \leqslant 0.002(1+\langle z\rangle )$ is achievable with ∼10–15k spectra in total (roughly half of which already exist).

6.2.1. Is Filling the Map with Spectroscopy Necessary?

The number of spectra needed derived above assumes that at least one spectrum per SOM color cell is necessary to estimate the $\langle {z}_{i}\rangle$ for that cell. However, if a particular region of color space is very well understood and maps smoothly to redshift, sparser spectroscopic sampling in that region together with interpolation across cells might be sufficient. Equivalently, groups of neighboring cells with low redshift uncertainty that map to roughly the same redshift could potentially be merged using a secondary clustering procedure, thus lowering the overall number of cells and the number of spectra required. These considerations suggest that, while the exact number of spectra required to meet the calibration requirement is uncertain, the results presented above are likely to represent upper limits.

6.3. Estimating the True Uncertainty in the Color–Redshift Mapping

6.4. Effect of Photometric Error on Localization in Color Space

6.5. Cosmic Variance

6.6. Galaxies in Under-sampled Regions of Color Space

From the preceding analysis, a reasonable step toward calibration of the photo-z's for cosmology is to target the regions of multicolor space currently lacking spectroscopy (the gray regions in Figure 6). It is therefore important to understand the nature of the galaxies in these regions, in order to predict the spectroscopic effort needed.

Of the 11,250 cells in the SOM presented here, roughly half currently have no objects with high-confidence spectroscopic redshifts. The distribution of these cells on the map, as well as their photometric redshift estimates, are displayed on the left side of Figure 9. The right side of Figure 9 shows the overall magnitude and photometric redshift distribution of the unsampled cells of color space. Most unsampled cells represent galaxies fainter than i = 23 (AB) at redshifts z ∼ 0.2–1.5, and ∼83% of these are classified as star-forming by template fitting. These magnitude, redshift, and galaxy type estimates directly inform our prediction of the spectroscopic effort that will be required to calibrate the unsampled regions of galaxy color space.

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Generally speaking, these galaxies have not been targeted in existing spectroscopic surveys because they are faint and not considered critical for galaxy evolution studies. However, they are abundant and thus important for weak lensing cosmology. In Appendix A we give a detailed estimate of the observing time that would be needed to fill in the empty parts of color space with a fiducial survey with Keck, making use of the Low Resolution Imaging Spectrograph (LRIS), DEIMOS, and MOSFIRE. We find that ∼40 nights would be required if we reject the 1% most difficult cells—a large time allocation, but not unprecedented in comparison with other large spectroscopic surveys. This is significantly less than the ∼100 nights needed to obtain a truly representative sample without prior knowledge of the color distribution (Newman et al. 2015). For both LSST and WFIRST the calibration sample required is likely to be significantly larger, due to the greater photometric depths of these surveys in comparison with Euclid. Therefore, methods to improve the sampling as proposed here will be even more important to make the problem tractable for those surveys.

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Statistically well-understood photometric redshift estimates for billions of galaxies will be critical to the success of upcoming Stage IV dark energy surveys. We have demonstrated that self-organized mapping of the multidimensional color distribution of galaxies in a broadband survey such as Euclid has significant benefits for redshift calibration. Importantly, this technique lets us identify regions of the photometric parameter space in which the density of galaxies $\rho ({\boldsymbol{C}})$ is non-negligible, but spectroscopic redshifts do not currently exist. These unexplored regions will be of primary interest for spectroscopic training and calibration efforts.

Applying our SOM-based analysis to the COSMOS field, we show that the regions of galaxy parameter space currently lacking spectroscopic coverage generally correspond to faint (i-band magnitude (AB) ≳ 23), star-forming galaxies at z < 2. We estimated the spectroscopy required to fill the color space map with one spectrum per cell (which would come close to or achieve the required precision for calibration) and found that a targeted, ∼40 night campaign with Keck (making use of LRIS, DEIMOS and MOSFIRE) would be sufficient (Appendix A). It should be noted that this analysis is specific to the Euclid survey. The calibration needs of both LSST and WFIRST are likely to be greater, due to the deeper photometry that will be obtained by those surveys.

We demonstrated that systematically sampling the color space occupied by galaxies with spectroscopy can efficiently constrain the N(z) distribution of galaxies in tomographic bins. The precise number of spectra needed to meet the bias requirement in $\langle z\rangle$ for cosmology depends sensitively on the uncertainty in the color–redshift mapping. Template-based estimates suggest that this uncertainty is rather high in some regions of Euclid-like color space. However, the smoothness of the spectroscopic redshift distribution on the map suggests that the template-based uncertainties may be overestimated, which would reduce the total number of spectra needed for calibration.

Assuming that the uncertainties in $P(z| {\boldsymbol{C}})$ from template fitting are accurate, we demonstrate that the Euclid requirement on ${\rm{\Delta }}\langle z\rangle$ should be achievable with ∼10–15k total spectra, about half of which already exist from various spectroscopic surveys that have targeted the COSMOS field. Understanding the true uncertainty in $P(z| {\boldsymbol{C}})$ will likely prove critical to constraining the uncertainty in $\langle z\rangle$ for the tomographic bins, and we suggest that developing efficient ways of constraining this uncertainty should be prioritized.

The topological nature of the SOM technique suggests other possible uses. For example, a potentially very useful aspect of the SOM is that it lets us quantify the "normality" of an object by how well-represented it is by some cell in the map. Rare objects, such as AGNs, blended sources, or objects with otherwise contaminated photometry could possibly be identified in this way. We also note that the mapping, by empirically constraining the galaxy colors that appear in the data, can be used both to generate consistent priors for template fitting codes as well as test the representativeness of galaxy template sets. These applications will be explored in future work.

We thank the anonymous referee for constructive comments that significantly improved this work. We thank Dr. Ranga Ram Chary, Dr. Ciro Donalek, and Dr. Mattias Carrasco-Kind for useful discussions. D.M., P.C., D.S., and J.R., acknowledge support by NASA ROSES grant 12-EUCLID12-0004. J.R. is supported by JPL, run by Caltech for NASA. H.Ho. is supported by the DFG Emmy Noether grant Hi 1495/2-1.S.S. was supported by Department of Energy grant DESC0009999. Data from the VUDS survey are based on data obtained with the European Southern Observatory Very Large Telescope, Paranal, Chile, under Large Program 185.A-0791. This work is based in part on data products made available at the CESAM data center, Laboratoire d'Astrophysique de Marseille.

Given the <0.2% accuracy in $\langle z\rangle$ required for the Euclid tomographic bins, and following the analysis presented above, a nearly optimal approach would be to obtain one spectrum per SOM cell, while rejecting ∼1% of the cells requiring the longest spectroscopic observations. Taking existing spectroscopy into account, a total of ∼5k new spectroscopic redshifts would be needed. We estimate that these spectra could be obtained in ∼40 nights with Keck, as outlined below.

To quantify the required exposure time, we constructed a fiducial survey on the Keck telescope with the LRIS (Oke et al. 1995), the DEIMOS (Faber et al. 2003), and the MOSFIRE (McLean et al. 2012) instruments. This telescope/instrument combination was chosen because the full redshift range of the calibration sample can be optimally probed with these instruments, and their performance in obtaining redshifts for i ∼ 24.5 galaxies has been demonstrated in numerous publications (e.g., Steidel et al. 2004; Newman et al. 2013; Kriek et al. 2014). For LRIS we follow Steidel et al. (2004) and assume the 300 groove mm⁻¹ grism blazed at 5000 Å on the blue side and the 600 groove mm⁻¹ grating blazed at 10,000 Å on the red side with the D560 dichroic. With DEIMOS the 600 grove mm⁻¹ grating tilted to 7000 Å was assumed. MOSFIRE was assumed to be in its default configuration. Sensitivities were estimated using the official exposure time calculators (ETCs) provided by Keck by scaling from a 24th magnitude flat spectrum object. We assume 1'' seeing, a 1'' wide slit, an airmass of 1.3, and we include appropriate slit losses. For all instruments we scaled the SNR to a binning of R ∼ 1500, the minimum required resolution for calibration redshifts. The assumed SNRs in a one hour exposure at 24th magnitude (AB) are given in Table 2.

We assume that the galaxies in the cells needing spectroscopy have the redshifts, galaxy spectral types, and reddenings derived from template fitting with Le Phare. The modeled galaxy spectral types, redshifts, and observed magnitudes were used to determine the required SNR and the instrument such that a >99% reliable redshift can be obtained. For star-forming galaxies at z < 2.7 we require SNR = 2 on the continuum because bright rest-frame optical emission lines will be used to determine the redshift. For star-forming galaxies at z >= 2.7 we require SNR = 3 on the continuum to clearly detect the Lyman break and the rest frame ultraviolet (UV) absorption features with LRIS or DEIMOS (e.g., Steidel et al. 2003). For galaxies classified as passive, we require SNR = 5 on the continuum (e.g., Kriek et al. 2009; Onodera et al. 2012), while objects intermediate between passive and star-forming were allowed to linearly scale between an SNR of 5 and 2 with an increasingly star-forming spectral template, because the spectral feature strength increases with star formation rate.

The magnitude measured in the band closest to the most prominent spectral feature was assumed for the SNR calculation, and the instrument with the highest sensitivity at that feature was assumed. For passive galaxies it was assumed that the 4000 Å break must be targeted at z < 2.3 and the 1216 Å Lyman forest break at higher redshifts, with DEIMOS used at z < 1.3, LRIS at 1.3 < z < 1.4, MOSFIRE at 1.4 < z < 2.3, LRIS at 2.3 < z < 3.5, and DEIMOS at z > 3.5. For other galaxies, the strongest of Hα, Hβ, O[iii], and O[ii] was targeted at z < 2.7, with DEIMOS at z < 1.5 and MOSFIRE at 1.5 < z < 2.7. The 1216 Å Lyman forest break was targeted at higher redshifts, with LRIS at 2.7 < z < 3.5 and DEIMOS at z > 3.5.

Objects were then grouped into masks by instrument and exposure time, assuming a multiplexing of 70 for DEIMOS and 20 for LRIS and MOSFIRE, making the assumption that deep observations could be obtained for rare faint objects by observing them in multiple masks. Assuming nights are 10 h long, overheads are 10%, 20% of the objects need to be observed by more than one instrument to confirm the redshift, and 30% are losses due to weather, we obtain the estimate of required observing time given in Table 2.

An exploratory program in early 2015 used samples from poorly sampled regions of color space as fillers on 2–4 hr Keck DEIMOS slit masks, finding that >98% of sources were readily identified from strong [O ii], [O iii], and/or Hα emission, while the non-detected sources had photometric redshifts for which no line detection was expected by DEIMOS.

We note that an additional ∼12 nights would be required to get to 99.8% completeness in color cells, and ∼49 (for a total of ∼100) more nights to reach 99.9% completeness. This confirms the difficulty in obtaining truly complete samples noted by previous work, as well as the importance of systematically rejecting sources (Newman et al. 2015).

Figure 10 shows two alternate maps generated with the same COSMOS data, but with different starting conditions and training orders. Note that the overall topological features are the same. The representativeness of these maps (in the sense described in Section 4.3) is essentially identical and the map shown throughout the paper. However, the positions and orientations of different photometric clusters are random.