Photometric Redshifts with the LSST. II. The Impact of Near-infrared and Near-ultraviolet Photometry

The CMNN photometric redshift estimator that we use for this work was first introduced and described by Graham et al. (2018).⁶ It is not intended to provide the "official" or "best" LSST photo-z but instead to provide sets of photo-z results in which the accuracy and precision of the photo-z estimates are directly related to the precision of the input photometry. In this way, it is particularly useful for comparing the relative photo-z quality from different survey strategies that affect the LSST photometric quality of the LSST.

The CMNN photo-z estimator requires "training" and "test" galaxy catalogs: the former is equivalent to a set of galaxies with "known" or spectroscopic redshifts, and the latter is equivalent to a sample of galaxies for which photo-z will be estimated. The full catalog of simulated galaxies that we use for this work is described in Section 2.2. For this work, instead of splitting the simulated catalog into test and training sets, we employ a leave-one-out strategy: every catalog galaxy is considered as the test galaxy in turn, and the training set is composed of all of the other galaxies in the simulated catalog (Section 2.2). For the test galaxy, the estimator identifies a subset of training galaxies that are its nearest neighbors in color-space using the Mahalanobis distance⁷

$$\begin{eqnarray}&&{D}_{M}=\sum _{1}^{{N}_{\mathrm{colors}}}\displaystyle \frac{{\left({c}_{\mathrm{train}}-{c}_{\mathrm{test}}\right)}^{2}}{{\left(\delta {c}_{\mathrm{test}}\right)}^{2}},\end{eqnarray} \tag{ 1 }$$

where c_train is the color of a training-set galaxy, c_test is the color of a test set galaxy, and δc_test is the measurement error in the color of the test set galaxy (Section 2.2.2). If a galaxy is undetected in a given filter, the associated color does not contribute to D_M. We enforce the number of colors to be N_colors ≥ 3 (i.e., a galaxy must be detected with at least three colors to obtain a photo-z estimate).

Once D_M is measured for all training-set galaxies, we use a percent-point function (PPF) value of PPF = 0.68 and the degrees of freedom, N_colors, to define a threshold value, ${T}_{{D}_{M}}$. Training-set galaxies with D_M values below ${T}_{{D}_{M}}$ are identified as the set of CMNNs. For example, for ${N}_{\mathrm{colors}}=5$, PPF = 68% of all training galaxies consistent with the test galaxy will have D_M < 5.86. The CMNN subset galaxies are then subjected to a weighted (${D}_{M}^{-1}$)⁸ random selection, and the redshift of the chosen training-set galaxy is adopted as the photo-z for the test galaxy. For test galaxies with less than 10 training-set galaxies in their CMNN subset (${N}_{\mathrm{CM}}\lt 10$), the 10 nearest neighbors are used. If N_CM ≥ 10, the photo-z uncertainty (δz) is the standard deviation in the true redshifts of the color-matched subset; if N_CM < 10, we calculate the PPF threshold value associated with the 10th nearest neighbor, and we inflate the uncertainty by $\times {\mathrm{PPF}}_{10}/0.68$.

Note that this is different from the application in Graham et al. (2018), which also used a PPF = 0.68 but chose the nearest neighbor in color-space and did not have a modification for galaxies with N_CM < 10 matches (such that galaxies with zero matches simply failed to obtain a photo-z). We furthermore do not apply an optical magnitude or color "pseudo-prior" to our photo-z estimates, as was done in Graham et al. (2018), so as to not give the optical an additional influence on the photo-z results in this paper, which focuses on the addition of UV and NIR photometry. The other major difference compared to Graham et al. (2018) is that, here, the entire simulated set of galaxies is used as both a test and training set instead of assigning each galaxy to one or the other before simulating photo-z results. This has allowed us to use much larger test sets and achieve more accurate statistical measures for the photo-z results in high-redshift bins.

Incorporating the NIR filter(s) and/or UV passbands causes an increase in ${N}_{\mathrm{colors}}$, which increases the number of degrees of freedom in Equation (1). This can also increase the size of the CMNN subset of training galaxies and, at times, degrade the overall photo-z quality. This is an important aspect of the CMNN estimator to keep in mind as we explore the impact of adding NIR and UV photometry. The photometric quality of the data must be good enough to overcome the increase in degrees of freedom in the estimator in order to see a beneficial impact on the photo-z results. In this work, we encounter several instances where the photometric quality for some simulated galaxy populations is insufficient to benefit from the addition of NIR or UV filters, and the photo-z results are deteriorated. Generally, we will mitigate this issue by identifying these populations using their observed characteristics (i.e., not using true catalog redshift), and then excluding the additional filters from the photo-z estimates. Each of these instances is discussed in detail in the relevant analysis section.

As in Graham et al. (2018), we use a galaxy catalog based on the Millennium simulation (Springel et al. 2005)—specifically, a catalog based on the galaxy formation models of Gonzalez-Perez et al. (2014) and fabricated using the lightcone construction techniques described by Merson et al. (2013).⁹ This simulated galaxy catalog was designed to model the optical and NIR properties of galaxies, and with appropriate limits, it also serves as a realistic representation of future LSST catalogs. "True" catalog apparent magnitudes for optical and NIR filters ugrizy and YJHK are included in this simulated galaxy catalog (for simplicity, we refer to WFIRST's F184 filter as K in this work because it fulfills a similar role), but we must synthesize magnitudes to represent the proposed CASTOR passbands.

2.2.1. CASTOR Passbands and Simulated Photometry

To simulate "true" catalog photometry for the CASTOR passbands, we start by using the same process as in Graham et al. (2018, their Section 5.1) to assign each galaxy an SED type from the catalog of Brown et al. (2014). First, we find the SED template that best matches each catalog galaxy's rest-frame g − r color. The SEDs are sequenced from early- to late-type, numbered from 0 to 128, and we include a modest amount of intrinsic scatter by simply adding a small uniform-random integer to the assigned SED number (within ±2). Next, we flux-scale the SED to match the catalog's LSST u- and g-band apparent magnitudes and then apply the four CASTOR bandpasses¹⁰ (see Figure 1) to calculate "true" catalog magnitudes in u_c, u-wide (u_w), UV, and UV-dark (UV_d). We do not recalculate a CASTOR g_c-band true catalog magnitude because the passband shape is quite similar to the LSST g filter, as shown in the top panel of Figure 1. Instead, we will simply apply the CASTOR magnitude limits to the catalog's LSST g-band apparent magnitudes.

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To validate that our synthesized CASTOR UV magnitudes are providing independent information about redshift, we plot the simulated galaxies in the color–color plane of CASTOR UV − LSST u versus the LSST u − g in the bottom panel of Figure 1. Plotted point color represents the true catalog redshift for each galaxy, and we can see that redshift is correlated with the UV − u color in a way that is unique from the u − g color. This confirms that our synthesized CASTOR UV photometry should have an influence on the estimated photometric redshifts. Since the UV magnitudes are the only passbands being synthesized in this work, we do not show similar color–color plots for Euclid or WFIRST NIR. The impact of CASTOR photometry on LSST photo-z is presented in Section 5.

2.2.2. Photometric Errors

To incorporate an LSST-like observational uncertainty into our catalog, we simulate observed apparent magnitudes from the true catalog magnitudes by adding a normal random scatter¹¹ with a standard deviation equal to the predicted magnitude error for each galaxy. Predicted magnitude errors for the LSST, as described in Section 3.2.1 of Ivezić et al. (2019), depend on the galaxy's magnitude and the total survey exposure time elapsed in a given filter (for this work, we assume no additional components from, e.g., deblending, or the different image resolutions of the four facilities). We assume standard observing conditions, an airmass of 1.2, and a uniform survey progression that accumulates a total of 56, 80, 184, 184, 160, and 160 visits in filters ugrizy by year 10 (where each visit is 30 s of integration time). We use the LSST simulations software package described by Connolly et al. (2014) to calculate the magnitude errors; it includes a systematic floor of 0.005 in all filters. After 10 yr of the LSST survey, these accumulated visits result in 5σ detection limits for each filter of u < 26.1, g < 27.4, r < 27.5, i < 26.8, z < 26.1, and y < 24.9 (as also listed in Table 1). In the bottom panel of Figure 2, we show the magnitude error as a function of observed apparent magnitude, with magnitude cuts applied at the 5σ limiting magnitude in filters ugrzy and at i ≤ 25 mag.

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Table 1.  Signal-to-noise Ratio = 5 Limiting Magnitudes

Filter	LSSTa	Euclidb	WFIRSTc	CASTORd
UV	...	...	...	27.4
u	26.1	...	...	27.4
g	27.4	...	...	27.1
r	27.5	...	...	...
i	26.8	...	...	...
z	26.1	...	...	...
y/Y	24.9	24.0	26.7	...
J	...	24.2	26.9	...
H	...	23.9	26.0	...
K	...	...	25.8	...

Notes.

^aLimits for a nominal 10 yr survey. ^bLimits from J.-C. Cuillandre, private conversation. ^cLimits from B. Jain, private conversation. ^dLimits for the primary (wide-area) survey.

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To predict the photometric errors for the NIR filters of Euclid and WFIRST, and the UV/blue passbands of CASTOR, we use the expected 5σ limiting magnitudes listed in Table 1. We note that the predicted WFIRST limiting magnitudes have slightly evolved since our simulations were run. Doré et al. (2019) reports that the expected 5σ point-source imaging depths in Y, J, H, and K (F184) are 26.9, 26.95, 26.9, and 26.25, respectively—not too far off from the values we have used (except for the H band). For all three future facilities, we use the same prescription for error as described in Section 3.2.1 of Ivezić et al. (2019), namely,

$$\begin{eqnarray}&&{\sigma }_{\mathrm{rand}}^{2}=(0.04-\gamma )x+\gamma {x}^{2},\end{eqnarray} \tag{ 2 }$$

where $x={10}^{0.4(m-{m}_{5})}$, m₅ is the 5σ limiting magnitude, m is the magnitude of the galaxy, and we use the fiducial value of γ = 0.04, which sets the impact of, e.g., sky brightness, to be zero. As with the LSST photometry, we impose a systematic floor of σ_rand ≥ 0.005 mag. The magnitude error as a function of apparent magnitude, down to the 5σ detection limit of each filter for the surveys considered in this work, is shown in the bottom panel of Figure 2.

The uncertainty in the apparent observed color of a galaxy is calculated as the root of the sum of the squares of the magnitude uncertainties in the two filters (i.e., the magnitude uncertainties from the two filters are added in quadrature, under the assumption of uncorrelated errors). In some parts of our analysis, we extend the cut on apparent magnitude from the 5σ to the 1σ limiting magnitude, and in all cases, we use ${m}_{1\sigma }={m}_{5\sigma }+1.75$ mag. In practice, obtaining matched aperture photometry for images with very different resolutions can lead to additional uncertainties in the combined catalogs' photometry (e.g., see Section 5.1 in Spergel et al. 2015). For this work, we must assume that this potential issue has been resolved by a sophisticated joint pixel analysis.

2.2.3. Catalog Demographics

In the top panel of Figure 2, we show the cumulative distribution of true apparent magnitudes in our simulated catalog. The line style switches from solid to dashed when the distribution passes the 5σ limiting magnitude for detection in the filter. From this, we can see that for the LSST photometry, ∼25% of the galaxies in our sample remain undetected in the u band, but all galaxies are detected in the grz bands and almost all in the y band. All galaxies are detected in the i band by design,¹² since we impose a cut¹³ of i < 25 mag, which is brighter than the 5σ detection limit of 26.8 mag.

In Figure 2, we have included the Euclid NIR photometry in our plots of true catalog magnitude distributions (top panel) and error versus magnitude (bottom panel). We can see that the cumulative distributions of true magnitudes are very similar for Euclid Y and LSST y (top panel) but that the 10 yr LSST photometry has significantly smaller errors than Euclid's (bottom panel). Our treatment of these two y-bands in our photo-z estimates is discussed in Section 2.2.4. The photometric quality offered by WFIRST is also illustrated in the lower panel of Figure 2. The WFIRST limiting magnitudes are sufficiently deep that all catalog galaxies are detected in all four WFIRST filters.

In Figure 3, we show the redshift distributions of the simulated galaxy catalog. Since we use each catalog galaxy as the "test" galaxy in turn, the "test" and "training" sets have the same redshift distributions. This similarity helps to avoid redshift bias in the photo-z results but is not a likely property of future spectroscopic training sets, as discussed below. In the top panel of Figure 3, we show the true redshift distribution of our simulated catalog, with applied cuts of i ≤ 25 and ≤26.8 mag (most of our simulations use the former).

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In the middle panel of Figure 3, we show the fraction of catalog galaxies detected in the CASTOR u_c and UV passbands as a function of true redshift, compared to LSST u, for detection limits of 1σ and 5σ. We can see that the fraction of galaxies detected in the CASTOR u passband at 5σ is approximately equivalent to that detected by the LSST u filter at 1σ (thick solid and thin dashed purple lines, respectively). The sharp drop in the fraction of galaxies detected in CASTOR UV at 5σ indicates that adding CASTOR UV photometry will probably only be able to improve photometric redshift estimates for z ≲ 1.7.

In the bottom panel of Figure 3, we show the fraction of catalog galaxies detected in the Euclid J and H filters as a function of true redshift, for detection limits of 1σ and 5σ. Like the CASTOR UV passband, the impact of including Euclid 5σ photometry in the photo-z estimates will drop for z ≳ 1.7. WFIRST detection fractions are not shown in any panel because, as with Euclid 1σ photometry, all LSST galaxies with i < 25 mag are completely detected in the four WFIRST filters.

Throughout most of this work, we will use a training set that has the same distributions of redshift and magnitude as the test set, and we emphasize that this is an ideal case (but appropriate for this work, because our focus is on the impact of photometry, exclusively, on the photo-z results). In reality, spectroscopic catalogs have some inherent bias and are not perfectly representative of the test set (which is an extremely difficult data product to build; e.g., Newman et al. 2015). In some of our simulations, we consider a training set that has better or deeper photometry than a test set from a wide-field imaging campaign, and in some cases, we also change the limiting magnitudes such that the test and training sets are no longer matched—but only when the proposed surveys seem likely to provide such a data product.

2.2.4. LSST y versus Euclid Y

The LSST y- and Euclid Y-band filters will not have exactly the same transmission function (the Euclid Y-band filter will be somewhat shifted redward and have sharper edges¹⁴ ), but for the purposes of this work, we consider the two filters to contribute equivalent information about a galaxy's SED to our photo-z estimator, and we use only one or the other. This is the most appropriate choice for our particular photo-z estimator, because including colors z − y, z − Y, y − J, and Y − J when calculating the Mahalanobis distance in color-space would give that region of the SED twice as much influence in Equation (1). We leave a study of whether two nearly superimposed filters could provide additional SED information, e.g., emission line galaxies, for future work.

In Figure 4, we plot the apparent magnitude error versus apparent magnitude for Euclid Y-band and the LSST y-band at survey years 1, 2, 5, and 10. It is clear that LSST's y-band photometric quality will be approximately equivalent to Euclid's Y-band with the LSST 2 yr data release. In this work, we do not simulate the LSST photo-z results at phases <2 yr, so in all cases, our simulations use the LSST y-band instead of the Euclid Y-band.

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As a final note, the y-band filter of any ground-based facility like LSST will be impacted by a variable water absorption feature. However, it should be possible to correct the y-band colors to a fiducial filter system to within acceptable uncertainties, given a measure of the precipitable water vapor and a rough SED shape determined from the other five LSST bands (S. Schmidt 2020, private communication). Thus, we have not explicitly included the effects of water vapor on the y-band photometry in this work.

To the results of our simulations, we apply a similar evaluation methodology as used in Graham et al. (2018). Since we are using the CMNN estimator to predict the relative improvement in photo-z due to changes in the input photometry (and not to predict absolute qualities), we do not extend our analysis to include predictions for cosmological parameters but suggest that readers consult Nakajima et al. (2012) and/or Mandelbaum (2018) for discussions of how photo-z errors impact cosmological analyses. Below, we describe first our statistical measures of photo-z quality and then the common plot styles that we will use to represent our results in this work.

2.3.1. Statistical Measures

2.3.2. Plot Styles

To visualize our photo-z results, we create plots that compare the true versus photometric redshifts. We use a two-dimensional histogram with log-normalized shading (such that black is always the most populated bin) in the densely sampled areas of z_true versus z_phot space, and we over-plot outlier galaxies with transparent red dots (see, e.g., Figure 5). We also draw a solid line representing z_true = z_phot to guide the eye. These plots are useful for obtaining a global sense of the photo-z quality and structure in the outliers positions, especially the features that are perpendicular to the z_true = z_phot, which represent photo-z degeneracies caused by the Balmer break passing between filters.

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To directly compare the statistical measures of photo-z quality for our different simulations, we generate plots of the robust standard deviation, robust bias, and fraction of outliers as a function of binned z_phot (see, e.g., Figure 6). We use a bin width of 0.3 and a bin spacing of 0.15 in z_phot, such that bins overlap. Small vertical error bars mark the error in the statistical measurements, and wide horizontal bars mark the value of the statistic over the full range of 0.3 ≤ z_phot ≤ 3.0. Dashed lines are often used to represent the SRD's target values in these plots.

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Both the LSST and the Euclid surveys will include smaller areas in which a significantly larger number of images are obtained and deeper imaging stacks will be created. For the LSST deep-drilling fields (DDFs), Ivezić et al. (2019) describes a potential observing strategy that would obtain ∼25 30 s consecutive visits in the griz filters every other night for four months in order to find high-z supernovae. This would generate up to an additional 1500 visits in each filter compared to the WFD main survey. There will be at least four LSST DDFs, totaling ∼40 deg². Since Graham et al. (2018) demonstrated that the u- and y-band filters are important for the quality of photometric redshifts, we assume that the observing strategy for any DDF to be used as a photo-z spectroscopic training set would include, e.g., 500 visits in each of the filters u and y, and 1000 visits in each of the filters griz. The 5σ limiting magnitudes of such a DDF deep stack would be 27.3, 28.8, 28.4, 27.8, 27.1, and 25.5 in filters ugrizy, respectively. Euclid is projected to have at least two deep fields that total ∼ 40 deg² in which the detection limit extends by ∼2 mag (e.g., Joachimi 2016). The 5σ limiting magnitudes of the Euclid deep fields would be 26.0, 26.2, and 25.9 mag in each of the filters YJH.

We simulate a deep training set with apparent observed magnitudes using these 5σ limiting magnitudes and apply cuts in apparent magnitude at these 5σ limits in all filters except the i band, where we retain the cut of i < 25 mag because obtaining a representative set of spectroscopic redshifts for galaxies fainter than this is difficult and not guaranteed.¹⁶ The i < 25 mag restriction also keeps this deep training set approximately matched to the test set in terms of its i-band magnitude and redshift distribution, which helps to avoid introducing a bias in the CMNN photometric redshift results. The photometric quality of the training set is thus improved in two ways: (1) the photometric errors and observational scatter are reduced and (2) more galaxies are detected in more filters. We use a test set of galaxies with simulated photometric quality equivalent to a 10 yr LSST survey and the main Euclid survey. For both test and training sets, the LSST y-band filter is used instead of the Euclid Y band.

Figure 8 demonstrates the impact of using a deep training set on the photo-z results by plotting the statistical measures of standard deviation, bias, and fraction of outliers as a function of redshift. As in the previous section, the thin lines represent the results when the NIR photometry is included for all galaxies without a restriction, and the fraction of galaxies for which the NIR photometry is included in the photo-z estimates for this simulation is shown in the bottom panel of Figure 7. As expected, the standard deviation, bias, and fraction of outliers are all improved by the use of a deeper training set and the inclusion of Euclid photometry. The greatest impact is seen in the intermediate- to high-z galaxy bins (z_phot > 1), where the standard deviation is improved by ∼25% with the use of a deep training set and then an additional ∼25% by the inclusion of Euclid photometry—but there is also significant improvement over the full redshift range of 0.3 < z_phot < 3.0. Note that the bias in the high-z bins switches from a slight overestimate (bias < 0) to a slight underestimate (bias > 0) when a deep training set is used. The alleviation of overestimates with a deeper training set is likely because the CMNN subsets of training galaxies will contain fewer high-z interlopers when the photometric quality is improved.

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To explore the potential impact of a deeper training set on the photo-z results earlier in the LSST survey, we repeat the simulation with test sets of a photometric quality equivalent to 2 and 5 yr, with and without Euclid JH photometry. We find that the deeper training set alone (no Euclid included) mitigates the bias at low- and intermediate redshifts at years 2 and 5 (the unmitigated bias can be seen in the middle plots of Figure 6). The use of a deeper training set at earlier times also helps to lower the fraction of outliers, but not as much as the addition of Euclid data. The biggest impact of the deep training set at early times is in the standard deviation: the year 2 results are significantly reduced across the full redshift range, approaching the year 5 results for z_phot > 1.7, and they are better than the year 5 results for z_phot < 0.6. When we use both a deep training set and Euclid photometry, the photo-z standard deviation at year 2 (5) is equivalent to or better than the results at year 5 (10) across the full redshift range (0.3 < z_phot < 3.0).

Rhodes et al. (2017) propose that LSST cover an additional northern survey area of ∼ 3000 square degrees in order to increase the amount of overlap with Euclid. This proposed northern survey is shallower than the LSST main survey, with a total of 43 visits in filters griz. We test what the photo-z results would be for this northern survey by simulating photometry with 6, 13, 13, and 11 visits in each of filters g, r, i, and z, respectively (i.e., distributing the 43 visits using the same relative fractions for each filter as the WFD survey). The LSST 5σ detection limits in griz for this shallow survey would be 26.0, 26.1, 25.4, and 24.6 mag, respectively. This is significantly deeper than PanSTARRS, which goes to 23.3, 23.2, 23.1, and 22.3 mag in its griz filters (Chambers et al. 2016). For now we ignore the degradation of the photometry due to the fact that most of these visits will be at higher airmass (all photometric errors used in this work are for airmass 1.2, as described in Section 2.2.2). This is acceptable here because our main goal is to evaluate how NIR photometry affects the photo-z in this shallow northern survey, and not the absolute quality of photo-z results for such a survey.

We simulate a test set with apparent observed magnitudes using the LSST 5σ limiting magnitudes quoted above and the Euclid 5σ limiting magnitudes for filters YJH. Note that Euclid Y-band is used here because there would be no LSST y-filter coverage in this shallow northern survey. We apply cuts in apparent magnitude of i < 25 mag and at the 5σ detection limit for all other LSST and Euclid filters. For the training set we simulate photometry using the LSST 10 yr and Euclid main survey 5σ limiting magnitudes, and apply cuts in apparent magnitude of i < 25 mag and at the 5σ limits for the other filters. Note that in this simulation the training set is deeper than the test set, which could lead to a bias in the photo-z results.

In Figure 9 we show the statistical measures of standard deviation, bias, and fraction of outliers for photometric redshifts based on the photometry of this proposed shallow northern survey (and for context we also show the LSST 10 yr equivalent results). As in previous sections, the thin lines represent the results when the NIR photometry is included for all galaxies, regardless of whether it might deteriorate the photo-z estimate. We find that the shallow northern survey is similar in depth to the LSST year 1 results (not shown in this work, but for reference see the year 2 results in Figure 6). Since plenty of science goals will be achievable with the LSST 1 yr data release photometric redshifts (e.g., as demonstrated by Graham et al. 2018), the same applies to a shallow northern LSST survey area. As also seen for the LSST year 2 results, the photo-z results for galaxies with z_phot > 1 are all improved when Euclid NIR is added to LSST photometry from a shallow northern survey. In particular, at redshifts 1 < z < 2.5 the addition of Euclid photometry to a shallow northern LSST survey significantly improves the photo-z results, decreasing the standard deviation, fraction of outliers, and absolute bias by up to ∼30% (Figure 6).

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Another option for this region will be to include photometry from the the Canada–France Imaging Survey (CFIS), which has been covering this area of the northern sky in the u-band with the Canada–France–Hawaii Telescope (CFHT) as a continuation of the Legacy for the U-band All-sky Universe¹⁷ program, for a wide variety of scientific applications. Ibata et al. (2017) present the first results from the CFIS u-band component, which has a 5σ limiting magnitude of ∼24.4 mag (i.e., approximately equal to 2 LSST u-band visits). To evaluate the impact of CFIS-u coverage for this shallow northern survey, we simulated photo-z results and include them in Figure 9. We find a small decrease in the standard deviation in the lowest redshift bin (z_phot ∼ 0.5) compared to using LSST and Euclid photometry only, and a more significant improvement at z_phot > 1.7 where the u-band data helps to resolve degeneracies between low- and high-z galaxies. Similar results for the impact on photo-z of adding CFHT u-band photometry to grizy survey data—in this case from the Hyper Suprime-Cam Subaru Strategic Program—were presented by Sawicki et al. (2019). They demonstrated that the photo-z were most improved at z < 0.75 and z > 2, e.g., with ∼25% and ∼40% reductions in standard deviation, respectively.

As a final note, if this shallow northern survey was done as an extension of the WFD survey area (and not, e.g., as a separate mini-survey), then it would remove approximately 1.2 visits per filter from all fields within the main survey's 18,000 square degrees. This is a small enough fraction that we do not simulate the overall impact of this loss of depth on the photo-z for the LSST main survey.

The survey area of overlap between LSST and WFIRST may only be 2200 square degrees, but this area will be observed to an unprecedented depth, which might enable alternative science goals. Here, we explore the impact on photo-z of adding WFIRST photometry to a deeper set of LSST galaxies that extends to i < 26.8 mag, the LSST i-band 5σ detection limit, instead of i < 25 mag as applied in the rest of this work. In this simulation, we keep the i < 25 mag cut applied to the training set because it is difficult to obtain large representative samples of galaxy spectra much deeper than that. We expect that this mismatch in depth between test and training set might induce a bias in the photo-z results: that the photo-z's are biased toward low-z_phot (under-predicted, on average) because the brighter training set has a lower average redshift.

In all of our simulations in this work, we impose the constraint that a galaxy must have three colors in order to obtain a photo-z estimate, but here, we furthermore constrain that those colors must be optical. This constraint prevents the situation where the photo-z results are deteriorated by a large number of faint, i > 25 mag galaxies that are detected only (or mainly) in the WFIRST filters—a population that would be cut from all of our other simulated test sets. This optical colors constraint ensures we are "comparing apples to apples" in this simulation and evaluating only the impact of adding WFIRST photometry to an LSST galaxy catalog (and not adding WFIRST-detected galaxies to an LSST catalog).

In Figure 11, we show the standard deviation and bias for a test set of i < 26.8 mag galaxies with LSST photometry only and when WFIRST YJHK photometry is included. For comparison, we also show the results for a test catalog with i < 25 mag. As expected, the overall photo-z results are poorer when fainter galaxies are included, but the addition of WFIRST photometry reduces the standard deviation by ≳60% for 1 < z_phot < 2. As also expected, using an i < 25 mag training set for this i < 26.8 mag test set induced a bias in the photo-z results at z_phot ∼ 1.5 and in the highest-z bins, but we find that the inclusion of the WFIRST photometry alleviates the bias. We do not show the plot for the fraction of outliers but find that, as expected, the addition of WFIRST photometry to a faint test set reduces the fraction of outliers by ∼60% at intermediate to high redshifts (z_phot > 1.5).

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In Section 3, we found that the additional degrees of freedom from adding NIR photometry could deteriorate the photo-z results, especially for test sets with poorer photometric quality. Despite this simulation including fainter i < 26.8 mag galaxies, we find no such deterioration when adding WFIRST photometry, and so, we do not need to apply any restrictions as was done in Section 3.

To explore which UV/u bandpass pair would be best for photometric redshifts, we simulate photo-z results with the CASTOR-Primary survey limits (Table 1), imposing i < 25 mag for the test and training set as usual. We show the standard deviation as a function of binned z_phot in Figure 12. As we found with the addition of Euclid photometry (Section 3), including NUV photometry for all galaxies caused a slight deterioration in the standard deviation for galaxies with 1.5 < z_phot < 2 (thin lines in Figure 12). To mitigate this, we only include the CASTOR photometry if it lowers the estimated photo-z uncertainty and is thus likely to improve the photo-z estimate. The statistical results after this restriction has been imposed are represented by the thicker lines in Figure 12. Figure 13 shows the fraction of galaxies that are detected by CASTOR and have their CASTOR photometry included in the photo-z estimate. We find that the CASTOR UV passband is useful for up to ∼40% of all simulated galaxies, and the deeper u band is useful for ∼40%–60% of simulated galaxies.

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The statistical measure of the standard deviation shown in Figure 12 demonstrates that all of the proposed CASTOR bandpasses could provide an improvement of ∼10% across the full redshift range. At low redshifts, z < 0.6, we see that the deeper u-band photometry provides a larger reduction in the standard deviation than the addition of the UV passband alone. This is because for low-z galaxies, information about the Balmer break at 4000 Å is conveyed via the u − g color, and the UV − u color is less indicative of redshift.

We find that, surprisingly, our restriction to only include CASTOR photometry when it decreases the photo-z uncertainty actually causes a small increase in the standard deviation in the lowest-z bins, when either the CASTOR UV/u_c or UV_d/u_w filter pair is used. This indicates that the CASTOR photometry can improve the reported precision of some photo-z estimates while degrading their accuracy (this issue is also discussed in Section 5.2). This could be mitigated by applying further conditions to our restriction, such as always including CASTOR photometry when it indicates z_phot < 0.5, but since the effect is minor, we avoid such additional complications at this time.

At intermediate redshifts (1 < z < 1.5), we see that all CASTOR filters provide an equivalent improvement to the standard deviation, a reduction of ∼15% compared to using the LSST filters alone, and in the highest-redshift bins, the UV/u_c performs a little better than UV_d/u_w. Since the UV/u_c pair provides a slightly lower standard deviation across the full redshift range of 0.3 < z < 3.0 compared to the UV_d/u_w pair, we continue the rest of this analysis with UV/u_c.

To further investigate the potential photo-z results in the CASTOR-Primary survey area with the UV/u_c passband pair, we add the CASTOR UV photometry and replace the LSST u-band with the CASTOR u_c photometry for all test- and training-set galaxies. As in the previous section, for all filters, we use the surveys' 5σ limiting magnitudes to simulate apparent observed magnitudes and errors, and we apply cuts at i < 25 mag and the 5σ limits. We do not replace the LSST g band with the CASTOR g_c band because, as is shown in Table 1, the 10 yr LSST main survey is predicted to be slightly deeper than the CASTOR-Primary survey in the g band.

In Figure 14, we show three panels representing the true versus photometric redshifts when: (1) only LSST ugrizy photometry is used, (2) the CASTOR-Primary UV and u_c photometry is included for all galaxies, and (3) the CASTOR photometry is included with the restriction that it must decrease the photo-z uncertainty (as described in Section 5.1). By comparing the bottom two panels with the top panel, we can immediately see how adding CASTOR data resolves degeneracies in galaxy color-space that cause outliers in certain regions of the z_true–z_phot plane (brown boxes). By comparing just the bottom two panels, we can see that our restriction inhibits the reduction of outliers in the low-z spurs at z < 0.5, indicating that in some cases, the CASTOR photometry improves the photo-z accuracy but not its precision, as discussed above.

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The statistical measures of photo-z quality are shown in Figure 15, which plots the standard deviation and fraction of outliers in bins of photo-z. The bias is not significantly impacted by CASTOR photometry and so is not shown. As described in the previous section, the thin lines represent results when the CASTOR photometry is included for all galaxies, without the restriction that the addition must result in a smaller photo-z uncertainty.

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The top panel of Figure 15 illustrates how adding CASTOR to LSST photometry reduces the standard deviation by ∼10% for galaxies with 0.3 < z_phot < 3.0. At low redshifts (z_phot ≲ 0.6), we find that the impact of CASTOR is equivalent to that of Euclid but that applying CASTOR photometry without restriction is more impactful and could reduce the standard deviation by ∼30%. At intermediate redshifts, 1 < z_phot < 2, we see that CASTOR provides a reduction of ∼15% in standard deviation compared to using the LSST filters alone. In the redshift range 1 < z < 1.5, the UV filter is passing over the UV-upturn from evolved hot stars (e.g., Greggio & Renzini 1990, 1999), and the flux from this UV-upturn provides redshift information via the UV − u color. By redshift z ∼ 1.5, the Lyα break has entered the UV bandpass, but the UV − u color continues to provide redshift information until z ∼ 2. At intermediate redshifts (1.4 < z_phot < 2), we find that the addition of Euclid is more effective at reducing the standard deviation.

In even the highest-redshift bins, we continue to see an improvement to the standard deviation when CASTOR UV photometry is included, despite the fact that Figure 13 shows that the UV only contributes to the photo-z estimate $\sim 10 \%$ of the time for galaxies with true redshifts z_true > 2. This is the result of the CASTOR photometry assisting with the identification of truly lower-z galaxies that have optical colors that are degenerate with higher-z galaxies. These galaxies would otherwise be assigned a high photo-z based on their optical colors alone, where they increase the scatter in the high-z_phot bins. In the bottom panel of Figure 15, we show that the improvements to the fraction of outliers are generally of a similar scale, except that CASTOR could reduce this metric by ∼50% for z < 0.5, and Euclid is more impactful for all galaxies with z > 1.

Generally, for both statistical measures and across all redshift bins, Figure 15 demonstrates how the addition of both CASTOR and Euclid could provide further improvements than either could on its own. This indicates that the correlations between galaxy UV and NIR photometry and redshift, which are put to use when we add CASTOR and Euclid data to the photo-z estimates, are independent of each other. The fact that each survey provides a similar impact on the photo-z results when added individually is just a coincidence, and CASTOR and Euclid would each deliver complementary information to the photo-z estimates.

The proposed CASTOR-Cadence survey would cover 20 square degrees and might overlap with the LSST DDFs, Euclid's deep fields, and/or the WFIRST survey area. Similar to Section 3.1, we evaluate the impact of using this region to build a deeper spectroscopic training set. For this simulation, we use a test set of galaxies limited to i < 25 mag, with photometric quality based on the 5σ limiting magnitudes of a 10 yr LSST survey, the CASTOR-Primary survey, and/or the Euclid main survey (all as quoted in Table 1). We use a training set of galaxies also limited to i < 25 mag, with photometric quality based on the 5σ limiting magnitudes of an LSST DDF (Section 3.1), the CASTOR-Cadence deep field (m_UV ∼ 29.25, m_u ∼ 28.95, and m_g ∼ 28.45 mag; Section 1), and the WFIRST survey (Table 1). Two aspects that are different about this simulation from all of those preceding it are that (1) the CASTOR g_c is replacing the LSST g band for the first time because the Cadence survey would be deeper than a stacked LSST deep-drilling field, and (2) the two NIR surveys' photometries are being mixed, with WFIRST used for the training set and Euclid for the test set.

In Figure 16, we demonstrate the impact of using a deeper training set based on the LSST DDFs only, when the CASTOR-Cadence photometry is included and when the WFIRST photometry is also included. We can see that a deeper training set that includes CASTOR photometry can significantly decrease the standard deviation and fraction of outliers at low (${z}_{\mathrm{phot}}\lt 1.5$) and high (z_phot > 2) redshifts. As expected, including the WFIRST photometry in this deep training set provides further improvements across all redshifts.

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We have simulated photometric redshift results for LSST galaxies that overlap with the Euclid-Wide survey area. We found that Euclid mainly improves the photo-z estimates at redshifts z > 1, as expected, because this is where the Balmer break is redshifted beyond the optical filters. Quantitatively, we found that the addition of Euclid 5σ detections to the LSST 10 yr catalog can reduce the standard deviation for galaxies with z > 1 (or within the full redshift range of 0.3 < z < 3.0) by ∼20% (∼10%), and the fraction of outliers by ∼40% (∼25%). We showed that Euclid would have a relatively larger positive impact on the photo-z estimates when added to the 10 yr LSST photometry than at earlier years of the survey and/or when Euclid detections down to 1σ are included (instead of 5σ). We conclude that reducing the standard deviation in the photo-z results is more efficiently done with deeper LSST ugrizy photometry than with Euclid data, but we emphasize that Euclid does offer the unique benefits of reducing the standard deviation at z > 1 and especially of reducing the fraction of outliers. Furthermore, we point out that observing for a longer time period with the LSST z and y filters would not recreate the benefits of including Euclid photometry. We demonstrated how a spectroscopic training set from overlapping LSST and Euclid DDFs can significantly improve the photo-z results and how the photo-z for a shallow northern LSST extension that overlaps with the Euclid-Wide survey would be scientifically useful (i.e., it would have an average standard deviation just ∼2 times higher than the LSST year 10 survey).

Compared to Euclid, WFIRST will provide deeper NIR photometric catalogs and one additional, redder filter (F184, which we have referred to as K), but it will have a significantly smaller overlap area with LSST. Within the overlapping area, we have demonstrated that the addition of WFIRST NIR photometry would provide a drastically improved photo-z: the standard deviation is reduced by ≳50% (∼25%) for galaxies with z > 1.5 (0.3 < z < 3), the bias becomes negligible for z > 1 (and is reduced by ∼30% for z < 0.5), and the fraction of outliers is reduced to just ∼2% for galaxies within 0.3 < z < 3.0. In particular, the catastrophic outliers caused by a degeneracy between optical colors of galaxies at low and high redshifts are almost entirely removed. We also considered the impact of adding WFIRST to a deeper LSST catalog (i < 26.8 instead of i < 25 mag) and found that WFIRST would provide critical improvements to the standard deviation, lowering it by 50% for intermediate redshifts.

The proposed CASTOR mission might provide deep UV-, u-, and g-band photometry that overlaps with the LSST main survey. We found that including the CASTOR-Primary survey photometry improves the LSST photo-z estimates at all redshifts, as expected, because detections in the bluer passbands help to break degeneracies in the optical colors of low- and high-redshift galaxies by identifying galaxies that are truly low z. In particular, we showed that including CASTOR photometry could reduce the standard deviation by ∼30% and the fraction of outliers by ∼50% at z < 0.5. We also demonstrate how further improvements to the photo-z estimates might be attained by building a deeper spectroscopic training set from a combination of the CASTOR-Cadence and LSST DDFs. Although we illustrated how both of CASTOR's potential UV- and u-band pairs could provide similar benefits to the LSST photo-z, at the time of this work's writing, the CASTOR passbands and surveys were still in the proposal stage, and they might continue to evolve from what we have assumed in this work.

Generally, we have found that increased photometric depth provides the largest potential reduction in the standard deviation for predicted LSST photo-z estimates, compared to including additional filters beyond the optical. However, we have also found that the addition of UV and/or NIR filters can significantly reduce the standard deviation, in particular, redshift ranges and, more importantly, that additional filters are absolutely necessary for reducing the fraction of 3σ and catastrophic outliers. We have demonstrated how the addition of NIR or UV photometry from Euclid or CASTOR, respectively, would each result in similar reductions for the standard deviation and fraction of outliers when included in LSST photo-z estimates individually. In addition, we have shown how Euclid and CASTOR each provide complementary information about a galaxy's redshift and, thus, how additional significant improvement to the photo-z estimates could be obtained by including data from both surveys. As a final note, the deeper NIR photometry from WFIRST should deliver superb photo-z estimates for LSST galaxies, albeit in the smaller area of overlap between the LSST and WFIRST surveys.

The Euclid, WFIRST, and CASTOR missions are all space telescopes, and there are additional merits to their data aside from just the improved photometric quality and expanded spectral range (e.g., spatial resolution). These aspects can also be used to improve the LSST photo-z, over and above what we have demonstrated (e.g., Pasquet et al. 2019). Doing so requires a photo-z estimator that has been constructed to ingest, and properly apply, the additional features of the data. Here, we suggest several options for obtaining even greater improvements for the LSST photo-z from the Euclid, WIFRST, and/or CASTOR surveys, which are beyond the scope of this work.

• 1.  
  With the CMNN estimator, all galaxies must have a photometric detection with appropriate errors in order to simulate photometric redshifts. A photo-z estimator that fully utilized non-detections might be better able to quantify the improvements offered by surveys that are significantly shallower than the LSST.
• 2.  
  With the CMNN estimator, adding extra filters can deteriorate the photo-z quality if the photometry is not constraining enough to balance the additional degrees of freedom. In this work, we have mitigated this issue by only adding extra filters when they are likely to improve the photo-z estimate. A photo-z estimator that takes a more sophisticated approach to the provisional addition of photometry might be able to show an even greater improvement in the photo-z quality.
• 3.  
  Photometric differences between filters that are similar, such as LSST's y and the Euclid or WFIRST Y, could be used to constrain the redshift of strong emission lines and, thus, provide a more precise photo-z for a subset of galaxy types.
• 4.  
  Euclid and WFIRST will have grism data, which we have not considered in this work but which could be used to further improve the photo-z estimates—either by providing additional SED information for test galaxies, or perhaps enhancing the training-set photometry.
• 5.  
  All space-based imaging surveys offer significantly better spatial resolution than ground-based imaging, and this could improve the photo-z by, for example, providing galaxy size and shape priors and/or improving the treatment of blended objects. This may require pixel-level analysis, whereas, here, we have worked only with mock catalogs.