Galaxy Ellipticity Measurements in the Near-infrared for Weak Lensing

Weak gravitational lensing (WL) is a slight deflection of light rays from distant galaxies when they propagate through the tidal gravitational field of intervening large-scale structure. The amplitude of the WL distortion can be used to map dark matter and measure dark energy by statistically quantifying the shear distortions encoded in the observed shapes of background galaxies, namely, galaxy ellipticities (e.g., Kaiser et al. 1995; Bartelmann & Schneider 2001). The ellipticities of galaxies are typically distorted only about 1% by WL (Troxel & Ishak 2015), so the WL signal in individual galaxies is challenging to detect. WL measurements thus rely on averaging over a very large sample to obtain the distortions and sufficiently unbiased estimates of galaxy shapes, which in turn require a correction for the impact of the point-spread function (PSF) of the telescope. In that sense, WL observations demand high-quality images because it requires a large number density of resolved galaxies and high signal-to-noise ratio (S/N), while minimizing the PSF corrections and related systematic uncertainties, with well-sampled PSFs (Massey et al. 2013; Schrabback et al. 2018).

Euclid (Laureijs et al. 2011) is a survey mission designed to understand the expansion and growth history of the universe and is scheduled to launch in the next decade. Euclid will image 15,000 deg² of sky in one broad optical band VIS spanning 550–920 nm and three additional near-infrared (NIR) bands (Y, J, and H). Euclid will detect cosmic shear with VIS by measuring ellipticities of ∼30 resolved galaxies per arcmin² with a resolution better than 018 (PSF FWHM) with 01 pixels. The near-infrared bands will primarily be used to derive photometric redshifts for the weak lensing (WL) sample, in conjunction with ground-based observations at visible wavelengths. The Euclid wide survey is expected to provide WL galaxy shape measurements for 1.5 billion galaxies with space-quality resolution.

To measure WL through surveys, one should measure galaxy ellipticities and its uncertainty, including systematics, very accurately. In particular, it is necessary to measure the shapes of typically faint and small, distant galaxies with high-S/N observations. In this work, we demonstrate that NIR bands result in a comparable or more precise galaxy ellipticity measurement compared to optical bands for WL studies despite their worse spatial resolution (03 compared to 018) and pixel sampling (03 versus 01 pixel scale). There are several advantages to using NIR bands (Tung & Wright 2017); first, NIR wavelengths sample the rest-frame optical light, which traces the older stellar population (hence the bulk of stellar mass) and is less affected by dust extinction. The VIS band covers the rest-frame UV and blue wavelengths, which predominantly traces emission from star-forming regions (Dickinson 2000). In particular, the shapes of galaxies as seen in the rest-frame UV are more clumped and irregularly distributed than older stellar populations. The second advantage is that galaxies in the NIR bands have an intrinsically smoother light distribution resulting in a lower shape noise than in the optical (Schrabback et al. 2018). Third, NIR images of galaxies have a higher surface brightness with more than nine times the number of source photons per pixel, based on a calculation using images in this study; this is at least partly due to the relative importance of the bulge compared to the disk as a function of wavelength. Finally, we find that the NIR bands are sensitive to a larger number density of distant galaxies than the VIS band (see Section 2).

In this paper, we study the shapes of the galaxy sample expected from Euclid-quality imaging and forecast how we can improve the shape measurement by using co-added NIR images.³ To do that, we select galaxies from HST/Cosmic Assembly Near-infrared Extragalactic Legacy Survey (CANDELS; Grogin et al. 2011; Koekemoer et al. 2011) observations satisfying the Euclid sensitivity limits and simulate Euclid-resolution images. The structure of this paper as follows. The sample selection using CANDELS data is introduced in Section 2. We describe the procedure of simulating Euclid-quality optical and NIR images from HST images in Section 3. In Section 4, we explain how GALFIT (Peng et al. 2002) is used to measure the ellipticity of galaxies after accounting for the PSF and compare the ellipticities obtained from GALFIT in simulated Euclid and CANDELS images. Finally, we present our conclusions in Section 5.

We select a sample of galaxies at optical and near-infrared (NIR) wavelengths from the HST/CANDELS survey that closely resembles the Euclid WL sample. Among the five CANDELS fields, we use the GOODS-S and GOODS-N fields that include the CANDELS Deep survey and cover about 340 arcmin² in V (0.606 μm), I (0.814 μm), J (1.25 μm), and H (1.6 μm). These fields are several magnitudes deeper than the Euclid survey. We use CANDELS photometric redshifts measured for all galaxies by Dahlen et al. (2013), unless spectroscopic redshifts are available. For the WL shape measurement, the Euclid survey will detect galaxies in a broad optical R+i+z band (VIS: 0.55–0.92 μm) down to 24.5 mag (10σ). It will use three additional NIR bands (Y, J, H in the range of 0.92–2.0 μm) reaching AB mag 24 (5σ) in each. To achieve the required dark energy figure of merit through WL, the surface density of resolved galaxies needs to be at least 30 arcmin⁻² (Euclid Red book; Laureijs et al. 2011).

We start by replicating the Euclid expected sensitivity selection on the CANDELS catalogs. We find that I < 24.5 AB mag results in about 30 galaxies per arcmin² with a mean redshift of ∼0.9, which is consistent with the Euclid requirement. Applying the Euclid H < 24 mag selection on the CANDELS NIR sample, results in a mean z ∼ 1.1 with about 37 galaxies per arcmin². The redshift distribution of each sample selection is shown in Figure 1. One clear advantage of the NIR is at z > 1, where the NIR bands select many more galaxies than the optical. This suppresses the shape noise induced by the intrinsic ellipticities of distant galaxies if the individual ellipticity uncertainties were similar to that in the optical; we assess the veracity of this in the following sections. In this study, we specifically use galaxies at a redshift range of 0.5 ≤ z < 3 to compare ellipticities estimated from optical and NIR images.

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Table 1.  Number of Galaxies per 340 arcmin² (and Number of Galaxies per arcmin²) in Five Redshift Bins between $0.5\leqslant z\lt 3.0$

	Total	$0.5\leqslant z\lt 0.7$	$0.7\leqslant z\lt 1.0$	$1.0\leqslant z\lt 1.3$	$1.3\leqslant z\lt 1.9$	$1.9\leqslant z\lt 3$
	(N/arcmin2)
VI (I < 24.5 and Re > 01)	4634	1318	1571	899	529	317
	(13.6)	(3.9)	(4.6)	(2.6)	(1.6)	(0.9)
JH (H < 23.5 and Re > 015)	4770	1042	1352	1004	903	469
	(14.0)	(3.1)	(4.0)	(2.9)	(2.6)	(1.4)

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We simulate Euclid VIS and NIR images using HST/CANDELS V (0.606 μm), I (0.814 μm), J (1.25 μm), and H (1.6 μm). Each of the Euclid VIS, Y, J, and H bands will have four images taken per unit area of sky with 01 pixel scale in the VIS and 03 pixel scale in the NIR bands (Laureijs et al. 2011). For the VIS images, we combine the CANDELS V- and I-band images that span the bandwidth of the Euclid VIS band, 0.55–0.92 μm. In both the VIS and NIR bands, there will be significant correlated noise if co-added images are made on a finer pixel scale with just four frames. By combining the J- and H-band images, we can both increase the signal to noise in the NIR and drizzle on a factor of two oversampled pixel scale with a point kernel (e.g., CANDELS WIDE survey, Grogin et al. 2011; HST/ACS COSMOS, Rhodes et al. 2007), thereby minimizing the impact of correlated noise. We therefore combine the CANDELS J- and H-band images (Euclid J: 1.16–1.58 μm, H: 1.52–2.04 μm). We estimate that the Euclid survey strategy results in a median of between 10 and 11 valid frames per pixel when combining all three bands, Y, J, and H. However, the PSF undersampling in the shortest-wavelength band and color gradients across such a wide wavelength range may introduce other systematics. The impact on the undersampled PSF as a result of the drizzling and the Euclid dither strategy is beyond the scope of this work and is currently being investigated. Furthermore, since the CANDELS Y-band imaging does not cover the entire GOODS-S and -N fields, we avoid including the Y-band in this analysis.

The step-by-step procedure to simulate the Euclid-quality images is outlined below:

• 1.  
  Produce cutouts of science and noise images (rms) for each galaxy from the large HST/CANDELS V, I, J, and H mosaics.
• 2.  
  Combine V and I or J and H by weighting each pixel according to the weight map (inverse variance), i.e., ${f}_{\mathrm{comb}}=({f}_{1}{w}_{1}+{f}_{2}{w}_{2})/({w}_{1}+{w}_{2})$, where ${f}_{\mathrm{1,2}}$ and w_1,2 are the pixel values of science image and weight map, and w_1,2 ∼ 1/rms_1,2² . Because each pixel value has noise associated with it, and the noise is somewhat heterogeneous due to the observing strategy, an inverse variance weighting is the optimal approach to combine images and increases the signal to noise of co-added images (Koekemoer et al. 2011).
• 3.  
  Rebin the pixel scale from 006 of CANDELS to 01 (V+I) or 015 (J+H). By using co-added J+H images, which will double the number of images, the co-added NIR images can be drizzled onto a 015 pixel scale, half of the original NIR pixel scale. This is challenging to do for VIS since only four frames will be taken.
• 4.  
  Smooth the combined images with a Gaussian kernel to correct for the difference in PSF FWHM between HST and Euclid (for optical, 01 versus 018; for NIR, 018 versus 03).
• 5.  
  Make noise maps for the V+I and J+H following a random Gaussian distribution with 1σ measured from the quoted sensitivity of Euclid VIS and NIR images. For the VIS images, the sensitivity is 24.5 mag at 10σ (estimated from an extended source with a 03 radius; Cropper et al. 2012). For the NIR images, the sensitivity is 24 mag at 5σ in each band (measured from a point source), respectively. By combining the J and H bands, the effective sensitivity is therefore 24 AB mag at 7σ.
• 6.  
  Add the noise maps to the images to obtain the simulated Euclid VI(V+I) and JH(J+H) images. Since CANDELS' background noise is negligible (more than 50 times smaller than that of Euclid), we do not remove the noise in the CANDELS images before adding Euclid noise maps.

A few galaxies in the sample (1% and 3% in JH and VI, respectively) are not observed in J and V bands because the CANDELS coverage of the field at different bands varies slightly. Thus, after excluding these sources, we have 7248 galaxies for VI and 9887 galaxies for JH. For illustrative purposes, the images of five galaxies in the CANDELS I and H bands, the simulated Euclid VI and JH images, and their GALFIT fits are shown in Figure 2.

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4.1. Masking Sources

4.2. GALFIT

4.3. Comparison of Ellipticities between CANDELS and Euclid-quality Images

Typically, only galaxies with a size comparable or larger than the PSF have a well-measured shape, so the shape of the smallest galaxies becomes ill defined. Also, as the S/N decreases, the ability to measure galaxy shapes decreases since the imaging data are only sensitive to the highest surface brightness regions of the galaxy. Therefore, the galaxy samples for shape measurement require a lower limit to the S/N of about 10, and the radius of the galaxy is larger than 1.25 times the PSF FWHM (Euclid Red book; Laureijs et al. 2011). In order to satisfy those requirements, we restrict the galaxy sample in the NIR band to have H < 23.5 mag (∼S/N > 11 for J+H), and a half-light radius (R_e) measured from GALFIT on the simulated Euclid JH image of larger than half of the NIR PSF FWHM of Euclid (${R}_{e}\,\gt$ 015). For the optical, we select galaxies having I < 24.5 mag and S/N(I) > 10 with a size limit, R_e measured from VI images >01. This is about half the FWHM of the PSF in the VIS band. In Figure 3, we show the distribution of R_e as a function of a redshift for the optical and NIR sample with $I\lt 24.5$ mag (left) and $H\lt 23.5$ mag (right), respectively. About 2.2% of optical and 9.8% of NIR-selected galaxies have ${R}_{e}\lt 0\buildrel{\prime\prime}\over{.} 1$ and ${R}_{e}\lt 0\buildrel{\prime\prime}\over{.} 15$, respectively. As a final sample for analyzing ellipticities, we use 4634 and 4770 galaxies for the optical and NIR, respectively. In Table 1, the number densities of the galaxy samples in the optical and NIR are listed at five different redshift bins. The relatively high S/N cut of $H\lt 23.5$ mag results in a similar total number density of galaxies with the VI band as also shown in Figure 1, but still translates to a higher number density by a factor of 1.4 at $z\gt 1$ (6.9 versus 5.1 arcmin⁻²).

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In order to study galaxy shapes, the complex galaxy ellipticity is typically used in WL studies (Miller et al. 2013; Schrabback et al. 2015). Using axis ratio (q) and position angle estimated from GALFIT, we compute the complex galaxy ellipticity (e) of our final sample, which is defined as

$$\begin{eqnarray}&&e={e}_{1}+{{ie}}_{2}=| e| {e}^{2i\phi },\end{eqnarray} \tag{ 1 }$$

where the modulus of the ellipticity ($| e|$) is defined as (1–q)/(1+q) and ϕ corresponds to the position angle of the major axis. We then compare complex galaxy ellipticities (e) measured from Euclid-quality images with the CANDELS values. Here, we consider CANDELS-measured ellipticity as the original ellipticity of an observed galaxy because of the much larger depth and better resolution compared to Euclid data. Through this comparison, we can investigate the robustness of the galaxy shape measurements on the simulated Euclid images.

In Figure 4, we plot differences in ellipticities between CANDELS and Euclid-quality data, Δe_α = (e_α of Euclid quality—e_α of CANDELS), as a function of e_α of CANDELS for both ellipticity components, $\alpha =1,2$, in the range of $-1\lt {e}_{\alpha }\lt 1$. Most galaxies scatter systematically around ${\rm{\Delta }}{e}_{\alpha }=0$ for both optical (purple) and NIR (green) with a median ∼0 at all redshift ranges considered in this study. The uncertainty in Δe quantifies how well the galaxy shapes are recovered with Euclid-quality images. The lowest redshift bin has a measured scatter (σ, the standard deviation of Δe_α) in the NIR, which is a factor of 1.2 larger than that in the VI band. However, the measured scatter in the NIR is similar to that in the VI at all other redshifts, $z\gt 0.7$. Overall, we find that the ellipticities of individual galaxies can be measured with a similar scatter from the Euclid VIS- and NIR-like images. There is a weak trend that the scatter of ${\rm{\Delta }}{e}_{\alpha }$ increases with redshifts in both selections. Galaxies are fainter and smaller at higher redshift, so the limited Euclid spatial resolution and sensitivity will result in a larger scatter in the measured shape. In particular, at higher redshifts, galaxies with larger ellipticities (in absolute values) tend to have larger discrepancies (see diagonal trends at $z\gt 1.3$). This is likely because highly elongated galaxies in CANDELS appear to be rounder and with less-constrained position angles at Euclid-quality resolution. This trend appears to be a bit stronger for the NIR high-z sample due to the pixelization in the JH data.

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In Figure 5, we compare the modulus of the ellipticity, $| e|$ from Equation 1, derived using the CANDELS I band to the simulated Euclid VI and CANDELS H band to the simulated Euclid JH. The ellipticities derived from the simulated Euclid images are correlated very well with the ellipticities derived from CANDELS images with a median of ${\rm{\Delta }}| e| =(| e|$ of Euclid quality—$| e|$ of CANDELS) ∼0 for both optical and NIR imaging. At z > 1.0, NIR yields a similar to lower scatter than the optical, while the scatter in VI-derived ellipticities is significantly smaller at $z\lt 0.7$. This trend of the uncertainty in ${\rm{\Delta }}{\text{}}e$ is more obvious in Figure 6(a). We compare the standard error (SE) of ${\rm{\Delta }}| e| (=\sigma ({\rm{\Delta }}| e| /\sqrt{N}$) at each redshift bin for JH (blue) and VI (red). At z < 1.0, the standard error of JH is 1.2–1.1 times larger than one of VI. But the trend reverses at z > 1.0 so that the standard error of JH is significantly smaller than VI by a factor of 1.5–1.3. As shown in Figure 5, the measured $\sigma ({\rm{\Delta }}| e| )$ of galaxies in JH is very similar to that of VI over the redshift range considered here; thus, the higher number density of galaxies at z > 1.0 drives a lower standard error of ellipticity differences in JH. In Figure 6(b), we show the median fractional error of ellipticity, which is defined as $\left(\mathrm{median}\ \mathrm{of}\ \tfrac{{\rm{\Delta }}| e| }{| e| [\mathrm{CANDELS}]}\right)/\sqrt{N}$, as a function of redshift. The small values indicate that the derived ellipticity with Euclid-quality JH and VI imaging is very close to the true value from CANDELS on average. We find that the performance of the JH band in galaxy ellipticity measurements is comparable to the VI at all redshifts despite the significantly worse spatial resolution of JH. This result is very similar to that derived by Tung & Wright (2017), who found that for a 1.2 m class telescope, the K_s band yields an ellipticity measurement error that is a factor of ∼3 smaller than in an R-band selected catalog, while the J band is a factor of 1.5–2.5 worse than the K_s-selected catalog. This is also consistent with the results of Schrabback et al. (2018), who found that ground-based K_s imaging with a PSF FWHM ∼ 035 yields an ellipticity dispersion for $z\gtrapprox 1.4$ galaxies, which is 0.76 times that of optically selected galaxy samples with single-orbit HST imaging. A comparison between ellipticities derived from the simulated Euclid JH data and CANDELS I-band data indicates a correlation; however, the I-band ellipticities are larger than that in the NIR and the scatter is larger than shown in Figure 5. This is likely because the I-band ellipticities are dominated by disk light while the NIR ellipticities are tracing a combination of disk and more-compact bulge light. Thus, if systematics arising from PSF undersampling and dither strategy on the NIR images can be accounted for in future work, the shape noise can be minimized by including ellipticity measurements from the NIR bands.

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We investigate galaxy ellipticities in simulated Euclid-quality optical (VIS) and near-infrared (NIR) images constructed from HST/CANDELS co-added V+I (VI) and J+H (JH) images. We select galaxies in CANDELS GOODS-S and -N fields (covering about 340 arcmin²) with photometry in the I and H bands at similar depths as the planned Euclid survey. In this study, we specifically use galaxies at a redshift range of 0.5 ≤ z < 3. After applying an S/N ⪆ 10 cut, the total number density of galaxies at 0.5 ≤ z < 3 is comparable in the NIR and optical; however, the NIR bands select 1.4 times higher number density of galaxies relative to the optical selection at z > 1.0, which enable us to reduce the statistical uncertainties in the shape measurements of distant galaxies. By co-adding Euclid-quality J- and H-band images, which double the number of frames and the exposure time, we can generate images at 015 pixel scale (half of the original Euclid NIR detector pixel scale) and achieve ${\rm{S}}/{\rm{N}}\gtrapprox 7$ at H < 24 mag and ⪆11 at $H\lt 23.5$ mag. Using GALFIT, we measure ellipticities of galaxies in CANDELS I- and H-band images with 006 pixel scale and the simulated Euclid VI and JH images with 01 and 015 pixel scale, respectively. We then compare ellipticities between CANDELS and simulated Euclid-quality images while considering HST/CANDELS ellipticity as the original ellipticity of a galaxy due to its superior depth and resolution.

A comparison between ellipticities derived from CANDELS and Euclid-quality VIS and NIR imaging shows that both wavelength ranges provide similar performance in measuring galaxy ellipticities at all redshifts included in this study despite the worse spatial resolution and pixel sampling of the NIR imaging. When combined with the higher source density in the NIR selection, we find that the standard error in NIR-derived ellipticities is about 30% smaller than the optical bands at z > 1.0, which implies a more precise ellipticity measurement than in the optical alone. Since the VIS and NIR galaxy shape measurements with Euclid have different fractional contributions of the bulge and disk, a combination of the two can improve the precision with which galaxy ellipticities are measured. The next step that is required before the NIR data can be used for WL studies is to assess how the drizzling affects both the undersampled telescope PSF and the correlated noise (see, e.g., Rhodes et al. 2007). However, even though the FWHM of the Euclid NIR imaging does not quite reach HST or WFIRST resolution, the NIR data provide a major advantage for WL measurements compared to optical ground-based observations that typically achieve a PSF FWHM ∼ 06–07 in good seeing conditions (e.g., Kuijken et al. 2015; Mandelbaum et al. 2018); while the latter provides good sensitivity to the WL signal with a median redshift of the sample of z ∼ 0.85, about half the galaxy sample will be unresolved due to the small size of galaxies, as shown in Figure 3, implying a higher statistical uncertainty in their ellipticities.

In conclusion, by using co-added J+H band Euclid-quality images, we show that the galaxy sample selected at NIR wavelengths yields a more precise ellipticity measurement, especially at high redshifts. This suggests that a careful evaluation of NIR shape systematics for future weak gravitational lensing surveys, such as with Euclid and WFIRST, should be undertaken.

This work is partly funded by NASA/Euclid grant 1484822 and is based on observations taken by the CANDELS Multi-Cycle Treasury Program with the NASA/ESA HST, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. The authors thank the anonymous referee for very useful comments that helped to improve the presentation of the paper. We also thank Lance Miller, Stefanie Wachter, Peter Schneider, Henk Hoekstra, and Jason Rhodes for thoughtful comments that improved this manuscript.