Milky Way Cepheid Standards for Measuring Cosmic Distances and Application to Gaia DR2: Implications for the Hubble Constant

Measurements of cosmic distances from standard candles form a cornerstone of our cosmological model. Yet even the best available standard candles require calibration of their absolute brightnesses via geometric distance measurements. Trigonometric parallax is the "gold standard" of such geometric distance measurements—the simplest, the most direct, and the most assumption-free. Previously, most Milky Way (MW) stars, including known examples of rare stars, were well out of range of even state-of-the-art 0.3–1 mas parallax measurements from Hipparcos and the Fine Guidance Sensor (FGS) aboard the Hubble Space Telescope (HST). Now we are entering a "golden age" of parallax determinations as most of the MW's stars will come into parallax range from relative astrometry measured at the μas level by the ESA mission Gaia (Gaia Collaboration et al. 2016a, 2016b, 2018).

The parallaxes of long-period Cepheids are among the most coveted because these variables can be seen with the HST in the host galaxies of Type Ia supernovae (SNe Ia) at D < 50 Mpc and used to calibrate their luminosities and the expansion rate of the universe (Riess et al. 2016, hereafter R16). Benedict et al. (2007) used the FGS on the HST to measure parallaxes to nine of the 10 Cepheids in the MW known at D < 0.5 kpc with individual precision of 8% and a sample mean error of 2.5%. However, all but one of these had periods <10 days, a range where Cepheids are too faint to be observed in most SN Ia hosts. Nearly all of the long-period Cepheids live at D > 1 kpc, demanding a parallax precision better than 100 μas for a useful measurement. Spatial scanning with the HST's WFC3 has provided relative astrometry with 30–40 μas precision to extend the useful range of Cepheid parallaxes to 2–4 kpc, measuring eight with P ≥ 10 days with an error in the mean distance of 3% and providing a calibration more applicable to extragalactic Cepheid samples (Riess et al. 2014; Casertano et al. 2016; Riess et al. 2018).

The Gaia mission is expected to measure the parallaxes of hundreds of MW Cepheids with a precision of 5–10 μas by the end of the mission. Such parallax measurements would support a ∼1% determination of the Hubble constant (H₀) provided the possible precision of the calibration of Cepheid luminosities is not squandered by photometric inaccuracy. To retain the precision of Gaia's Cepheid parallaxes when they are used as standard candles it is necessary to measure their mean brightness on the same photometric systems used to measure their extragalactic counterparts. By using such purely differential flux measurements of Cepheids along the distance ladder, it is possible to circumvent systematic uncertainties related to zeropoints and transmission functions which otherwise incur a systematic uncertainty of ∼2%–3% in the determination of H₀, nearly twice the target goal, even before including additional uncertainties along the distance ladder.

To forge this photometric bridge, in HST Cycle 20 (2012) we began observing MW Cepheid "standards" among the set of 70 with P > 8 days, A_H < 0.4 mag, V > 6 mag, and expected distances of D < 7 kpc, criteria which yield the most useful sample for calibration of the thousands of extragalactic Cepheids observed in the hosts of SNe Ia. These extragalactic Cepheids across the hosts of 19 SNe Ia and in NGC 4258 have all been observed in the near-infrared (NIR) with the WFC3-IR in filter F160W (similar to the H band) to reduce systematics caused by reddening and metallicity, and in optical colors F555W (similar to the V band) and F814W (similar to the I band), to form a reddening-free distance measure (Hoffmann et al. 2016; Riess et al. 2016). To measure the much closer and brighter MW Cepheids on the same photometric system and to mitigate saturation, we use very fast spatial scans, moving the telescope during the observation so that the target covers a long, nearly vertical line over the detector. We use a scan speed of 75 s⁻¹, corresponding to an effective exposure time of 0.005 s in the visible and 0.02 s in the IR, much shorter than the minimum effective exposure times possible with the WFC3 hardware. Scanning observations are also free from the variations and uncertainties in shutter flight time (for F555W and F814W with WFC3-UVIS) that affect very short pointed observations (Sahu et al. 2015). Spatial scans offer the additional advantage of varying the position of the source on the detector, which averages down pixel-to-pixel errors in the flat fields, and can also be used to vary the pixel phase, reducing the uncertainty from undersampled point-spread-function photometry. Finally, unlike ground-based photometry, which relies on calibrators in the same region of the sky, the HST can measure the photometry of MW Cepheids over the whole sky, without concern about regional variations in calibrators. Most of the observations were obtained in HST "SNAP" mode, which selects a subset of the targets to be observed based on scheduling convenience, thus essentially randomly with respect to intrinsic Cepheid properties. Observations were obtained for a total of 50 of the 70 Cepheids, which therefore constitute an unbiased subset of the full sample.

In Section 2 we present the three-filter spatial scan photometry of the 50 MW Cepheids observed in our HST programs, and compare them internally as well as with ground-based measurements in corresponding passbands. In Section 3 we carry out an analysis of the recently released Gaia DR2 parallaxes for our targets; using the precise and accurate HST photometry, we verify the existence and magnitude of a zeropoint offset for the Gaia parallaxes, and at the same time test current measurements of H₀. In Section 4 we discuss these results and the nature of the zeropoint issue.

The 50 MW Cepheids collected here were observed photometrically in several HST programs: GO-12879, GO-13334, GO-13686, GO-13678, GO-14206, and GO-14268 include photometric and astrometric measurements for 18 of the targets, while GO-13335 and GO-13928 are purely photometric SNAP programs. For eight of these targets, the photometric measurements have been reported in Riess et al. (2018); the photometry of the other 42 targets follows the same procedures. Here we summarize the key steps for convenience; the full description is in Riess et al. (2018).

• 1.  
  Fluxes are measured from the amplitude of the fits of the line-spread function to the extracted signal at every position along the scan; a 15 pixel minirow across the scan is used to perform the fit. The flux is divided by the effective exposure time, i.e., the pixel size divided by the scan rate. Pairs of direct and scanning mode images are used to calibrate out possible errors in the pixel size and scan rate, and provide the aperture correction applicable between scanning and staring mode observations. This offset has an error in the mean of 0.002–0.003 mag, depending on the filter.
• 2.  
  We multiply the measured flux by the local (relative) pixel area using the same pixel area map used for photometry of all point sources in staring mode; this corrects from flux per unit area to actual flux.
• 3.  
  We then need to correct for the differing sizes of the pixel length along the scan (Y) direction, which changes the effective exposure time seen at each location along the scan. This step partially reverses the correction in step 2; the net result of steps 2 and 3 is to multiply the fitted amplitude by only the relative pixel size perpendicular to the scan direction. Riess et al. (2018) compare pairs of scans of MW Cepheids in back-to-back exposures, and demonstrate a mean photometric error per scan observation of 0.007, 0.003, and 0.001 mag in F160W, F555W, and F814W, respectively. For the sample of 50 Cepheids presented here, the mean number of epochs per filter is between 2 and 3.
• 4.  
  Finally, we apply a correction for the light-curve phase, i.e., the difference between each Cepheid's magnitude at the observed phase and the magnitude at the epoch of mean intensity of its light curve. These phase corrections are derived from ground-based light curves of these Cepheids in filters with wavelengths best corresponding to the WFC3 filters with their sources given in a table in the Appendix. The phase corrections are calculated in the HST system after the ground-based light curves are transformed to this system using the transformations given in R16. Because the phase corrections are relative quantities, they do not have a zeropoint and they do not change the zeropoint of the light curves, which remain on the HST WFC3 natural system. The uncertainties in these phase corrections depend on the quality of the ground-based light curves; the average uncertainty in the magnitude corrections is 0.024, 0.020, and 0.018 mag per epoch in F555W, F814W, and F160W, respectively. The empirical scatter between multiple measurements for the same target—typically 4–5 for the targets in Riess et al. (2018)—is consistent with the estimated uncertainties. The mean uncertainty in the light-curve mean magnitude for these 50 Cepheids is 0.021, 0.018, and 0.015 mag in F555W, F814W, and F160W, respectively. The internal agreement between individual epochs, each corrected to the mean (for Cepheids with three or more epochs), is shown in Figure 1 and includes both the photometry errors and the phase-correction uncertainties.

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For distance measurements and for the determination of H₀, it is useful to convert these three bands to the reddening-free Wesenheit magnitudes (Madore 1982) used by R16 for measuring extragalactic Cepheids in the hosts of SNe Ia:

$$\begin{eqnarray}&&{m}_{H}^{W}={m}_{F160W}-0.386({m}_{F555W}-{m}_{F814W}).\end{eqnarray} \tag{ 1 }$$

These 50 ${m}_{H}^{W}$ values have a mean uncertainty of 0.019 mag, including photometric measurement errors, phase corrections, and error propagation to the Wesenheit magnitude, corresponding to approximately 1% in distance; at the mean expected parallax of 400 μas this represents a mean uncertainty of 4 μas in the predicted parallax. At this level of precision, both the breadth of the instability strip at 0.04–0.08 mag in ${m}_{H}^{W}$ (Persson et al. 2004; Macri et al. 2015) and the expected parallax uncertainties by the end of the Gaia mission (5–14 μas) will still dominate the determination of individual Cepheid luminosities. Some of these Cepheids have been suggested as possible binaries, but in general we do not automatically exclude possible binaries from consideration. At a typical MW Cepheid distance of 2.5 kpc, companion separations of less than 01 for the HST WFC3 UVIS channel or <400 au are unresolved and thus included with the measured Cepheid flux. This contribution, while small, is statistically matched in extragalactic Cepheids and thus cancels in the use of Cepheid fluxes along the distance ladder. For wider binaries, 400–4000 au, Anderson & Riess (2017) estimate that the effect on the photometric calibration of Cepheids is on the order of 0.004% (in distance) and thus negligible.

In Table 1 we provide the photometric measurements of these 50 Cepheids for WFC3 F555W, F814W, F160W, and ${m}_{H}^{W}$.⁸

Table 1.  Photometric Data for MW Cepheids

Cepheid	Log Period/	F555W	σ	n	F814W	σ	n	F160Wa	σ	n	${m}_{H}^{W}$,b	σ	${\pi }_{{\rm{R}}16}$ c	σ
	day	(mag)		(epochs)	(mag)		(epochs)	(mag)		(epochs)	(mag)		(mas)
AA-GEM	1.053	9.9130	0.029	1	8.542	0.025	1	7.348	0.017	1	6.871	0.023	0.254	0.005
AD-PUP	1.133	10.015	0.028	1	8.675	0.023	1	7.488	0.020	1	7.023	0.024	0.210	0.005
AQ-CAR	0.990	8.9836	0.020	2	7.854	0.009	2	6.766	0.007	3	6.382	0.011	0.350	0.007
AQ-PUP	1.479	8.8671	0.018	2	7.120	0.014	2	5.487	0.013	4	4.864	0.016	0.338	0.007
BK-AUR	0.903	9.5609	0.036	1	8.220	0.038	1	7.015	0.021	1	6.549	0.029	0.369	0.008
BN-PUP	1.136	10.051	0.033	1	8.505	0.017	1	7.198	0.015	1	6.653	0.021	0.248	0.005
CD-CYG	1.232	9.1207	0.011	3	7.468	0.012	3	5.900	0.012	5	5.314	0.014	0.398	0.008
CP-CEP	1.252	10.757	0.015	1	8.638	0.052	1	6.871	0.022	1	6.105	0.030	0.268	0.006
CR-CAR	0.989	11.750	0.019	1	9.973	0.018	1	8.384	0.014	1	7.750	0.017	0.187	0.004
CY-AUR	1.141	12.052	0.012	1	9.953	0.020	1	8.106	0.025	1	7.348	0.027	0.179	0.004
DD-CAS	0.992	10.036	0.007	3	8.523	0.011	3	7.108	0.012	4	6.576	0.013	0.319	0.006
DL-CAS	0.903	9.1059	0.019	1	7.569	0.022	1	6.238	0.018	1	5.697	0.021	0.547	0.011
DR-VEL	1.049	9.7083	0.034	1	7.770	0.020	1	6.183	0.021	1	5.487	0.026	0.484	0.011
GQ-ORI	0.935	8.7199	0.020	1	7.632	0.024	1	6.523	0.032	1	6.155	0.034	0.422	0.010
HW-CAR	0.964	9.2782	0.016	2	8.007	0.013	2	6.798	0.005	3	6.359	0.009	0.368	0.007
KK-CEN	1.086	11.598	0.017	1	9.862	0.021	1	8.292	0.015	1	7.674	0.018	0.167	0.003
KN-CEN	1.532	10.062	0.023	2	7.924	0.017	2	5.856	0.006	5	5.083	0.013	0.282	0.005
RW-CAMd	1.215	8.8673	0.015	1	7.044	0.014	1	5.451	0.021	1	4.799	0.022	0.517	0.011
RW-CAS	1.170	9.3719	0.021	1	7.863	0.016	1	6.483	0.022	1	5.952	0.024	0.326	0.007
RY-CAS	1.084	10.075	0.019	1	8.333	0.040	1	6.715	0.010	1	6.094	0.020	0.347	0.007
RY-SCO	1.308	8.2067	0.012	3	6.206	0.010	3	4.408	0.010	3	3.688	0.012	0.751	0.014
RY-VELd	1.449	8.5234	0.036	1	6.757	0.016	1	5.211	0.017	1	4.581	0.023	0.403	0.009
S-NOR	0.989	6.5779	0.011	1	5.410	0.012	1	4.391	0.012	1	3.992	0.014	1.053	0.021
S-VUL	1.839	9.1668	0.008	3	6.862	0.012	3	4.885	0.010	4	4.047	0.011	0.287	0.006
SS-CMA	1.092	10.121	0.012	4	8.444	0.008	4	6.894	0.011	3	6.299	0.012	0.312	0.006
SV-PERd	1.046	9.2186	0.016	1	7.760	0.014	1	6.435	0.027	1	5.924	0.028	0.397	0.009
SV-VEL	1.149	8.7316	0.026	1	7.302	0.009	1	6.024	0.010	1	5.524	0.015	0.409	0.008
SV-VUL	1.653	7.2675	0.047	1	5.648	0.033	1	4.214	0.027	1	3.641	0.035	0.457	0.011
SY-NOR	1.102	9.8284	0.023	1	7.925	0.038	1	6.214	0.013	1	5.531	0.022	0.438	0.009
SZ-CYG	1.179	9.6209	0.013	2	7.756	0.017	2	6.004	0.008	3	5.336	0.012	0.427	0.008
T-MONd	1.432	6.0680	0.023	1	4.828	0.016	1	3.725	0.021	1	3.298	0.024	0.746	0.016
U-CAR	1.589	6.3852	0.038	1	4.967	0.023	1	3.768	0.019	1	3.272	0.026	0.596	0.013
UU-MUS	1.066	9.9212	0.024	1	8.457	0.025	1	7.108	0.010	1	6.595	0.017	0.283	0.006
V0339-CEN	0.976	8.8402	0.024	1	7.321	0.016	1	5.990	0.024	1	5.455	0.026	0.548	0.012
V0340-ARA	1.318	10.460	0.024	1	8.554	0.014	1	6.808	0.012	1	6.124	0.016	0.241	0.005
VW-CEN	1.177	10.379	0.031	1	8.718	0.023	1	7.158	0.010	1	6.569	0.018	0.243	0.005
VX-PER	1.037	9.4589	0.008	4	7.906	0.006	3	6.470	0.009	5	5.922	0.010	0.403	0.008
VY-CAR	1.276	7.6162	0.014	5	6.253	0.007	4	4.991	0.004	6	4.517	0.007	0.538	0.010
VZ-PUP	1.365	9.7715	0.033	1	8.262	0.022	1	6.931	0.017	1	6.400	0.023	0.198	0.004
WX-PUP	0.951	9.1909	0.030	1	7.944	0.012	1	6.807	0.010	1	6.378	0.016	0.372	0.007
WZ-SGR	1.339	8.2021	0.012	6	6.481	0.013	6	4.858	0.009	4	4.245	0.011	0.554	0.011
X-CYG	1.214	6.5295	0.020	1	5.230	0.049	1	4.080	0.033	1	3.630	0.039	0.887	0.023
X-PUP	1.414	8.6949	0.019	3	7.128	0.010	3	5.628	0.008	4	5.075	0.012	0.338	0.006
XX-CAR	1.196	9.4627	0.027	1	8.067	0.015	1	6.833	0.022	1	6.346	0.025	0.261	0.006
XY-CAR	1.095	9.4660	0.011	4	7.927	0.009	3	6.455	0.006	6	5.913	0.008	0.371	0.007
XZ-CAR	1.221	8.7725	0.017	3	7.217	0.006	3	5.770	0.007	4	5.221	0.010	0.422	0.008
YZ-CAR	1.259	8.8644	0.016	3	7.401	0.007	3	5.991	0.013	5	5.478	0.015	0.354	0.007
YZ-SGR	0.980	7.4662	0.021	1	6.176	0.014	1	5.103	0.020	1	4.657	0.022	0.786	0.017
Z-LAC	1.037	8.5686	0.022	1	7.157	0.015	1	5.917	0.018	1	5.424	0.021	0.507	0.011
Z-SCT	1.111	9.7535	0.019	2	8.079	0.021	2	6.513	0.008	3	5.918	0.014	0.362	0.007

Notes.

^aDoes not include addition of 0.052 ± 0.014 mag to correct CRNL 6.4 dex between MW and extragalactic Cepheids. ^bIncludes addition of 0.052 ± 0.014 mag to correct CRNL 6.4 dex between MW and extragalactic Cepheids. ^cπ_R16 = 10^{−0.2(μ−10)} where $\mu ={m}_{H}^{W}-{M}_{H}^{W}$ and M^W_H is the absolute Wesenheit magnitude determined from the Cepheid period and the distance scale from R16 where H₀ = 73.24 km s⁻¹ Mpc⁻¹ as discussed in the text. ^dNot used in the final analysis; see the text.

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The best-fit solution from R16 yields a calibration of the Cepheid P–L relation of

$$\begin{eqnarray}&&{M}_{H}^{W}=-5.93-3.26(\mathrm{log}P-1).\end{eqnarray} \tag{ 2 }$$

Employing the derived periods in Table 1 yields the values of M^W_H, and combined with the apparent Wesenheit magnitudes on the WFC3 system (${m}_{H}^{W}$) we derive distance moduli of

$$\begin{eqnarray}&&\,\mu ={m}_{H}^{W}-{M}_{H}^{W}\end{eqnarray} \tag{ 3 }$$

and the expected parallax

$$\begin{eqnarray}&&{\pi }_{{\rm{R}}16}={10}^{-0.2(\mu -10)}\end{eqnarray} \tag{ 4 }$$

in mas given in Table 1. With negligible uncertainties in the periods, the mean uncertainties in the predicted parallaxes are ∼2%–3% in distance due to the width of the instability strip. These expected parallaxes are on the scale in which H₀ = 73.24 km s⁻¹ Mpc⁻¹ as obtained in the same best-fit solution from R16.

This set of photometry offers a number of distinct advantages over ground-based magnitudes. By measuring all Cepheids along the distance ladder (and in both hemispheres) with a single, stable photometric system, the HST WFC3, we can largely eliminate the propagation of zeropoint and bandpass uncertainties among Cepheid flux measurements. This is especially important in the NIR where individual system zeropoints are typically based on only a handful of historical standards, systematic uncertainties are ∼0.02–0.03 mag (Riess 2011), and the relative systematic differences between two systems can be expected to be ∼0.03–0.04 mag. To illustrate these differences, we compare the HST WFC3 system photometry with ground-based equivalents, transformed into the same system using the conversions of R16. Ground-based observations in V, I, J, H were obtained from the sources listed in the Appendix, and are rather inhomogeneous but the best available; the NIR measurements are primarily from three sources: Monson & Pierce (2011), Laney & Stobie (1992), and new observations obtained at CTIO. In Figure 2 we compare the ground-based mean magnitudes to the HST WFC3 values. We find mean differences (in the direction ground–HST) and a sample dispersion (SD) in F555W, F814W, F160W, and ${m}_{H}^{W}$ of 0.024 mag, SD = 0.032 mag, 0.038 mag, SD = 0.027 mag, −0.056 mag, SD = 0.048 mag, and −0.051 mag, SD = 0.052 mag, respectively; a few outliers (4, 2, 1, and 1, respectively) are marked in Figure 2. In the following we use only the reddening-free Wesenheit magnitude ${m}_{H}^{W}$.

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Restricting our analysis to HST magnitudes limits the sample of usable Cepheids; over 200 more have ground-based photometry, and in principle could be used for the same type of analysis, as was done for DR1 in Casertano et al. (2017). However, our sample is close to complete for the most relevant Cepheids, those with long periods. Moreover, at the much higher precision of DR2 versus DR1 parallaxes (roughly 40 versus 300 μas per Cepheid), further reduced by averaging across a Cepheid sample, photometric errors and systematics become dominant, as we will demonstrate in Section 3. Even at the current (DR2) precision, the quality of photometric information is paramount to obtain the best possible information from Gaia parallaxes.

The Gaia mission (Prusti 2012; Gaia Collaboration et al. 2016a, 2016b, 2018) is well positioned to revolutionize our knowledge of the luminosity scale of various stellar types, including those used to set the cosmic distance scale. By mission end, Gaia parallaxes for the Cepheids in our sample are expected to have errors of 5–14 μas, about 2% of their typical parallax; with tens of objects, the uncertainty in the Cepheid luminosity calibration will be ≪1%, negligible in the error budget for the local measurement of H₀ (R16).

With the release of DR2 (Gaia Collaboration et al. 2018, hereafter G18), the nominal statistical parallax errors for the Cepheids in our sample were expected to drop from ∼300 μas, typical of DR1, to ∼40 μas. These Cepheids are all in the brightness range 6.05 < G < 11.70 (mean magnitude; G is the natural passband of the Gaia astrometric detectors). They are fainter than the saturation limit at the shortest gating interval used (TDI gate 4, 16 lines; Gaia Collaboration et al. 2016b), and thus are not expected to be sigificantly affected by saturation effects. However, Lindegren et al. (2018; hereafter L18) and online material accompanying DR2⁹ identify significant systematic uncertainties which substantially reduce the present leverage of the DR2 Cepheid parallax measurements.

Perhaps the most significant issue with DR2 parallaxes identified in L18 is the existence of a significant parallax zeropoint error, i.e., a number which must be subtracted from all Gaia DR2 parallaxes. In principle, large-angle astrometric measurements, such as those carried out by Hipparcos and Gaia, yield an absolute parallax measurement, without the need for a correction from relative to absolute parallax. In contrast, narrow-angle parallax measurements, such as those using the HST (e.g., Benedict et al. 2007; Riess et al. 2014, 2018; Casertano et al. 2017; Brown et al. 2018), are only sensitive to relative parallaxes of stars within the same field, and require a correction to absolute parallax, often based on astrophysical information. However, as pointed out, e.g., by Michalik & Lindegren (2016), instrumental uncertainties associated with monitoring the large angle between observing planes can lead to systematic errors in the determined parallaxes. Specifically for Gaia, a variation in basic angle with period equal to the spin period of the satellite is difficult to correct on the basis of self-calibration procedures; in particular, Butkevich et al. (2017) show that the effect produced by a periodic variation of this nature is almost degenerate with a global shift of the parallaxes, resulting in a whole-sky systematic offset, i.e., a zeropoint error. Indeed, L18 consider the measured parallaxes for a carefully selected sample of over 500,000 quasars, whose parallaxes are expected to be completely negligible (<0.001 μas), and find that they have a mean value of −29 μas, with a small dependence on color and ecliptic latitude (their Figure 7). According to L18, "the actual offset applicable for a given combination of magnitude, color, and position may be different by several tens of μas." The quasars are primarily faint (G > 17 mag); thus, a possible magnitude dependence, suggested in their Figure 7 (left panel), is difficult to investigate. The distribution of corrected parallaxes π_corr = π_meas + 29 μas is fairly consistent with a normal distribution if their nominal errors are increased by ∼8% (see L18, Figure 8). We will return to the issue of the parallax zeropoint error later in this section.

Other potential systematics identified by G18 and L18 include the following.

• 1.  
  Uncharacterized systematic errors dominate over the ideally available precision in the post-fit astrometric residuals (see L18, Figure 9) in DR2 by a large factor for G < 12 mag, with the discrepancy increasing for brighter magnitudes. Note that a systematic deviation of parallax measurements as a function of Cepheid brightness would be somewhat degenerate with a luminosity scale determination because brightness is partially correlated with distance.
• 2.  
  A small proportion of individual parallaxes are "corrupted"; they can generally be identified by large positive or negative values and must be discarded.
• 3.  
  The statistical uncertainties may be underestimated by up to ∼30% for stars with G < 12 mag.

We note that the spatial correlation of parallax errors on the sky for DR2 is not very significant for the Cepheids in our sample, for which only two pairs are separated by less than 10°.

In light of these issues, there is likely no unique way to model the Cepheid sample while using the DR2 results to determine their luminosity scale. Rather we take a cautious, "common sense" approach to illustrate what such an approach can reveal at present. We anticipate a reduction to these systematic uncertainties through independent analyses of other classes of objects and from future Gaia data releases.

As a first, exploratory step we plot the DR2 parallaxes of the Cepheids in Table 1 against their uncertainties in Figure 3. It is immediately obvious that three of the 50 (RY-Vel, RW-Cam, and SV-Per) have anomalously high formal uncertainties, and two of their parallaxes define the extrema for the set. (Note that because of the excess noise formalism (Lindegren et al. 2012), large errors are often indicative of poor adherence to the model used, in this case a five-parameter, single-star astrometric model.) One of the three (RW-Cam) was also an outlier in the lower-precision DR1 data (Casertano et al. 2017). For SV-Per and RW-Cam, our HST spatial scan data demonstrate the presence of a companion within 02 of the Cepheid; see the insets in Figure 3. Both have reported UV excess from IUE spectra consistent with B8III companions (Evans 1994). The companions are the likely source of the anomalous astrometric solution. All three objects are excluded from further analysis.

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An additional, independent test can be carried out thanks to the existence of HST parallax measurements for 19 Cepheids, obtained using the FGS (Benedict et al. 2007) or WFC3 spatial scanning (Riess et al. 2018). The comparison between HST and Gaia DR2 parallaxes is shown in Figure 4 versus their mean Gaia G magnitude; error bars combine the errors in the HST and DR2 parallaxes. Two Cepheids, Y-Sgr and Delta-Cep, were excluded from this comparison because their Gaia DR2 values were negative, indicating they are likely corrupted. Delta-Cep is also a binary (Anderson et al. 2015), and its orbit will eventually be included in the Gaia solution in later releases. For one, ℓ-Car, its DR2 G mag was unrealistically faint, and so we plotted it at its ground-based value. The agreement is good for Cepheids with G > 6 mag, seven of eight of which fall within 1σ, but it becomes quite poor for Cepheids with G < 6 mag (even excluding the three mentioned), just one of 11 within 1σ. This suggests as expected that around G ≈ 6 mag, where the Gaia detectors are known to saturate, the DR2 parallaxes become much less reliable. We conclude that even with the possible maximum 30% enhanced Gaia DR2 errors the Cepheid parallaxes at G < 6 mag are not yet sufficiently well understood and should not be used in any quantitative analyses. To be safe, we also exclude from further comparisons the Cepheid T-Mon: with a mean magnitude G ≈ 6.1, it is the brightest in our sample (brighter by 0.3 mag in our own F555W data than the next one). Because of brightness variations in Cepheids, T-Mon is likely to have exceeded the saturation limit during some of its epochs of astrometric observations. A comparison of the eight Cepheids with G > 6 mag yields a DR2 parallax offset of −90 ± 21 μas but the comparison is strongly impacted by SS CMa. Excluding SS CMa yields an offset of −55 ± 25 μas. While informative and consistent with subsequent analyses, we do not make explicit use of the HST parallaxes when comparing the Gaia parallaxes to their photometric predictions to retain independence with the expectations based on R16 or Riess et al. (2018).

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After excluding four Cepheids with too large formal uncertainties (RY-Vel, RW-Cam, and SV-Per) or too close to the saturation threshold (T-Mon), we are left with 46 Cepheids with HST photometry and reliable Gaia DR2 parallaxes and uncertainties. Following the approach of Casertano et al. (2017), we determine for each Cepheid the expected parallax based on its photometry and the absolute magnitude derived from the Leavitt law (Leavitt & Pickering 1912), calibrated by R16 in the same photometric system. This parallax has a typical uncertainty of only a few percent. Figure 5 compares the measured DR2 parallax with the expected value; the comparison is made in parallax space to avoid issues related to the conversion of low signal-to-noise ratio parallaxes to magnitudes (Hanson 1979) which otherwise skews their likelihood in magnitude space (see also recommendations in Luri et al. 2018).

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As expected, the DR2 parallaxes are offset, on average, with respect to the predicted values. However, a cursory examination of Figure 5, and basic statistics on the differences between predicted and measured values, suggest a zeropoint offset in the same direction but somewhat larger (in absolute value) than the value of −29 μas for quasars reported by L18. Therefore, we proceed to constrain the parallax zeropoint internally from our sample, as discussed above. Fortunately, the Cepheids in our sample have a fairly narrow range of color (a dispersion in $F555W-F814W$ of 0.28 mag) and magnitude, so we will assume zeropoint variations are small across our sample.

At the same time, we consider a possible rescaling of the photometrically predicted distances (parallaxes) because these make direct use of the calibration of the distance ladder and attendant value of the Hubble constant, H₀ = 73.24 ± 1.7 km s⁻¹ Mpc⁻¹, from R16. A degree of tension exists between this value of H₀ and that determined from Planck cosmic microwave background (CMB) data in concert with the ΛCDM model, which yields H₀ = 66.93 ± 0.62 km s⁻¹ Mpc⁻¹ (Planck Collaboration et al. 2016). To the extent the DR2 data permit an independent determination of the Cepheid luminosity calibration, they can also help distinguish between these values of H₀.

Therefore we seek to optimize the value of

$$\begin{eqnarray}&&{\chi }^{2}=\sum \displaystyle \frac{{({\pi }_{\mathrm{DR}2,i}-\alpha {\pi }_{{\rm{R}}16,i}-{zp})}^{2}}{{\sigma }_{i}^{2}},\end{eqnarray} \tag{ 5 }$$

with the two free parameters, α and zp, representing the cosmic distance scale from DR2 relative to H₀ = 73.24 km s⁻¹ Mpc⁻¹ and the parallax zeropoint appropriate for the DR2 Cepheid measurements, in the direction of measured minus predicted parallaxes (consistent with the definition of L18).

To determine the individual σ_i we add in quadrature the photometric parallax uncertainty (mean of 0.02 mag or 4 μas in parallax), the intrinsic width of the NIR Wesenheit P − L relation (0.05 mag or a mean of 18 μas in parallax), and the nominal parallax uncertainty as given in the DR2 release. The mean of these σ_i is 39 μas (median 35 μas).

Minimizing the value of χ² gives zp = −46 ± 13 μas and α = 1.006 ± 0.033, with a value of χ² = 62.5 for 44 degrees of freedom. Confidence regions for the two parameters are shown in Figure 6. Although these two parameters are correlated, the range of Cepheid parallaxes, from 0.2 to 1 mas, breaks to some extent the degeneracy between α and zp and allows for their separate determination.

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This parallax zeropoint error, somewhat different (larger in absolute value) from the value determined by L18, is significant in comparison with the formal DR2 uncertainties of our sources. The uncertainty in the parallax zeropoint error has the potential to impact significantly any astronomical analysis based on DR2 parallaxes, especially when multiple sources are used, thus in principle reducing statistical uncertainties. The potential dependence of the zeropoint error on source properties suggested by L18, such as color or (possibly) magnitude, is especially relevant, as it suggests that applying the nominal L18 zeropoint correction, which was determined for blue, faint objects, may not be optimal for objects with different characteristics. Indeed, in L18 the offset measured from quasars appears to increase at the brighter end (14 < G < 16 mag) and at the redder end (G_BP − G_RP > 1 mag), where the offset fluctuates around a higher value of −50 μas, both directions that apply to Cepheids. The online documentation also indicates that the estimated parallax zeropoint depends on the sample of sources examined (Arenou et al. 2018), and the value determined in L18 should not be used to "correct" the catalog parallax values.

A more precise constraint on the zeropoint offset for use in other studies may be derived by fixing the value of α (e.g., to unity based on other geometric distance measurements to Cepheids from R16 which have a mean error of 1.4%) which results in a constraint of −46 ± 6 μas. That this value is more than 3σ from the value derived from relatively bluer and fainter quasars from L18 reinforces their finding that the parallax zeropoint offset can vary with sources' position, magnitude, and color, all quite different between MW Cepheids and quasars but in the right direction as suggested by the brightest and reddest quasars.

The value of α is quite consistent with unity, indicating that the predicted parallaxes, after accounting for the offset, are in good agrement with DR2, affirming the cosmic distance scale or the value of H₀ used to predict the parallaxes from R16. On the other hand, this value of α is inconsistent with the value of α = 0.91 needed to rescale the parallaxes to match the Planck CMB + ΛCDM value of H₀ at the 2.9σ confidence level (99.6% likelihood). Including the eight Cepheids with HST parallaxes from Riess et al. (2018) to help constrain the parallax offset gives a result of α = 1.035 ± 0.029 and α = 1.010 ± 0.029 after excluding the one Cepheid with a large difference between HST and Gaia DR2, SS CMa. These are 4.4σ and 3.4σ from the Planck CMB + ΛCDM value of H₀, respectively.

We also note that the value of χ² appears somewhat high for the 46 Cepheids and two fitted parameters (44 degrees of freedom), a value which would be exceeded by chance in 3.5% of trials. The bottom of Figure 6 shows the residuals from the best-fit versus Gaia DR2 G magnitudes with a dispersion of 43 μas. No trend is apparent nor any outliers (largest deviation is 2.3σ, expectable for 44 Cepheids and below the threshold for outlier rejection; Chauvenet's criterion would suggest a threshold of 2.6σ for outlier rejection for a sample with 46 objects).

If we consider the high χ² to be an indication of additional variance in the data, a promising source is suggested by L18 and other material accompanying DR2, which states that for bright targets (G < 12 mag) formal errors may be underestimated by up to 30%. Rescaling the DR2 parallax errors by ${{\chi }_{\mathrm{dof}}^{2}}^{1/2}\approx 1.19$ raises the mean error to 46 μas. We refit the model and find zp = −47 ± 16 μas and α = 1.008 ± 0.039 with a value of χ² = 45.0; the inconsistency with Planck CMB + ΛCDM is 2.6σ. For the expanded errors there is now no Cepheid with a deviation >2σ. Because we would expect between two and three such Cepheids, one might argue that the expanded errors are now too large. However, we think this identifies the range of reasonability for fitting these data.

Unfortunately, the cost of needing to measure the appropriate zeropoint offset from the Cepheid sample is (painfully) large. The marginalized uncertainty in α is 0.033, providing a 3.3% independent calibration of the cosmic distance scale, 2.5 times what would otherwise result from the formal parallax uncertainties and full knowledge of the parallax zeropoint, better than it has ever been determined in the local universe. As an illustration, in Figure 7 we use the constraint of −53 ± 2.6 μas on the parallax zeropoint offset calculated from 3475 red giants with Kepler-based asteroseismic estimates of radii and parallaxes from Zinn et al. (2018). The mean color of this red giant sample well matches the Cepheids (greater optical extinction of the Cepheids compensates their bluer color). The red giant mean magnitudes are a few magnitudes fainter than the Cepheid mean but much closer than the quasar sample used by L18. It is therefore not surprising that the constraint from the red giants is quite consistent with the Cepheids. We have chosen not to formally include it in the determination of H₀ due to its model-dependence. However, this external constraint, which is five times more precise than the internal constraint, demonstrates the value of independent knowledge of the DR2 offset term. Making use of it reduces the uncertainty in the distance scale for the HST Cepheid sample to ∼1.3%.

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We are optimistic that future Gaia data releases will resolve the uncertainty of the parallax zeropoint offset while producing parallax measurements near the expectations for the end of the mission. With these and the HST photometry presented here we would expect to reach the full potential precision of ∼0.5% from this Cepheid sample.

3.1. Ground-based Sample: Caveats

To improve the constraint on zp and α we might consider using a larger sample of MW Cepheids, though the augmentation of the sample would need to rely exclusively on ground-based photometry. There are compilations of ground-based Cepheid photometry that could augment the HST sample by an additional ∼150–250 Cepheids, for example from van Leeuwen et al. (2007).

However, because the HST sample of 50 presented here was selected to have P > 8 days, A_H < 0.4 mag, V > 6 mag, D < 6 kpc, and is largely (>70%) complete (with selection made randomly by the scheduling of the HST), an expanded sample would be dominated by Cepheids that necessarily violate these criteria. Most would have A_H > 0.4 mag or P < 8 days with resulting negative consequences.

On average, such shorter-period Cepheids are bluer in their mean B − V by ∼0.2 mag, which might alter their parallax zeropoints relative to the redder HST sample. In addition, these Cepheids are a couple of magnitudes fainter, so that their astrometric observations are more easily contaminated by a companion.

Further, Cepheids with P < 8 days would be shorter than the period range they would be used to calibrate, i.e., those visible in distant SN Ia hosts, putting too great a reliance on the linearity of the P–L relation. Adding Cepheids with A_H > 0.4 mag would lead to larger magnitude errors due to variations in the reddening law.

Additional loss in precision is expected from the use of ground-based photometry in lieu of HST photometry for the expanded sample. Ground-based photometry covering two hemispheres is by necessity quite inhomogeneous, especially in the NIR where limited standards are available. The few truly wide-angle surveys to date in the NIR lack the time sampling needed to determine the mean of the light curves. Moreover, in the NIR, the correspondence between the ground-based H-band filter and the WFC3 F160W filter is particularly poor, as indicated by the large color term of ∼0.2 mag per mag of J − H color which has been measured between the two (Riess 2011; R16). We find systematic errors are readily apparent between different sources of ground-based NIR Cepheid photometry. A comparison of the 79 Cepheids in common between the two most recent compilations, Monson & Pierce (2011) and van Leeuwen et al. (2007), shows a gradient of 0.015 mag per mag at >3σ confidence in H and a mean difference of 0.08 mag in J.

We also note that the comparison of the HST system photometry for the 50 Cepheids presented here with ground equivalent (see the Appendix for the source) indicated a systematic difference of −0.051 mag (ground–HST) in the Wesenheit magnitudes used to measure distances. The size of this systematic error would likely depend on the specific ground-based system used, or their mixture for heterogeneous collections.

Lastly, without the use of the high-resolution HST data one would lose the means to test for contamination of parallaxes by nearby companions as illustrated in two of three cases in Section 2.

To better compare the size of the uncertainties associated with the HST Cepheid sample and an augmentation to it from ground-based data we produced a sample comprised of all the Cepheids with photometry from a single source; NIR mean magnitudes from Monson & Pierce (2011) for Northern Cepheids and V- and I-band mean magnitudes from Berdnikov et al. (2000), excluding objects in the HST sample, leaving 86 additional Cepheids. For these we included the −0.051 mag offset identified between the ground and HST measurements of ${m}_{H}^{W}$ and the reduced CRNL of 0.036 mag (reduced to the 4.5 dex that applies from extragalactic Cepheids to HST system standards; see the notes in Table 1) that would apply between the ground and extragalactic Cepheids. A basic comparison to the DR2 parallaxes is shown in Figure 8. The augmented sample, though in rough agreement with the HST sample, has far greater errors with a dispersion of differences (after removing the parallax zeropoint offset) of 99 μas, 2.3 times that of the HST sample. Even removing the most deviant points leaves a high dispersion of 60–70 μas. This level of uncertainty is far greater than we can model by increasing the Gaia DR2 parallax errors, even by the maximum suggested value of 30% as it would require ∼70%. It is hard to realistically characterize the source of this additional variance and whether it may belie other important dependencies, and we therefore decided not to make further use of an augmented sample.

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4.1. The Zeropoint Error in DR2 Parallaxes

4.2. Implications for Determination of the Hubble Constant

We are grateful to the entire Gaia collaboration for providing data and assistance which made this project possible. We congratulate them on their tremendous achievement to date.

Support for this work was provided by the National Aeronautics and Space Administration (NASA) through programs GO-12879, 13334, 13335, 13344, 13571, 13678, 13686, 13928, 13929, 14062, 14394, 14648, 14868 from the Space Telescope Science Institute (STScI), which is operated by AURA, Inc., under NASA contract NAS 5-26555. B.B. and M.G.L. are indebted to the Italian Space Agency (ASI) for their continuing support through contract 2014-025-R.1.2015 to INAF, and to R. Morbidelli (INAF) and R. Messineo (ALTEC S.p.A, Torino) of the Italian Data Processing Center (DPCT) for their help with the Gaia data. A.V.F.'s group at UC Berkeley is also grateful for financial assistance from the TABASGO Foundation, the Christopher R. Redlich Fund, and the Miller Institute for Basic Research in Science (UC Berkeley). S.C. and A.G.R. gratefully acknowledge support by the Munich Institute for Astro- and Particle Physics (MIAPP) of the DFG cluster of excellence "Origin and Structure of the Universe."

This research is based primarily on observations with the NASA/ESA Hubble Space Telescope, obtained at STScI, which is operated by AURA, Inc., under NASA contract NAS 5-26555. This publication makes use of data products from the Wide-field Infrared Survey Explorer (WISE), which is a joint project of the University of California (Los Angeles) and the Jet Propulsion Laboratory/California Institute of Technology, funded by NASA. It has also made use of the SIMBAD database, operated at CDS, Strasbourg, France. Research at Lick Observatory is partially supported by a generous gift from Google.

The HST data used in this paper are available at the MAST archive  doi:10.17909/T9G40B.

Sources for Ground-based Phase Corrections as given in Tables 2 and 3. Compared to the Riess et al. (2018) analysis, we make use of additional ground measurements from the ASAS-SN (Shappee et al. 2014) web interface (Kochanek et al. 2017), Berdnikov et al. (2000, 2007, 2015), and van Leeuwen et al. (2007).