Inspecting the Cepheid Distance Ladder: the Hubble Space Telescope Distance to the SN Ia Host Galaxy NGC 5584

The current expansion rate of the universe, known as the Hubble constant or H₀, is one of the fundamental parameters of the standard model of cosmology and of any viable cosmological model. A few decades after the initial estimate of around 500 km s⁻¹ Mpc⁻¹ by Hubble (1929), H₀ became a point of debate with values of either ≈100 km s⁻¹ Mpc⁻¹ (e.g., van den Bergh 1970; de Vaucouleurs 1972) or ≈50 km s⁻¹ Mpc⁻¹ (e.g., Sandage & Tammann 1975). The debate was finally settled after the findings of the Hubble Space Telescope (HST) H₀ Key Project, whose final result was H₀ = 72 ± 8 km s⁻¹ Mpc⁻¹ (Freedman et al. 2001). This value was found to be in agreement with the subsequent results from the observations of the cosmic microwave background (CMB) by the Wilkinson Microwave Anisotropy Probe (WMAP; e.g., Spergel et al. 2003) based on the standard Lambda-cold-dark-matter (ΛCDM) model. However, in recent years and with the improved precision of the measurements of H₀, a significant tension has again arisen, this time between the so-called early-universe cosmology-dependent approaches finding H₀ ≈ 67 km s⁻¹ Mpc⁻¹ and late-universe direct measurements mostly finding H₀ ≈ 73 km s⁻¹ Mpc⁻¹.

On the one hand, using the precise observations of the CMB from the Planck satellite, Planck Collaboration et al. (2020) concluded that the ΛCDM provides an excellent explanation of the CMB data and reported a model-dependent prediction of the Hubble constant, H₀ = 67.4 ± 0.5 km s⁻¹ Mpc⁻¹. This result is in good agreement with other early-universe measurements. For example, by combining baryon acoustic oscillation (BAO) and big bang nucleosynthesis (BBN) data, Addison et al. (2018) reported a CMB-independent value of H₀ =66.98 ± 1.18 km s⁻¹ Mpc⁻¹, and by combining BBN and BAO data with galaxy-clustering and weak-lensing data, the Dark Energy Survey reported ${H}_{0}={67.4}_{-1.2}^{+1.1}$ km s⁻¹ Mpc⁻¹ (Abbott et al. 2018). In a recent study and based on new high-resolution CMB observations from the Atacama Cosmology Telescope, Aiola et al. (2020) reported H₀ = 67.9 ± 1.5 km s⁻¹ Mpc⁻¹, consistent with previous early-universe results. All of these results are based on the ΛCDM model.

On the other hand, the SH0ES (Supernovae H₀ for the Equation of State) project, which uses Cepheid-calibrated Type I a supernova (SN Ia) data, finds a significantly higher locally measured H₀ value. Cepheids are one of the most reliable distance indicators and, using more than a decade of observations (see, e.g., Riess et al. 2005), the SH0ES project measures H₀ = 73.5 ± 1.4 km s⁻¹ Mpc⁻¹ for the Hubble constant (Riess et al. 2019; Riess 2019). This is one of the most precise determinations of H₀ and is in more than 4σ tension with the early-universe results.

There are also other direct but Cepheid-independent methods for measuring H₀. Using time-delay data of gravitationally lensed quasars, the H0LiCOW (H0 lenses in COSMOGRAIL's Wellspring) project reported ${H}_{0}={73.3}_{1.8}^{+1.7}$ km s⁻¹ Mpc⁻¹ (Wong et al. 2020). In another gravitational lensing study, Birrer et al. (2020) employed a different lens mass profile modeling approach and found ${H}_{0}={67.4}_{-3.2}^{+4.1}$ km s⁻¹ Mpc⁻¹ and ${H}_{0}={74.5}_{-6.1}^{+5.6}$ km s⁻¹ Mpc⁻¹ using two different data sets. Using yet another independent method of geometric distance measurements to megamaser-hosting galaxies, Pesce et al. (2020) reported H₀ = 73.9 ± 3.0 km s⁻¹ Mpc⁻¹. Another late-universe method to measure the H₀ is to calibrate SNe Ia with the tip of the red giant branch (TRGB), using which the Carnegie-Chicago Hubble Program (CCHP) found H₀ = 69.8 ± 1.4 km s⁻¹ Mpc⁻¹ (Freedman et al. 2019).

Although some late-universe results are in agreement with those from the early-universe approaches, the absolute majority of direct methods find H₀ values larger than the early-universe model-dependent methods, and currently, different combinations of the late-universe measurements are in 4σ to 6σ tension with the early-universe ΛCDM predictions (Verde et al. 2019; Riess 2019), and removing any one method does not appear to resolve the tension. In fact, the general consistency of the direct methods on the one hand, and of those of the early-universe methods on the other hand, significantly reduces the possibility that systematics in one method, data, or analysis would solve this problem.

However, because the persistence of the H₀ tension would mean the failure of the base ΛCDM model, and given the generally understood success of the ΛCDM in explaining the CMB and the large-scale structure data, it is absolutely necessary to not only take different approaches for measuring H₀, but also, as emphasized by Riess et al. (2020), to scrutinize in detail all the data, methods, and the studies that have led to this cosmic discordance.

The three main rungs of the Cepheid distance ladder are the (i) calibration of the period–luminosity relation using geometric distance measurements to nearby Cepheids, (ii) calibration of the SN Ia absolute magnitude using Cepheids in SN Ia host galaxies out to around 40 Mpc, and (iii) using SN Ia out to redshift 0.15 to measure H₀.

On the first rung, different studies (e.g., Nardetto 2018; Borgniet et al. 2019; Kervella et al. 2019b; Gallenne et al. 2019; Anderson 2019; Hocdé et al. 2020a, 2020b; Musella et al. 2021) have focused on enhancing our understanding of the different properties of Cepheid variables, and in a recent study, Breuval et al. (2020) presented a new calibration of the period–luminosity (PL) relation for Milky Way (MW) Cepheids using their companion parallaxes from Gaia (Kervella et al. 2019a). In addition, the high-precision measurement of the distance to the Large Magellanic Cloud (LMC) by Pietrzyński et al. (2019) provides an accurate calibration of the Cepheid period–luminosity relation in this satellite galaxy of the MW.

On the third rung, the impact of the SN Ia environment (Roman et al. 2018) and in particular that of the star formation rate (Rigault et al. 2015) of their host galaxies on distance measurements has been investigated. Jones et al. (2018), however, conclude that the environmental dependence of SN Ia properties has a negligible effect on the H₀ measurements. Dhawan et al. (2018) and Burns et al. (2018) used near-infrared observations, where SN Ia luminosity variations and extinction by dust are less than in the optical observations and concluded that the H₀ tension is likely not caused by systematics like dust extinction or SN Ia host-galaxy mass. Also, Hamuy et al. (2020) reported that different methods for standardization of SN Ia light curves yield consistent results with a small standard deviation, concluding that SNe Ia are robust calibrators of the third rung.

In this study, we scrutinize the second rung, that is, the Cepheid calibration of SN Ia host galaxies. This intermediate rung plays the important role of connecting the geometric distance calibrations of the ladder to the cosmic scales. Therefore, it is vital to also investigate the observations and analysis involved in the second rung of the Cepheid distance ladder independently of SH0ES, which has so far been the only program undertaking this effort. Riess et al. (2016, hereafter R16) used HST data to measure the Cepheid distances to 19 SN Ia host galaxies for the calibration of the luminosity of larger-distance SNe Ia. R16 present near-infrared (NIR) observations, whereas a companion paper by Hoffmann et al. (2016, hereafter H16) reports the complete optical observations of Cepheid variables in these SN Ia host galaxies. Out of these 19 galaxies, 10 have been observed in the earlier stages of the SH0ES project (Riess et al. 2009) and with older HST instruments, i.e., NICMOS ⁷ and WFPC2 ⁸ , using the F555W (V), F814W (I), and F160W (H) bands for measuring mean magnitudes and periods of their Cepheid variables ⁹ . However, for 9 out of 19 of these galaxies, the photometric time series necessary for identifying the Cepheids, measuring their light curves, and estimating their periods have been obtained using the wideband HST F350LP filter available on WFC3/UVIS. The wavelength range of this so-called "white light" filter covers those of the V and I bands, and hence is suitable for the detection of faint sources. The nine galaxies mentioned above currently have much fewer random phase observations in the V and I bands, not sufficient for light-curve analysis. NGC 5584, however, has time-series observations in all three bands. Using the data of this galaxy, the SH0ES team obtained a relation between the periods of the Cepheids and the ratio of their amplitudes in V and I relative to those in the F350LP band. Then, assuming that these relations derived from NGC 5584 also hold in other SN Ia host galaxies, the SH0ES project corrects for the effect of random phase observations of the Cepheids in the galaxies with few V and I observations. Therefore, the Cepheids in the galaxy NGC 5584 played a key role in obtaining the periods and mean magnitudes of the Cepheids in almost half of the current SH0ES sample of SN Ia host galaxies and, in turn, in the final measurement of H₀.

For our inspection, we use the same observations of NGC 5584 that were used by SH0ES. Hence, this work is not a complete reproduction of the original experiment, because we do not repeat the observations themselves. However, where possible, we intentionally explore numerical methods and tools different from those used by SH0ES to provide an independent insight into the H₀ problem. The goal of this work is to inspect the foundations of the Cepheid distance scale, independently of any input on our analysis from the SH0ES team.

This paper is organized as follows. In Section 2, we briefly outline the method. Section 3 presents a full description of the data. We describe our analysis in Section 4, present our results in Section 5, and finally conclude in Section 6.

A standard approach for distance measurements using the Cepheid variables (Leavitt & Pickering 1912) is to use the reddening-free "Wesenheit" index (Madore 1982) in the H band defined by R16 as

$$\begin{eqnarray}&&{W}_{H}=H-{R}_{H}(V-I),\end{eqnarray} \tag{ 1 }$$

where H, V, and I are the mean magnitude of the Cepheids in F160W, F555W, and F814W, respectively. In our analysis, we adopt R_H = 0.386, which is derived from Cardelli et al. (1989) and Fitzpatrick (1999) and is also adopted by e.g., Riess et al. (2019); Bentz et al. (2019), and Breuval et al. (2020). The distance to a nearby SN Ia host galaxy can be measured by obtaining the relation between the pulsation period and W_H of its identified Cepheids, and by adopting a W_H versus period relation calibrated by the Cepheids in, e.g., the MW or the LMC. The observations of NGC 5584 for the measurements of periods and the mean magnitudes of its Cepheids in the above-mentioned bands are described in the next section.

3.1. Archival Observations

3.2. Calibrations

In all cases, we obtain the calibrated cosmic-ray-cleaned (i.e., the flc.fits files for WFC3/UVIS, and flt.fits files for the WFC3/IR observations) data provided by the MAST database. In the case of WFC3/UVIS, the flc.fits files are also corrected for the charge transfer efficiency loss ¹¹ . We list the full information regarding these observations in Appendix D.

Similar to H16, we use the TweakReg software ¹² for image registration and alignment. For all the images of all the bands, we achieve an alignment better than 0.1 pixels; the same precision is also reported by H16. We use the coordinates of the "local standard stars" provided by H16 (see Section 3.3 for more details on these stars) in order to have the same absolute astrometry as theirs. This provides an exact identification of the Cepheids using the R.A. and decl. reported by H16.

Each observation epoch consists of multiple exposures. For example, 11 out of 13 epochs in the F555W bands consist of six different exposures, and the other two consist of four different exposures (see Appendix D). We use the AstroDrizzle software⁶ to combine all of the exposures of each epoch (and of the same filter) to obtain final distortion-corrected drizzled science images for the purpose of our analysis.

Figure 1 shows examples of UVIS and IR images of NGC 5584.

Zoom In Zoom Out Reset image size

3.3. The Cepheids in NGC 5584

4.1. Photometry

Our precise alignment of the images using the local standard stars of H16 provides an exact identification of the Cepheids using the R.A. and decl. reported by H16. In the left panel of Figure 1, we mark the positions of the 199 optically identified Cepheids. Out of these, only 82 are identified in F160W and measured by R16. These are marked with red dots on both panels of Figure 1.

4.1.1. PSF Modelling

To measure the brightness of the Cepheids at each epoch, we use the PSF photometry routines of the Photutils package of Astropy (Bradley et al. 2019), which provides tools similar to, but also more general than, DAOPHOT (Stetson 1987), which is used by H16. For optical bands, we perform the PSF photometry on 100 pix × 100 pix (4 × 4 arcsec²) portions of the image centered on each Cepheid for all epochs. The background is locally estimated for each Cepheid and is automatically subtracted from the Cepheid flux. Similar to H16, we use the TinyTim package, which provides PSF models for various cameras and different HST bands (Krist et al. 2011). We checked various fitting algorithms and background estimators and found that the choice has a negligible effect on the flux measurement. Therefore, similar to R16, we use a Levenberg–Marquardt-based algorithm (provided by Astropy as LevMarLSQFitter) for determining the best-fit parameters, which are the (x, y) position and the flux (plus their uncertainties) for the Cepheids, and the MMMBackground routine, which calculates the background using the DAOPHOT MMM algorithm (Stetson 1987).

For IR photometry, i.e., for the F160W band, our procedure is the same as in the optical analysis, except that (similar to R16) the (x, y) positions of the Cepheids are fixed to their best-fit values from the F814W band and that the PSF photometry is performed on 50 pix × 50 pix (around 6.5 × 6.5 arcsec²) portions of the image centered on each Cepheid. The reason for fixing the (x, y) position is that the significantly lower resolution of IR images may lead the fitting algorithm to pick a wrong neighboring source rather than the Cepheids if (x, y) are allowed to vary as free parameters.

Examples of our PSF photometry results are shown in Figure 2. We choose to show those Cepheids that are presented as representative by H16 (in their Figure 4).

Zoom In Zoom Out Reset image size

4.1.2. Epoch-to-epoch Offset

The observation condition varies from epoch to epoch and would affect the flux of the Cepheids. To correct for this, we use the local standard stars that were introduced earlier. For each band, we perform a PSF photometry of these nonvariable stars and measure their average fluxes in all epochs, ${\bar{F}}_{\mathrm{all}}$, and also for each epoch, ${\bar{F}}_{\mathrm{epoch}}$. The Cepheid fluxes at each epoch are then scaled by ${\bar{F}}_{\mathrm{all}}/{\bar{F}}_{{\rm{epoch}}}$ to correct for the epoch-to-epoch offset ¹³ .

4.1.3. Magnitude Zero Points and Aperture Correction

The magnitude zero points, ZPs, for different HST bands are provided by Kalirai et al. (2009), Deustua et al. (2017), and on the STScI calibration pages. ¹⁴ These ZP values are based on the WFC3 standard aperture radius of 04. Therefore, the difference between this standard aperture and the PSF modeling should be measured and corrected for. A customary approach also adopted by H16 is to perform both PSF and aperture photometry on a sample of ideally isolated and relatively bright stars in the image and to obtain a statistical mean difference between the two.

In this work, we take a rather different approach. Ideally, for a single isolated star, the difference between the aperture and PSF photometry should be directly dependent on the aperture size and the PSF model, while the background should be the same. Here, given that the aperture size for the purpose of correction is fixed to 04, the difference is basically caused by the extra light captured by the tails of the PSF model beyond the 04 radius. Therefore, one way to directly obtain this difference is to measure the flux of the PSF model using a 04 radius aperture (10 pixels for UVIS and around 3 pixels for IR). The magnitude difference, hence the aperture correction (AP_cor), would then be

$$\begin{eqnarray}&&{\mathrm{AP}}_{\mathrm{cor}}=2.5{\mathrm{log}}_{10}\left[\displaystyle \frac{{F}_{\mathrm{ap}}}{{F}_{\mathrm{PSF}}}\right]-2.5{\mathrm{log}}_{10}[{EE}(r=10)],\end{eqnarray} \tag{ 2 }$$

where F_ap is the fraction of the PSF flux inside the aperture, F_PSF is the total flux of the PSF model, and EE(r) is the encircled energy for different aperture radius r (see Deustua et al. 2017 for further details). We compare the two methods of measuring the aperture correction in Appendix A.

The ZP and AP_cor values used in this study are listed in Table 1 for each band ¹⁵ .

Table 1. The Zero Point (ZP) and the Aperture Correction (AP_cor) Values, Both in Mag, for the Different HST Bands Used in This Study

	F555W	F814W	F350LP	F160W
ZP (mag)	25.737	24.598	26.708	24.5037
APcor (mag)	0.032	0.034	0.032	0.049

Note. See Section 4.1.3

Download table as:  ASCIITypeset image

4.2. Crowding Bias

4.3. Light-curve Fitting Using Templates from Galactic Cepheids

The data collected for each Cepheid consist of several epochs for different passbands. From these data, we need to derive the pulsation period, as well as the mean magnitudes in each band. In H16, this was done using template light curves from Yoachim et al. (2009). In this work, we use different template light curves and fitting strategies so that all bands are analyzed simultaneously.

We derive synthetic light curves in the HST photometric bands for various known Galactic Cepheids, covering the instability strip (in effective temperature and period). The radius, effective temperature, and period of these Cepheids are shown in Figure 3. We then use a dimensionality reduction algorithm to parameterize any light curve using only a few parameters.

Zoom In Zoom Out Reset image size

4.3.1. Data Set for the Templates

We choose to use observational data as the basis for our templates, fitted with our modeling tool SPIPS (Mérand et al. 2015) which synthesizes photometric observations based on variations of the stellar radius and effective temperature. We collect high-quality spectro-, photo- and interferometric data for many Galactic Cepheids and fit their SPIPS models. It should be noted that the knowledge of the distance and/or the projection factor of these Galactic Cepheid does not play a role in building the light-curve templates.

Our final sample comprises 28 stars with periods ranging from 12 to ∼90 days (Breuval et al. 2021). We do not include Cepheids with a period shorter than 12 days because (i) Cepheids observed in distant galaxies are biased toward the brightest ones, which result in an observational cut around ∼20 days, and (ii) Cepheids light curves change dramatically around 9–10 days, which has been long noticed ever since Fourier decomposition was applied to the Cepheids' light curves (see, e.g., Simon & Lee 1981). We do not apply any selection cut on radius and effective temperature, as our intent is to sample Cepheids in the instability strip.

The SPIPS models are based on radial and temperature variations enabling the synthesizing of any photometric light curve using the filter bandpass definition and atmospheric models. The advantage of this method is that it can accurately extrapolate light curves in passbands for which we do not have data. We use the HST bandpasses and ZPs defined at the Spanish Virtual Observatory's Filter Profile Service. ¹⁶

4.3.2. Reduction of Dimensions in Templates

Our 28 Galactic Cepheids light curves contain a lot of information, which needs to be reduced into parameterized templates. We reduce the dimensions of our template data set via a PCA, using the scikit-learn Python library (Pedregosa et al. 2011). The training data set for PCA are 28 vectors composed of the concatenated light curves (one for each band) over a single pulsation cycle, centered around their means (see Figure 4). When it comes to choosing how many components to keep to fit our light curves, it is customary to consider the amount of variance reproduced by a given number of the most significant components. In our data set, at any given phase and for any bands, the standard deviation is never greater than 0.2 mag, with a total standard deviation of 0.13 mag (around the average light curve). Using enough PCA components to reproduce 95% of the variance should reproduce light curves within ∼0.03 mag (on average), which we deemed enough for our application. Our main goal is to extract the average magnitude from sparse and irregularly sampled time series: even if the light curve is reproduced within 0.03 mag, the average is likely estimated with much higher accuracy. Keeping three principal components covers 93.8% of our training set variance, whereas using four components leads to 97.1% of the variance being reproduced, which corresponds to a standard deviation of 0.022 mag. See Figures 5 and 6 for different information regarding the PCA components.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

4.3.3. Fitting Strategy

For a given NGC 5584 Cepheid, we have a list of observations in various passbands and at different dates. The initial period is estimated by a computed periodogram on the F555W and F350LP data. Then, a full model is fitted to the data using the PCA light curves. Our model also includes reddening, using the formula contained in SPIPS and parameterized using the color excess E(B–V). We fix the reddening to E(B–V) = 0.035, which we estimate using DUST ¹⁷

We iterate on the initial parameter by randomizing the period (±5%) to account for the uncertainty of the periodogram estimation. The PCA coefficients are initialized to their mean value from the analysis of the template stars. During the least-squares fit, a uniform prior constrains the coefficient to only evolve inside the range of values observed on the template stars. From the randomized starting periods, we keep the fit with the global lowest reduced χ². Using the best-fit parameters and covariance matrix, we can compute the domain of uncertainty for the synthetic light curves and derive the average magnitudes and amplitudes.

Our fitting method has several differences from the one presented in H16 using Yoachim et al. (2009). First of all, we fit all data at once. This is feasible because our model includes realistic information about the offset between bands and the shape of the light curve. An example can be seen for star 347072 in Figure 7 (which we discuss further in Section 5). The F814W data of this Cepheid are very noisy, and the fitted light curve is constrained mostly by the F555W and F350LP data, which are of much better quality. Even if the modeled light curve in F814W is systematically above the data points, it is the most realistic within our hypothesis and priors derived from Galactic Cepheids.

Zoom In Zoom Out Reset image size

5.1. Light Curves, Mean Magnitudes, and Periods

Using the light-curve template fitting explained in Section 4.3, we obtain the periods and the mean magnitudes for all the identified Cepheids in the four HST bands. Figure 7 presents our results for the light curves of the Cepheids we showed in Figure 2—they are chosen by H16 as the representative Cepheids of NGC 5584. Our light curves can be directly compared with those H16 showed in their Figure 4. Most of the light-curve models nicely represent the data. One exception among these is the Cepheid 347072, which, as discussed earlier, has poor-quality data points in F814W. This Cepheid is not detected in the F160W band and therefore is not included in the measurement of distance by either SH0ES or by us in this work.

Figure 8 provides one-on-one comparisons of our results with those of SH0ES reported in H16 and R16. The top row provides comparisons for the mean magnitude measurements in the V, I, and H bands, as well as the (V − I) color. For the H band, we only show the uncertainties on the Y-axis (i.e., from our results), because R16 published only the so-called total uncertainties (σ_tot) and not those of the mean magnitudes in the H band. A generally good agreement can be seen between the two results, especially for the H band and the V − I color, both of which directly contribute to the distance measurement (see Equation (1)).

Zoom In Zoom Out Reset image size

The leftmost panel on the bottom row of Figure 8 provides a comparison for period measurements. As can be seen, although we use a different approach for template fitting and hence the period measurements, the two results are in general agreement with only a few exceptions.

Regrettably, H16 does not provide mean magnitudes in the F350LP band. We therefore cannot have a direct comparison for this quantity. However, we can compare amplitude measurements in the F350LP band as discussed in the next subsection.

5.2. Amplitude Measurements

We remind the reader that H16 uses the amplitude ratios versus period relation of the Cepheids in NGC 5584 to correct the random-phase observations of other SN Ia host galaxies in the V and I band. Therefore, an accurate and precise measurement of these relations can potentially impact the final H₀ measurements. The three panels (from the right) on the bottom row of Figure 8 provide comparisons for our amplitude measurements versus those of the SH0ES team. We perform two different measurements of the amplitudes: (1) peak to peak (PTP), which measures the magnitude difference between the maximum and minimum of the light-curve model, and (2) the rms, which is the standard deviation of the light curve (regularly sampled) from their mean value. While the PTP results (which are the ones shown in Figure 8) are in general agreement with the amplitude measurements of SH0ES, it is not robustly estimated in our method. Our PCA-based fits allow variations in the shape of the model, especially between phases 0.8 and 1.0, which is where the amplitude is measured (see, for example, the F350LP light curve of star 258671 in Figure 7). On the other hand, the amplitude is directly one of the template-fitting parameter in the SH0ES analysis. While PTP and rms differ by a factor of $2\sqrt{2}\approx 2.83$ for a pure sinusoidal wave, the value varies with the exact shape of the light curve. From our high-definition template sample star, we find that ${\left(\tfrac{\mathrm{PTP}}{\mathrm{RMS}}\right)}_{I}=3.08\pm 0.12$ and ${\left(\tfrac{\mathrm{PTP}}{\mathrm{RMS}}\right)}_{V}=3.14\pm 0.14$. We are interested in ratios between bands and the comparison with SH0ES' results. For the amplitude ratios, we find that $\left(\tfrac{{\mathrm{PTP}}_{I}}{{\mathrm{PTP}}_{V}}\right)=0.98\pm 0.03\left(\tfrac{{\mathrm{RMS}}_{I}}{{\mathrm{RMS}}_{V}}\right)$. In other words, the ratio of the amplitudes is almost independent of the amplitude measurement method, and our amplitude ratios computed from the rms (which we use in our subsequent analysis) are comparable to those of SH0ES with a scatter of 6%.

In Figure 9, we compare our results and those of SH0ES for the amplitude ratio versus period relation. The blue squares and red diamonds are our A_V /A_LP and A_I /A_LP , respectively. The gray plus and cross symbols are the same quantities as published by SH0ES in H16 and have significantly larger scatters. The dots and empty circles are, respectively, A_V /A_LP and A_I /A_LP for the MW Cepheids. The blue solid line and the red dashed line are the results of our linear fits on A_V /A_LP and A_I /A_LP versus $\mathrm{log}P$. While for the linear fitting we only used the data from the Cepheids in NGC 5584, and while only 28 of the MW Cepheids with period >12 days were used to build our template light curve, the fitted line also passes through the MW Cepheids data points even for those with small periods. The linear correlation coefficient measured for both of these relations is ≈0.3. From the linear fitting, we find

$$\begin{eqnarray}\begin{array}{rcl}{A}_{V}/{A}_{{LP}} & = & 1.167+0.073(\mathrm{log}P-1.5),{\sigma }_{\mathrm{fit}}=0.014,\\ {A}_{I}/{A}_{{LP}} & = & 0.757+0.090(\mathrm{log}P-1.5),{\sigma }_{\mathrm{fit}}=0.022,\end{array}\end{eqnarray} \tag{ 3 }$$

where σ_fit values are the standard deviations of the fits and are an order of magnitude smaller than those of SH0ES (see Table 2 of H16). The small scatter in this relation means that our amplitude ratio measurement is less noisy and is indicative of a high-quality light-curve modeling approach. We note that in H16 the light curves of different bands are fitted separately (using Yoachim et al. 2009 templates), and then the amplitudes resulting from the different fits are divided to yield the amplitude ratios. This could be the reason for the large scatter in their amplitude ratios. On the other hand, in our approach, all of the light curves (of all bands) are fitted simultaneously, hence the amplitudes are not estimated independently from one another, leading to a lower scatter.

Zoom In Zoom Out Reset image size

5.3. Uncertainties on the Wesenheit H Magnitudes

In their Section 2.2, R16 describe a σ_tot as the total uncertainty on their Cepheid distance measurements. They refer to the uncertainty of the crowding bias in the H band as σ_sky and that of the optical observations as σ_ct , and they add them as a single value for all the Cepheids in a given galaxy. Because we apply the crowding bias (in all bands) for each Cepheid before the template fitting, the values of mean magnitudes already include the effect of crowding bias and their uncertainties. In addition, our template light-curve fitting method analyses all the data together; therefore, the uncertainty on the H-band mean magnitudes already includes the effect of limited phase coverage.

Therefore, for the total uncertainty on W_H , we have

$$\begin{eqnarray}&&{\sigma }_{{WH}}={[{\sigma }_{H}^{2}+{R}_{H}^{2}({\sigma }_{V}^{2}+{\sigma }_{I}^{2})+{\sigma }_{\mathrm{int}}^{2}]}^{1/2},\end{eqnarray} \tag{ 4 }$$

where σ_int is the intrinsic dispersion due to the nonzero width of the instability strip. To estimate σ_int, we follow the procedure of Riess et al. (2019). Using the Cepheid observations in the LMC, Riess et al. (2019) present PL relations and their scatter in different HST bands. To estimate σ_int, they subtract (in quadrature) the mean Cepheid measurement errors from the scatter of the PL relation for a given band. Their mean measurement error for different bands are given in their Section 2.2 and the values for the scatter of the PL relations are listed in their Table 3. For W_H , the intrinsic dispersion is estimated to be σ_int = 0.069 mag.

5.4. The Period–Luminosity Relations

In addition to the PL relation in W_H , we also present PL relations for all of the bands F350LP, F555W, F814W, and F160W, as well as for optical Wesenheit index, W_I , in Figure 10. The latter is defined as W_I = I − R_I (V − I) with R_I = 1.3 (Riess et al. 2019). The uncertainties on individual Cepheids in this figure also include the contribution from the σ_int explained in the previous section ¹⁸ . We note that the data points in the PL relations shown in Figure 6 of H16 appear to contain only the measurement uncertainties, which are comparable in size to this work's results as shown in our Figure 8.

Zoom In Zoom Out Reset image size

The solid lines represent the results of fitting a linear relation of the form $m=\alpha \mathrm{log}P+\beta$, where m is the mean magnitude. We fix the slope α to the values given in Table 3 of Riess et al. (2019; which lists the PL relations from Soszynski et al. 2008, Macri et al. 2015, and R16), and fit for the intercept with a 3σ clipping. The slightly larger scatter in our PL relations compared to those found by SH0ES for NGC 5584 is most probably due to our different treatment of the crowding bias. As stated earlier in the text, SH0ES added a single value of crowding bias for all the Cepheids in a galaxy, which shifts the PL relation slightly toward fainter values. However, we add the crowding bias values estimated at the location of each Cepheid separately, which introduces a somewhat larger scatter in the PL relation ¹⁹ .

5.5. The Distance to NGC 5584

In this section, we derive the distance modulus of NGC 5584, based on the apparent Wesenheit magnitudes W_H of our sample of 82 Cepheids in this galaxy. By applying an existing W_H PL relation to the known period of our stars, we derive the absolute magnitude M^W _H for each Cepheid and then their individual distance modulus $\mu ={W}_{H}-{M}_{H}^{W}$.

We perform this calculation using two different PL relations: one calibrated in the MW (Breuval et al. 2020), ${M}_{H}^{W}=-5.432\ (\pm 0.029)-3.332\ (\pm 0.177)[\mathrm{log}P-0.84]$, and another calibration from the LMC (Riess et al. 2019), ${M}_{H}^{W}=15.898-3.26\mathrm{log}P$. For the slope of the latter relation, a 0.02 mag uncertainty is stated in Riess et al. (2019) while they mention no uncertainty on the intercept. Therefore, we assume a conservative uncertainty of 0.02 mag error also for the intercept (the intercept uncertainties in Macri et al. 2015 are much smaller than 0.02 mag). We then subtract the LMC distance modulus as measured by Pietrzyński et al. (2019). For both PL relations, the individual distance moduli obtained for each Cepheid are represented in Figure 11. The Galactic PL relation yields a weighted mean distance modulus of 31.810 ± 0.047 mag, while the LMC calibration results in 31.639 ± 0.038 mag. The 1σ confidence regions of these weighted mean values are also shown in Figure 11. The distance modulus from the Galactic PL relation is in agreement with μ = 31.786 ± 0.046 (mag) measured by SH0ES in R16. The distance modulus from the LMC PL relation, however, is smaller though still in agreement within 2.5σ with the SH0ES result.

Zoom In Zoom Out Reset image size

It is not surprising that different distances are obtained based on LMC and MW PL relations, given that the LMC has a smaller metallicity compared to the MW (Romaniello et al. 2008), i.e., the larger distance inferred from the LMC PL relation is consistent with its smaller metallicity. The difference in terms of distance modulus obtained with the MW and LMC PL relations highlights the need for a metallicity correction that has been extensively studied (see, e.g., Pietrzyński et al. 2004; Gieren et al. 2018; Groenewegen 2018; Ripepi et al. 2019, 2020; Breuval et al. 2021), though yet with no clear consensus. However, NGC 5584 is a spiral galaxy with a structure similar to that of the MW and, in fact, its metallicity gradient is very similar to that of the MW (see Table 6 of Balser et al. 2011 and Table 8 of R16). The MW PL relation, therefore, is more appropriate for measuring the distance to NGC 5584.

From our μ for NGC 5584 based on the MW PL relation, we calculate a distance of d_{NGC 5584} = 23.01 ± 0.05 Mpc.

The 4σ−6σ tension (Riess 2019) between the direct and the early-universe measurements of H₀ asks for detailed investigations in the different methods involved. NGC 5584 played a key role in the direct measurement of H₀ from the Cepheid distance ladder by the SH0ES team (Riess et al. 2016). Observations of this galaxy were employed to derive a relation between the ratio of the pulsation amplitude of Cepheids in the V and I bands relative to the wide F350LP HST band and the period. The F350LP band has been used by the SH0ES team for detection and light-curve measurement of Cepheids in around half of the current SN Ia host galaxies used for H₀ measurements, and the relation mentioned above has been used to obtain the mean V and I magnitudes from the sparse sampling of Cepheid light curves in these bands.

In this contribution, we provided an independent and detailed analysis of the HST data from NGC 5584. Where possible, we intentionally used methods and tools different from those used by SH0ES. This allowed the investigation of the possible influence of these methods on distance measurements. The key parts of our detailed analysis are listed as follows:

• 1.  
  applying PSF photometry routines of the Photutils package of Astropy (Bradley et al. 2019) instead of the classic DAOPHOT software (Stetson 1987),
• 2.  
  testing and finding the negligible influence of the choice of PSF modeling and background subtraction algorithms,
• 3.  
  applying a different aperture correction procedure for the PSF photometry,
• 4.  
  adopting a slightly modified approach for crowding bias estimation (using a sigma-clipping approach on the artificial stars flux measurement rather than directly removing bright estimated sources done by SH0ES),
• 5.  
  a different approach for applying the crowding bias compared to SH0ES (applying the bias separately for each Cepheid rather than averaging over the whole galaxy for the optical observations), and
• 6.  
  employing a completely different approach for Cepheid light-curve modeling for the measurement of mean magnitudes, amplitudes, and periods.

Our main results can be summarized as follows:

• 1.  
  Our measurements of Cepheids' mean magnitudes and period and those of SH0ES are in good agreement. In particular, we find no systematic difference in our H-band mean magnitudes and (V–I) color, both of which directly influence the distance measurements, compared to SH0ES.
• 2.  
  We derived a significantly tighter amplitude ratio versus period relation compared to the one derived by SH0ES.
• 3.  
  We measure two distance moduli for NGC 5584 using two different PL relations calibrated in MW and LMC. The result from the former is in agreement with the value from SHOES within 1σ, and the result from the latter is 0.147 ± 0.060 mag smaller than that of SH0ES, though still within 2.5σ.

We do not attempt at reporting a value for H₀ based on the distance to only one SN Ia host galaxy, and we only note that a smaller distance to NGC 5584 points toward a higher H₀ value. However, we consider the MW PL relation to be more appropriate for distance measurements to NGC 5584, due to the similar metallicity and structure of these two galaxies. Nevertheless, the effect of metallicity and its measurement methods (Bresolin et al. 2009; Kudritzki et al. 2012) on extragalactic Cepheid distances requires further investigations.

The main conclusion of the current study is that our inspection of the NGC 5584 Cepheids does not yield any systematic hints toward the resolution of the H₀ problem. However, it would be important to also independently inspect for systematics in the distance measurements to all the galaxies used for calibration of SN Ia absolute magnitude. For doing so, and until reasonably fine-sampled time-series data of all SN Ia calibrators become available, it would certainly be better to use our precise amplitude ratio versus period relations for light-curve analysis of Cepheids in SN Ia hosts with limited time-series data as they would potentially yield more accurate mean magnitudes in the V and I bands. This would also provide an investigation into the potential statistical effect of these relations in H₀ measurements.

While it is important to continue the investigations on the H₀ measurements, the current findings seem to be pointing toward a nontrivial solution to this problem. This could mean that our current understanding of the local or early-universe may require modifications or a complete change of paradigm. In the local universe, the presence of a large local underdensity (which is incompatible with LCDM; Haslbauer et al. 2020) has been presented (Shanks et al. 2019; Haslbauer et al. 2020) as a possible cause of the H₀ discrepancy (but see also Riess et al. 2018a and Shanks et al. 2018). In the early universe, various scenarios such as nonstandard recombination, dark matter/dark energy interaction, and self-interacting neutrinos have been presented; however, so far no consensus has been reached (for reviews and summaries, see Verde et al. 2019; Poulin 2020; Knox 2020).

While it is important to seek alternative ideas on the theoretical side, the improvement of current observational methods, as well as the development of new independent ones, are necessary for progress toward a solution to the H₀ problem. For the Cepheid distance ladder, the number of SN Ia calibrators observed by the HST is soon to be doubled by the SH0ES program (Riess et al. 2019), hence the statistical uncertainty on H₀ measured by this method would be reduced. In addition to the strong lensing, megamasers, and TRGB methods (see also Beaton et al. 2016; Kim et al. 2020) mentioned in the Introduction, other Cepheid-independent routes would also soon contribute to H₀ measurements. Huang et al. (2020) present Mira variables for the calibration of SN Ia absolute magnitudes. Also, using the advanced LIGO and Virgo gravitational wave detectors, The LIGO & Virgo Collaborations et al. (2021) have reported an H₀ measurement using standard sirens (see also Coughlin et al. 2020). As the number of detected standard sirens increases in the future, the currently large statistical uncertainty in their resulting H₀ measurement would decrease, making them an important independent way of measuring the cosmic expansion rate (Feeney et al. 2019).

One of the most promising contributions to the accuracy of the cosmic distance scale in the near future would be from Gaia. The impact of the first (see, e.g., Casertano et al. 2017; Gaia Collaboration et al. 2017) and second (see, e.g., Groenewegen 2018; Riess et al. 2018b; Clementini et al. 2019; Breuval et al. 2020; Ripepi et al. 2020) data releases of Gaia on the calibration of the Cepheid PL relation is already considerable. It is, however, still limited by the persistently uncertain value of the instrumental parallax ZP (see, e.g., Arenou et al. 2018; Khan et al. 2019). The early Gaia data release 3 (EDR3) published on 2020 December 4 (Gaia Collaboration et al. 2020) significantly improved the accuracy of the measured parallaxes of the MW Cepheids. Mitigation of the uncertainty due to the instrumental parallax ZP through an ad hoc position-, color- and magnitude-dependent calibration is also presented by Lindegren et al. (2020). As discussed by Riess et al. (2021; see also Breuval et al. 2021), this improvement brings the calibration of Cepheids luminosities to a 1% level, which makes them the most accurate distance indicators available to date. As the number of measurement epochs and the understanding of the Gaia instrument increase, DR3 and DR4 will eventually provide trigonometric reliable parallax measurements at a few percent level or better for hundreds of MW Cepheids. Combined with accurate photometry and extinction corrections from 3D extinction maps (see, e.g., Chen et al. 2019; Hottier et al. 2020), this set of absolutely calibrated distances will result in a very tight set of Cepheid PL relations, calibrated for solar metallicity. Our Galaxy therefore appears as a particularly appealing alternative to the Magellanic Clouds as the primary anchor for extragalactic Cepheid distances, thanks to the similarity of its metallicity with those of distant SN Ia host galaxies. Relying on MW Cepheids presents the advantage of reducing the possible bias introduced by the metallicity correction. This will effectively bypass the metallicity correction, thus increasing the overall robustness of the SN Ia calibration.

As also noted in Riess (2019), the precise measurement of H₀ provides a powerful end-to-end test of the LCDM standard model of cosmology. Future observational progress and inspections such as the current study would eventually conclude whether the H₀ tension is caused by a measurement error, or it means that the LCDM should be abandoned as a correct model of the universe.

We thank the anonymous referee for the constructive comments. The research leading to these results has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program under grant agreement No 695099 (project CepBin). W.G. and G.P. gratefully acknowledge financial support for this work from the BASAL Centro de Astrofisica y Tecnologias Afines (CATA) AFB-170002. G.P. also acknowledges the support from NCN MAESTRO grant UMO-2017/26/A/ST9/00446 and DIR/WK/2018/09 grants of the Polish Ministry of Science and Higher Education. P.K., N.N., V.H., and S.B. acknowledge the support of the French Agence Nationale de la Recherche (ANR), under grant ANR-15-CE31-0012-01 (project UnlockCepheids). We acknowledge the use of the HST observations of NGC 5584 performed by the SH0ES team (PI: Adam Riess, Cycle: 17, Proposal ID: 11570), which are publicly available on the MAST database. This research made use of Photutils, an Astropy package for the detection and photometry of astronomical sources (Bradley et al. 2019).

Facilities: HST - Hubble Space Telescope satellite, MAST. -

Software: DrizzlePac, Astropy (Bradley et al. 2019), scikit-learn.

Here, we compare aperture correction using the aperture photometry of the PSF model with the approach using aperture and PSF photometry of stars in the image. For this purpose, we compare the PSF and aperture photometry of 13 uncrowded stars using the stacked version of all F555W band images. For these stars, we crop a 50 pixel × 50 pixel portion of the image and perform PSF photometry in the same way as described in Section 4.1.1. All of the sources except those from the central star are removed using PSF modeling prior to the aperture photometry with an aperture radius of 10 pixels. The difference is then measured using Equation (2), and we find a mean value of AP_cor = 0.056 ± 0.024 mag for the F555W band. It is 1σ larger than the value obtained using the aperture photometry of the PSF model. We expect a similar result for the F814W band. For the F160W band, the AP_cor we measure using the PSF model, i.e., 0.049 mag, is also around 1σ smaller than the 0.06 ± 0.01 measured by Huang et al. (2020) for the F160W images of the SN Ia host NGC 1559. The difference between these two methods is most probably due to the imperfect subtraction of the background noise in the actual images and the absence of this noise in the PSF model. Therefore, by noting that the difference in the F160W band is most relevant for distance measurement (Equation (1)), measuring AP_cor using aperture photometry of the PSF model rather than using uncrowded stars in the image leads to around a 0.01 ± 0.01 mag decrease in the distance modulus.

In Section 4.2, we explained our method of estimating the crowding bias at the location of each Cepheid. Here we investigate the environmental dependence of crowding bias on the F555W band. Figure 12 shows the distribution of crowding bias across the galaxy (right panel), bias versus the projected angular distance d_cen in degrees from the center of the galaxy (left panel), and bias versus local surface brightness, μ_local (middle panel). The Cepheids that are most affected by the crowding bias are statistically closer to the center of the galaxy where the stellar density is generally higher. However, small bias values can also be found at a smaller d_cen, and we measure a correlation coefficient of r = −0.24 between the absolute value of the crowding bias and d_cen. The small correlation is possibly due to the spiral structure of NGC 5584, i.e., even at small galactocentric distances, there are less crowded regions.

Zoom In Zoom Out Reset image size

We also check the crowding bias versus μ_local. The latter is measured using the following three-step method: (i) we first measure the average sky background using around twenty 40 × 40 pixel square regions outside the parts of the image covered by NGC 5584, (ii) we then used the same size squares at the location of each Cepheid (where we already estimated crowding bias) and measured the total flux inside them, and (iii) in the end, the average sky background is subtracted from the local total fluxes, and the result is converted to surface brightness using the angular area in arcseconds and the magnitude ZP. We measure a correlation coefficient of r = −0.40 between the crowding bias and μ_local. This relatively larger correlation implies that the local surface brightness is a better proxy to crowding bias compared to galactocentric distance. Using a second-order polynomial fit, we find

$$\begin{eqnarray}&&\mathrm{Bias}=-0.013{\mu }_{\mathrm{local}}^{2}+0.61{\mu }_{\mathrm{local}}-7.4,\end{eqnarray} \tag{ B1 }$$

with the standard deviation of the model minus data being σ_fit = 0.066. This relation may be used to estimate the crowding bias in the F555W band from the local surface brightness instead of the artificial star injection approach. It may be useful to note that in the regions with μ_local ≳ 23 mag/arcsec², the effect of crowding bias is negligible.

To compare our crowding bias measurements with those of H16, we also show the bias as a function of period in Figure 13. This figure can be directly compared to Figure 14 of H16. Our results are (within 1σ) in agreement with those of H16. In particular, as can be seen in the lowest panel, the effect of the crowding bias on the V – I color measurement is very small. We remind the reader that unlike the approach of SH0ES (averaging over the galaxy for the optical observations), we apply the crowding bias estimated for individual Cepheids on their photometric results before the template fitting.

Zoom In Zoom Out Reset image size

Our results for the photometric properties of the identified Cepheids in NGC 5584 are listed in Table 2.

We list the information on all the data files we retrieved from the MAST database in Table 3.