A Cepheid systematics-free test of $H_0$ to $\lesssim2.5\%$ accuracy using SH0ES photometry
Doron Kushnir Amir Sharon


firstpage–lastpage
A robust approach for time-bin encoded photonic quantum information protocols
    Nora Tischler
    April 29, 2024
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The recent SH0ES determination of the Hubble constant, H_0=73.04±1.04 km/s/Mpc, deviates significantly by 5σ from the Planck value, stimulating discussions on cosmological model extensions. To minimize statistical uncertainty and mitigate sensitivity to systematic errors in any single anchor distance determination, SH0ES combines Cepheids from various observations, including those from Type Ia supernova (SNe Ia) host galaxies, NGC 4258, and closer galaxies (MW, LMC, SMC, and M31), although this mixed sample may introduce unknown or subtle systematic errors due to comparing distant and closer Cepheids. To address this, we propose a subset excluding Cepheids from the closer galaxies, retaining only the NGC 4258 water megamasers as a single anchor, circumventing potential systematic errors associated with observational methods and reduction techniques. Focusing solely on these Cepheids yields competitive statistical errors, approximately 2.5%, sufficient to identify a 3σ tension with the Planck H_0 value. Our approach offers an opportunity to utilize optical photometry with systematic uncertainty smaller than the statistical uncertainty, potentially achieving higher precision than NIR photometry, given the lower optical background. However, currently the optical photometry sample's fidelity does not match that of NIR photometry. The significant Hubble tension obtained is unrelated to Cepheids and we discuss other options.






cosmological parameters – distance scale – stars: variables: Cepheids– supernovae: general








§ INTRODUCTION



The most recent determination of the Hubble constant by the SH0ES collaboration <cit.>, H_0=73.04±1.04 (in units of km s^-1 Mpc^-1 hereafter), exhibits a significant deviation of 5σ from the Planck value <cit.>, H_0=67.4±0.5, commonly referred to as the Hubble tension. This discrepancy between the Cepheid- and Type Ia supernovae (SNe Ia)-based SH0ES measurement and the cosmic microwave background temperature and polarization anisotropies Planck measurement has spurred numerous proposals for extensions of the standard ΛCDM cosmology model <cit.>.

The absolute distance scale utilized by SH0ES is grounded on the period-luminosity relation of Cepheids <cit.>, measured in the Hubble Space Telescope (HST) F160W filter (similar to the NIR H band). These Cepheids are situated in 37 SNe Ia host galaxies (hereafter referred to as host Cepheids) along with other anchor galaxies with absolute distance measurements. The Hubble tension manifests as a discrepancy of 0.1-0.2 mag in the magnitudes of SH0ES Cepheids <cit.>, wherein the host Cepheids appear brighter than the ΛCDM prediction.

The approach adopted by the SH0ES collaboration to determine H_0 involves combining the host Cepheids with a diverse large sample of Cepheids obtained from various observations. This strategy offers several advantages, including minimizing the statistical uncertainty in H_0 and utilizing multiple independent geometrical distance anchors, which reduces sensitivity to potential systematic errors in any single anchor distance determination. The additional Cepheids reside at the anchor galaxy NGC 4258 <cit.>, as well at much closer galaxies such as the SMC and LMC  <cit.> and the Milky Way <cit.>. Cepheids from M31 (at 750 kpc) are also incorporated to better constrain the P-L relation. However, the comparison of distant host Cepheids (typically at distances of 15-50 Mpc and M101 at a distance of 6.5 Mpc) to Cepheids much closer to us involves a measurement range of ≥ 10 magnitudes which could in principle result in systematic errors between such measurements.

The measurement of host and N4258 Cepheids involves HST imaging of galaxies at multiple epochs using visual and NIR filters. This imaging data is utilized to identify and select Cepheids, determine their periods, extract their photometry at each epoch and filter, and amalgamate all epochs to derive a single mean magnitude value for each filter. These mean magnitudes are then employed to construct the Wesenheit index <cit.>[We follow the convention that a single Cepheid magnitude x is the magnitude of intensity mean, ⟨ x ⟩, and colors (x-y) stand for ⟨ x ⟩-⟨ y ⟩.]: 

    W_H=F160W-R_H^VI(F555W-F814W),

where F555W and F814W are the observed magnitudes in the corresponding HST filters (similar to the optical V and I band, respectively) and R_H^VI≈0.4 is chosen to achieve extinction-independent measurements (for some specific extinction law, see a detailed discussion in Section <ref>). On the other hand, the measurement of close Cepheids involves a distinct process <cit.>:


  * Close Cepheids are identified and selected based on archival observations.

  * These Cepheids, which are significantly brighter (by a factor of 10^3-10^9) than host Cepheids, necessitate a different observing technique with HST, often involving a rapid spatial scan or DASH mode, along with a correction for count-rate nonlinearity.

  * Due to the narrow field of view of HST, a limited number of close Cepheids can be observed in one orbit, typically resulting in observations at a single epoch, which are then phase-corrected to mean light using archival (usually ground-based) observations.

  * The blending of host Cepheids with their environment (either chance superposition of Cepheids on crowded backgrounds or light from stars physically associated with Cepheids) due to the HST's angular resolution[ [angle-symbol-over-decimal];;0.1 corresponds to 7-25 pc for the hosts (and 3 pc for M101).] requires a correction to photometry known as crowding correction, which is not applied to close Cepheids.

The SH0ES collaboration extensively studied the impact of these differences on the comparison between close and host Cepheids and compensated for them through various corrections (R22).

Additionally, the large distance to host Cepheids biases their selection towards longer periods and lower extinction. Since long-period Cepheids are rarer, the close Cepheid sample is biased towards shorter period Cepheids, see Figure <ref>. This discrepancy in period distributions is mitigated by assuming a specific shape of the P-L relation, which may include breaks. Furthermore, the extinction of close Cepheids (in the MW and M31) tends to be higher than that of host Cepheids (see discussion in Section <ref>). To address this, observations are conducted in the NIR and the Wesenheit index is utilized. Finally, the metallicity of LMC and SMC Cepheids is lower than that of host Cepheids, which reside in galaxies specifically selected to have similar, roughly solar, metallicity. This difference is accounted for by assuming a metallicity-dependent P-L relation.




The potential systematic errors arising from the aforementioned differences may introduce uncertainties as these effects lack a comprehensive theoretical understanding. Consequently, several studies have scrutinized the potential impact of such systematic errors on the determination of H_0 <cit.>. However, none of these works have presented solid evidence of the existence of systematic biases. Nonetheless, it remains challenging to conclusively rule out a systematic bias (though see the comprehensive assessments in R22), thereby directing many investigations into potential systematic errors in the SH0ES measurement towards the Cepheids. In this paper, we propose a methodology to utilize the SH0ES Cepheids for H_0 measurement in a manner that is robust against Cepheid-related systematic errors. Our approach focuses solely on the hosts and N4258 Cepheids, capitalizing on the fact that the relative distances between the N4258 Cepheids and the hosts Cepheids are less susceptible to many systematic errors that may arise when comparing the hosts Cepheids to other anchor Cepheids. In essence, the NGC 4258 anchor stands out because it is the sole anchor galaxy for which the Cepheids are measured (both in terms of observation technique and data reduction) precisely as the hosts Cepheids.

For instance, consider the correction for blending photometry applied to the hosts Cepheids. Previous works found no evidence that this correction introduces systematic biases, both employing HST <cit.> and JWST observations <cit.>. Nevertheless, it is challenging to rule out the possibility that a significant portion of long-period Cepheids (that are more massive and younger) are physically associated with stars. For example, <cit.> demonstrated by observing Cepheids in M31 that long period Cepheids have a higher chance of being in open clusters (see their Figure 13). The available data, however, are limited to Cepheid ages older than 50 Myr <cit.>. The ages of the long-period Cepheids are ≲20 Myr <cit.>, probably shorter than the dispersing time of open clusters. It is, therefore, reasonable to assume that a significant fraction of long-period Cepheids reside in open clusters. In such scenarios, the photometry of these long-period Cepheids would be biased, as the open clusters are not resolved for the hosts, and the crowding corrections applied by the SH0ES collaborations cannot rectify this effect. When comparing these Cepheids to the close Cepheids, where the cluster is resolved, their relative distances would be biased. While we do not claim the existence of such an effect <cit.>, it inherently cancels out when comparing the hosts Cepheids to the N4258 Cepheids, as the open clusters affect both in the same manner. Other potential systematic errors, such as those in identification, selection, period determination, photometry, and mean magnitude determination, are likewise eliminated or minimized. Therefore, we propose utilizing a subsample of SH0ES Cepheids that excludes the MW, LMC, SMC, and M31 Cepheids, effectively making the determination of H_0 impervious to Cepheid-related systematic errors.

By excluding the MW, LMC, SMC, and M31 Cepheids, the majority of Cepheids in the sample exhibit >1 (see Figure <ref>). Consequently, all Cepheids with <1 can be disregarded, effectively mitigating the sensitivity to breaks in the P-L relation. Moreover, Cepheids in the remaining sample exhibit similar metallicities, and the extinction towards them is minimal. Although systematic differences persist between M101 and NGC 4258 (hereafter nearby galaxies) and other (hereafter faraway) galaxies, such as the average shorter periods of Cepheids in nearby galaxies compared to those in faraway galaxies (see Figure <ref>), we demonstrate in Section <ref> that these systematic differences have negligible effects on the determination of H_0. With this refined sample, the remaining statistical errors in determining the Cepheid relative distance scale, specifically those associated with the zero point of the Cepheid brightness (attributed to intrinsic scatter, measurement error, and finite number of Cepheids) and the SNe Ia brightness (due to intrinsic scatter and measurement error), are sufficiently small to be competitive (Section <ref>). The statistical error is approximately 2.3% (or about 1.65 km/s/Mpc), which is adequate to identify a 3σ tension with the Planck H_0 value.

The systematic errors associated with Cepheids in our approach are significantly smaller than the statistical uncertainty. Specifically, we illustrate that uncertainties related to the reddening law and its free parameters, metallicity sensitivity, and breaks in the P-L relation (or even the existence of a global P-L relation) are either eliminated or substantially reduced. Since these sensitivities are the rationale for opting for NIR over optical photometry, there exists an opportunity to utilize optical photometry with significantly reduced systematic errors stemming from these effects. In this case, the relevant Wesenheit index becomes:

    W_I=F814W-R_I^VI(F555W-F814W),

where R_I^VI≈1.3 is chosen for W_I to be independent of extinction <cit.>. In Section <ref>, we demonstrate that the systematic uncertainty with optical photometry is markedly smaller than the statistical uncertainty, potentially allowing for higher precision than that achieved with NIR photometry, as the optical background is nearly an order of magnitude lower than that in the NIR, owing to higher resolution, smaller pixels, and lower flux from red giants (R22). The reduced sensitivity to extinction correction is attributed to the minimal extinction of Cepheids in our sample (Section <ref>). Upon scrutinizing the optical optical datasets for the 19 hosts of <cit.>, we find that, in practice, the photometry of the optical bands is not much improved compared to the NIR photometry and the fidelity of the optical photometry sample is not at the level of the NIR photometry sample (see Section <ref>). 

The key advantage of our approach is that the significant Hubble tension obtained is unrelated to Cepheids. Given the absence of any indications of SNe Ia-related systematic errors following detailed scrutiny <cit.>, our analysis suggests that either the standard cosmology is flawed or there exists a systematic error in the current distance determination to NGC 4258. We delve into this matter and summarize our findings in Section <ref>.









§ FITTING METHOD AND DATA



In this section, we describe our fitting technique (subsection <ref>) and the dataset (subsection <ref>) used for our main fits. These fits specifically exclude Cepheids from the MW, LMC, SMC and M31, along with those with periods <1.



 §.§ Fitting method



We adopt a fitting methodology similar to that of <cit.>, which bears resemblance to the approach outlined in R16. We express the Wesenheit magnitude, elaborated upon in Section <ref>, in the k band (either I for F814W or H for F160W) of the jth Cepheid within the ith galaxy as follows:

    m_k,i,j^W   =μ_i+M_k^W+b^W_k[P]_i,j+Z^W_kΔlog_10(O/H)_i,j.

Here, [P]_i,j=log P_i,j-1, M_k^W denotes the absolute Cepheid magnitude normalized to a period of =1 and a fiducial metallicity of log_10(O/H)=8.9, Δlog_10(O/H)_i,j=log_10(O/H)_i,j-log_10(O/H) represents the difference between the estimated Cepheid metallicity and the fiducial value, and μ_i=5log_10D_i+25 stands for the distance modulus to the ith galaxy, with D_i being the luminosity distance in Mpc. The nuisance parameters b^W_k and Z^W_k define the relationship between Cepheid period, metallicity, and luminosity. An alternative analysis employing period bins is conducted, where for each Cepheid in the l period bin, we express:

    m_k,i,j^W   =μ_i+M_k^W,l+b^W,l_k[P]_i,j+Z^W,l_kΔlog_10(O/H)_i,j.

Here, M_k^W,l, b^W,l_k and Z^W,l_k define the relation in the l period bin. An analysis incorporating a break in the P-L relation entails two period bins with M_k^W,1=M_k^W,2 and Z_k^W,1=Z_k^W,2.

The magnitudes of SNe Ia in the calibration sample are expressed as:

    m_B,i,j=μ_i+M_B,

Where m_B,i,j represents the maximum-light apparent B-band brightness of the jth SNe Ia in the ith host[or, possibly, to the jth measurement of the same SN Ia.] at the time of B-band peak, corrected to the fiducial color and luminosity. The distance to NGC 4258, μ_N2458=29.397±0.032 mag <cit.>, is incorporated as an additional data point.

We simultaneously fit for b^W_k, Z^W_k, M_k^W (or for b^W,l_k, Z^W,l_k, M_k^W,l in the case of the binned analysis), M_B, and the galaxy distances μ_i. As the system of equations is linear, the fit can be conducted analytically <cit.>. We employ a global outlier rejection threshold of 2.7σ. Our findings in Sections <ref>-<ref> demonstrate that our results are relatively insensitive to the method of outlier rejection. Given M_B, we determine H_0 through:

    H_0=10^M_B/5+a_B+5,

where a_B=0.71273±0.00176 mag represents the intercept of the SNe Ia magnitude-redshift relation (R16).



 §.§ Cepheid Data



Since the latest extragalactic Cepheid optical dataset from R22 has not yet been made publicly accessible, we perform our analysis using two separate datasets, as made available through the data release of R22 [<https://github.com/PantheonPlusSH0ES/DataRelease>. The tables exclusively include Cepheids that pass the 3.3σ outlier rejection criterion of R22.]: 



  * R22-W_H – This dataset incorporates m_H,i,j^W, [P]_i,j, log_10(O/H)_i,j, and σ_H,i,j^W (representing the total statistical uncertainty in m_H,i,j^W; we use the provided covariance matrix as well) for the full (37 hosts; 42 SNe Ia) sample.

  * R22-W_I – Here, we utilize m_I,i,j^W, [P]_i,j, log_10(O/H)_i,j, and σ_I,i,j^W (covariance matric is not provided in this case) for the 19 R16 hosts (and 19 SNe Ia; hosts distances 30 Mpc). We scale σ_I,i,j^W by a factor of 0.85, as recommended in the data release notes (see the discussion in Section <ref>).


The R22 datasets incorporate several improvements compared to earlier data releases, as outlined in Section 3.4 of R22. R22 data release includes m_B,i,j along with its corresponding covariance matrix.









§ W_H FITTING RESULTS



For our primary fit using the R22-W_H dataset, we adopt R_H^VI=0.386 <cit.>, resulting in H_0=72.68±1.67 (refer to case 1 in Table <ref> for detailed information). Figure <ref> illustrates the individual Cepheid P-L relations, where the solid red line indicates the best fit, and the dashed red lines represent one standard deviation of the residuals around the best fit within each galaxy. Cepheids failing the global 2.7σ outlier rejection criterion are highlighted in red. Our outcome closely aligns with the result of fit 10 from R22, which yielded H_0=72.51±1.54, utilizing NGC 4258 as the sole anchor while incorporating all (3500) Cepheids in the analysis.

The obtained error of approximately 2.3% in H_0 is linked to the error in M_B through δ M_Bln(10)/5, where 

    δ M_B^2≈δμ_N4258,anc^2+δΔμ_N4258^2+δ(Δμ_host+m_B)^2.

Here, δμ_N4258,anc≈0.032 mag denotes the absolute distance error to NGC 4258 from megamaser observations <cit.>. δΔμ_N4258 represents the NGC 4258 contribution to the error of the relative distance between NGC 4258 and the host galaxies based on the Cepheids P-L relation. δ(Δμ_host+m_B) signifies the weighted error in the mean of each host's contribution to the relative distance and m_B. For each galaxy, we can estimate δΔμ_i≈(∑_m,n C^-1_mn)^-1/2, where C is the covariance matrix for σ_H,i,j^W, and δ m_B,i≈(∑_m,n D^-1_mn)^-1/2, where D is the covariance matrix for δ m_B,i,j. This estimate leads to δΔμ_N4258≈0.028 mag, which can be approximated by δΔμ_N4258≈0.42/√(223)≈0.028 mag, where 0.42 mag is the mean statistical uncertainty of the 223 N4258 Cepgeids' Wesenheit magnitude. Similarly, we obtain δ(Δμ_host+m_B)=[∑(δΔμ_i^2+δ m_B,i^2)^-1]^-1/2≈0.021 mag, where the SNe Ia contribution to the error budget, δ_m_B=[∑δ m_B,i^-2]^-1/2≈0.016 mag, can be approximated by the 0.11 mag SNe Ia intrinsic scatter, δ_m_B≈0.11/√(37)≈0.018 mag. Our estimate yields δ M_B≈0.047 mag, resulting in an 2.2% error in H_0, consistent with the fit results.





We proceed to demonstrate that the systematic error associated with determining H_0 using our primary R22-W_H fit is considerably smaller than the statistical error. Following the methodology of R22, we evaluate the systematic uncertainty by investigating the impact of various fit variants on the derived value of H_0. Most of the examined variants exhibit a negligible effect on H_0 (|δ H_0|<0.6). These include variants such as incorporating Cepheids with <1 (with or without a different P-L slope; the effect is minimal due to the limited number of Cepheids in this range, approximately 250), exclusion of the outlier rejection, adjustment of the outlier rejection to eliminate the single largest outlier at a time, raising the outlier rejection threshold to 3.5σ (with either global or single-outlier-at-a-time rejection), adjusting the period cutoff to >1.2 (which aligns the period distributions of nearby and faraway galaxies more closely), disregarding the metallicity term (setting Z_H^W=0), adopting R_H^VI=0.3 <cit.>, employing R_H^VI=0.5 <cit.>, and including M31 Cepheids (the effect is minimal due to the limited number of Cepheids in M31 with >1, approximately 50). The sole variant exhibiting a non-negligible effect on the value of H_0 is the adjustment of the period cutoff to <1.6 (δ H_0≈-0.88). Given the minimal impact of all fit variants, we can safely disregard the systematic uncertainty in this case.

The robustness of our primary R22-W_H fit stems from the uniformity of the Cepheid dataset utilized. The consistent observational methodology employed to determine photometry across this sample mitigates many potential systematic errors associated with photometry determination. Generally, any systematic biases that are power-laws of the period are absorbed within the P-L relation, rendering them inconsequential. However, other functional forms or other systematic differences between nearby and faraway galaxies may still exert some influence. For instance, the average periods of Cepheids in nearby galaxies tend to be shorter compared to those in faraway galaxies (refer to Figure <ref>). To scrutinize such effects rigorously, we propose a robust, overarching test. This test directly compares Cepheids with identical periods across nearby and faraway galaxies, without presupposing a universal P-L relation. Before applying this test, it is insightful to revisit our primary fit within a narrow period bin. The outcomes of such fits within 0.1-wide  bins are depicted in Figure <ref>. Evidently, the fit results are distributed around the primary fit outcome, exhibiting no discernible period dependence, and all are consistent with the primary fit (though the precision within each bin is constrained by the number of Cepheids and SNe Ia available, indicated for each bin). Subsequently, we conduct a joint fit across all bins simultaneously, as illustrated in Figure <ref>, encompassing ten  period bins spanning the range [1,2]. This fit assumes only a local power-law P-L relation within each 0.1-wide  bin and remains unaffected by potential period distribution disparities between nearby and faraway galaxies. The resultant determination of H_0=72.50±1.69 fully aligns with our primary fit outcome (case 2 in Table <ref>), with no compelling evidence for a substantially better fit.





In conclusion, our primary W_H fit enables the determination of H_0 with an accuracy of approximately 2.3%, with a significantly reduced impact from systematic uncertainties.









§ W_I FITTING RESULTS





The minimal systematic uncertainties associated with extinction, metallicity, and the P-L relation shape, as demonstrated for our primary W_H fit (Section <ref>), suggest that comparable accuracy can also be achieved with W_I. In this section, we validate this assertion. Leveraging optical photometry offers the potential for enhanced precision compared to NIR photometry, given the significantly lower optical background attributed to higher resolution, smaller pixels, and diminished flux from red giants (R22). While σ_I^W is indeed smaller than σ_H^W (refer to Figure <ref>), the magnitude of this effect is not as substantial as anticipated (with the mean of the error distribution being approximately 0.45 mag for W_H and 0.35 mag for W_I). Further discussion on the fidelity of the R22-W_I dataset is provided towards the end of this section.



For our primary R22-W_I fit, we adopt R_I^VI=1.18 <cit.>, yielding H_0=76.16±1.69 (case 3 in Table <ref>). It is noteworthy that we obtain χ_ν^2≈0.901 for 1299 degrees of freedom, with a probability of 5×10^-3 to achieve a lower value of χ_ν^2. Using the initially published σ_I,i,j^W without the 0.85 multiplier (see Section <ref>) resulted in χ_ν^2≈0.750 for 1321 degrees of freedom, with an insignificant chance of obtaining such a low value of χ_ν^2[After we communicated this discrepancy to the SH0ES collaboration, they introduced the 15% correction.]. This relatively low value of χ_ν^2 might be attributed to an overestimate of σ_I,i,j^W due to the correlation between F555W and F814W background errors[SH0ES collaboration, private communication.], which is presently estimated at 15% (and accounted for in our analysis), although not explicitly discussed in any SH0ES publication. If we increase the assumed error overestimation to 20%, we obtain χ_ν^2≈0.97 that aligns perfectly with 1291 degrees of freedom, without altering any of our conclusions below. Consequently, we proceed with the recommended 15% error overestimation for the optical data, mindful that further quality checks may be necessary before achieving the same level of reliability as the NIR data.

The obtained 2.2% error in H_0 can be interpreted similarly to how it was estimated for the primary W_H fit. We find δΔμ_N4258≈0.019 mag, which can be approximated as δΔμ_N4258≈0.33/√(220)≈0.022 mag, where the mean statistical uncertainty of the 220 N4258 Cepheids' Wesenheit magnitude is 0.33 mag. Similarly, we derive δ(Δμ_host+m_B)≈0.027 mag, with SNe Ia dominates the error budget, where δ_m_B≈0.024 mag, which can be approximated as δ_m_B≈0.11/√(19)≈0.025 mag. Although the photometry of the Cepheids offers higher accuracy, reducing the error of the relative distance between NGC 4258 and the host galaxies based on the Cepheids P-L relation, the smaller number of SNe Ia increases the error, resulting in a total error similar to the W_H case. Our estimate yields δ M_B≈0.046 mag, leading to an error of 2.1% in H_0, consistent with the fit results. It is expected that the H_0 error with the complete 42 SNe Ia sample of R22 would be 1.9%, although inclusion of the full covariance matrix for the optical Wesenheit magnitude (not accounted for in our analysis) might slightly alter this estimate.

We proceed to assess the systematic error in determining H_0 with our primary R22-W_I fit. Most of the examined variants exhibit negligible effects on H_0 (|δ H_0|<0.55). These include incorporating Cepheids with <1, with or without a different P-L slope (the impact is minor due to the limited number of Cepheids, 200, with <1), exclusion of the outlier rejection, adjustment of the outlier rejection to eliminate the single largest outlier at a time, raising the outlier rejection threshold to 3.5σ (with either global or single-outlier-at-a-time rejection), implementing a period cutoff <1.6, disregarding the metallicity term (i.e., setting Z_I^W=0), adopting R_I^VI=0.95 <cit.>, adopting R_I^VI=1.4 <cit.>, and including M31 Cepheids (with minimal impact due to the limited number of 50 Cepheids in M31 with >1). The sole variant exhibiting a non-negligible effect on the value of H_0 is the adjustment of the period cutoff to >1.2 (δ H_0≈-1.26). Given the marginal effects of all fit variants, we conclude that the systematic uncertainty can be disregarded in this case as well. One particularly surprising outcome is the observed low sensitivity to R_I^VI, considering the conventional rationale for employing W_H due to the presumed high sensitivity of W_I to extinction correction. We elucidate the reason for this diminished sensitivity with our sample in the subsequent section.

We next replicate our binned analysis for the W_I scenario. The outcomes of the fits in 0.1- bins are displayed in Figure <ref>. As depicted in the figure, although the fit outcomes in each bin align with the primary fit result (albeit with accuracy limited by the number of Cepheids and SNe Ia in each period bin, indicated in the figure), there appears to be a discernible trend with period in this case. Upon fitting H_0 in each bin as a linear function of , we ascertain a slope of 11.4±3.5 km s^-1 Mpc^-1 dex^-1 (a significance of 3.3σ). This should be considered an upper limit to the trend, as we have not accounted for the correlation among some bins stemming from shared hosts, and all bins are influenced by the NGC 4258 distance error. Fitting only for >1.2, where each bins contains a significant fraction of the hosts (to minimize the variance of the SNe Ia sample compared with the Cepheid sample), we obtain a slope of 10.1±4.7 km s^-1 Mpc^-1 dex^-1 (a significance of 2.2σ). This observation might signify additional concerns with the optical photometry (in addition to the overestimated σ_I,i,j^W). Subsequently, upon fitting all bins collectively, we arrive at H_0=75.25±1.71, aligning with the primary fit result (case 4 in Table <ref>). We discern no substantial evidence for a superior fit, with a χ^2 smaller by approximately 59 from a single, [1,2], -bin fit and 37 fewer degrees of freedom.











§ LIMITED IMPACT OF EXTINCTION CORRECTIONS



In the preceding section, we illustrated the minimal impact of extinction correction on the W_I fitting results. Here, we delve into the rationale behind this observation. We commence with a discussion on extinction corrections (subsection <ref>), followed by an evaluation of the extinction distribution of extragalactic Cepheids (subsection <ref>). Subsequently, in subsection <ref>, we showcase a substantial sensitivity to extinction when incorporating MW Cepheids.



 §.§ Correction for extinction



The rationale behind employing the Wesenheit index lies in its minimal susceptibility to extinction <cit.>. Defined for three bands,  X, Y, and Z, the index is expressed as

    W=X-R_X^YZ(Y-Z),

measured in magnitudes. In the presence of extinction, this can be expressed as

    W   =   (X-X_0)+X_0-R_X^YZE(Y-Z)-R_X^YZ(Y-Z)_0
       =   X_0-R_X^YZ(Y-Z)_0≡ W_0,
   
where the subscript 0 represents the extinction-free magnitude, E(Y-Z)=(Y-Z)-(Y-Z)_0 denotes the selective extinction, and the final equality holds if R_X^YZ=(X-X_0)/E(Y-Z)=A_X/E(Y-Z)≡R̃^X_YZ, with A_X representing the total extinction in band X. An often-overlooked aspect is that the Wesenheit index doesn't correct X for extinction; instead, with the appropriate choice of R_X^YZ, it yields a quantity insensitive to extinction (also noted in R22). For any arbitrary value of R_X^YZ, the deviation of W from W_0 is given by

    W-W_0   =   A_X-R_X^YZE(Y-Z)
       =   (R̃^X_YZ-R_X^YZ)E(Y-Z),
  
such that even if R̃^X_YZ is not precisely known or varies across different lines-of-sight towards the Cepheids, the deviation in W-W_0 remains minimal as long as the selective extinction E(Y-Z) remains small. With zero extinction, E(Y-Z)=0, any value of R_X^YZ can be chosen. However, this value must be consistent across all Cepheids; otherwise, disparate quantities are being compared. <cit.> selected different values of R_X^YZ for different Cepheids, rendering their conclusion invalid for that analysis (also highlighted in R22).

An alternative approach involves directly adjusting X for extinction <cit.>:

    F   =   X-R_X^YZE(Y-Z),
       =   (X-X_0)+X_0-R_X^YZ[(Y-Z)-(Y-Z)_0]
       =   X_0,
 
where the last equality holds if R_X^YZ=R̃^X_YZ. Similar to the W case, for any given value of  R_X^YZ, the deviation of F from X_0≡ F_0 is given by

    F-F_0   =(R̃^X_YZ-R_X^YZ)E(Y-Z).
  
The distinction lies in the allowance for choosing different R_X^YZ values for different Cepheids, as the color term vanishes under zero extinction. However, this method assumes knowledge of (Y-Z)_0, which is not always available, such as for (V-I)_0 in the range >1.72 <cit.>, where a significant fraction of the extragalactic Cepheids is situated (see Figure <ref>). It is worth noting that if (Y-Z)_0 follows a linear relationship with , the analysis with F is identical to that with W (for a single global R_X^YZ value), as the difference between E(Y-Z) and (Y-Z) gets absorbed into M_k^W and b_k^W <cit.>. 

We infer that minimizing sensitivity to extinction corrections involves selecting Cepheids with minimal extinction. As demonstrated in Section <ref>, this holds true for both nearby and faraway galaxies, elucidating the reduced susceptibility of our dataset to extinction correction.



 §.§ The extinction distribution



We initiate the analysis by assessing the reddening distribution of extragalactic Cepheids (utilizing the R22-W_H sample unless specified otherwise). The top panel of Figure <ref> displays the (F555W-F814W) color of extragalactic Cepheids against their period. We confine this examination to <1.72, where an accurate determination of the intrinsic MW Cepheids colors is available <cit.>. In Appendix <ref>, we compute the intrinsic color for 1<<1.72 <cit.>:

    (F555W-F814W)_0   ≈   0.156+0.75,

represented by a solid black line. Additionally, we distinguish between nearby (M101 and NGC 4258; black circles) and faraway (the other hosts; red circles) galaxies, considering that Cepheids in nearby galaxies are measured with slightly higher precision (see Figure <ref>). The errors in (F555W-F814W) color (not depicted) primarily stem from dispersion due to blending, amounting to approximately 0.15,mag for the nearby sample and 0.2,mag for the faraway sample.



Subsequently, we estimate the host galaxy E(B-V) toward each Cepheid using (see Appendix <ref>):

    E(B-V)   ≈   1.42[(F555W-F814W)-(F555W-F814W)_0],

and then we correct for (the small, 0.1 mag) Galactic extinction <cit.>. The resulting E(B-V) distribution is depicted in the bottom panel of Figure <ref>. As illustrated in the figure, the E(B-V) distribution of the nearby sample (solid black line) exhibits a sharp change at E(B-V)≈0, indicative of a blue edge. This blue edge corroborates both Equation (<ref>) and the photometry of nearby Cepheids in (F555W-F814W), which were determined through entirely independent methods[For instance, deriving Equation (<ref>) necessitates knowledge of the extinction toward MW Cepheids, sourced from <cit.>.]. The small number of Cepheids with negative E(B-V) (physically implausible values) aligns with the scatter of σ_int≈0.1 mag around the intrinsic P-C relation <cit.>, as well as with color errors stemming from blending. The blue edge of the faraway sample's E(B-V) distribution (solid red line) appears somewhat less distinct, which is understandable given the larger color errors associated with faraway galaxies that can blur the blue edge. 

The distributions' shapes are influenced by various selection biases, notably favoring Cepheids with minimal extinction (E(B-V)≲0.5), resulting in a negligible impact of extinction corrections. To further illustrate this minimal impact, we perform our primary fit without the color term, setting R_I^VI=R_H^VI=0. For W_H, we derive H_0=72.39±1.66 (case 5 in Table <ref>), compared to H_0=72.68±1.67 with R_H^VI=0.386. Similarly, for W_I, we obtain H_0=74.42±1.66 (case 6 in Table <ref>), contrasting with H_0=76.16±1.69 with R_I^VI=1.18. The overall effect on H_0 amounts to δ H_0≈0.3 for W_H and δ H_0≈1.7 for W_I, validating our assertion of small systematic uncertainty in Sections <ref> and <ref>.



 §.§ Incorporating MW Cepheids



Here, we illustrate the significant sensitivity of the W_I fitting results to extinction that arises when including MW Cepheids. Initially, we repeat our primary W_H fit with the inclusion of MW Cepheids, employing the MW anchor instead of the NGC 4258 anchor. Following the methodology of <cit.>, we transform the residual Gaia parallax calibration offset, zp, into a linear parameter and fit for it. The data for MW Cepheids are sourced from Table 1 in <cit.>. Since a significant fraction of the MW Cepheids have <1, we do not restrict the Cepheid sample to >1 in this case (for all galaxies). We obtain H_0=74.43±2.21[We incorporate an intrinsic dispersion of σ_int=0.069 mag for W_H of the MW Cepheids <cit.>.] (case 7 in Table <ref>), consistent with the result using the NGC 4258 anchor, H_0=72.68±1.67. Similarly small impact is observed for our primary W_I fit[Here, we adopt σ_int=0.085 mag for the intrinsic dispersion <cit.>.] (H_0=75.67±2.24 for the MW anchor, case 8 in Table <ref>, compared to H_0=76.16±1.69 for the NGC 4258 anchor). Subsequently, we demonstrate the considerable systematic uncertainty of the W_I fitting results related to the extinction law, attributed to the substantial extinction associated with MW Cepheids.

The bottom panel of Figure <ref> illustrates the selective extinction affecting the MW Cepheids utilized in our analysis. These data are sourced from the catalog of <cit.>, with detailed references therein outlining the methods employed to estimate the extinction. As depicted in the figure, MW Cepheids, particularly those with >1, endure substantial extinction due to their location within the Galactic disc, where massive stars are prevalent, relative to the position of the Sun. When using the MW anchor instead of the NGC 4258 anchor and setting R_I^VI=R_H^VI=0, we obtain H_0=74.28±2.21 for W_H (case 9 in Table <ref>) and H_0=83.1±2.6 for W_I (case 10 in Table <ref>). While a negligible effect is obtained for the W_H case, the effect is notably pronounced in the W_I case (δ H_0≈7.5), indicating that even minor uncertainties in correcting for this effect would result in significant systematic uncertainties. The heightened effect observed for W_I primarily motivates observations with the F160W filter. However, as demonstrated in Section <ref>, this effect is considerably mitigated for our sample. 

A comparable scenario arises for M31. The selective extinction affecting M31 Cepheids is depicted in the bottom panel of Figure <ref>. This distribution is derived using the methodologies outlined in Section <ref>. As illustrated in the figure, M31 Cepheids experience substantial extinction due to the nearly edge-on orientation of M31's disk, but close enough to allows for the measurement of obscured Cepheids.









§ DISCUSSION AND SUMMARY



The SH0ES collaboration employs a comprehensive approach to ascertain H_0, combining host Cepheids with a diverse array of Cepheids from various observations. This strategy minimizes statistical uncertainty in H_0 and harnesses multiple independent geometrical distance anchors, mitigating sensitivity to potential systematic errors in any single anchor distance determination. Cepheids from NGC 4258, as well as from closer galaxies like the MW, LMC, SMC, and M31, are incorporated to allow the use of multiple anchors and to enhance the precision of the P-L relation. However, this Cepheid sample with a wide-distance range is susceptible to systematic errors due to the comparison of distant host Cepheids with those much closer. Host and NGC 4258 Cepheids are measured via HST imaging, while closer Cepheids are identified through archival observations and observed using different techniques. Corrections are applied to account for differences in observing methodologies and biases in period distributions and extinction levels between close and host Cepheids. Additionally, the metallicity discrepancy between LMC/SMC and host Cepheids is addressed by considering a metallicity-dependent P-L relation.

We proposed utilizing a subset of SH0ES Cepheids to derive H_0 independently of potential systematic errors associated with Cepheids. This subset excludes Cepheids from the MW, LMC, SMC, and M31, retaining only the water megamasers in NGC 4258 as a single anchor. By omitting these Cepheids, which are subject to different observational methods and reduction techniques compared to those in host galaxies and NGC 4258, potential systematic errors in Cepheid measurement are circumvented, encompassing identification, selection, period determination, photometry, template usage, and crowding correction. We demonstrate the elimination or substantial reduction of uncertainties pertaining to reddening law, metallicity sensitivity, and breaks in the P-L relation, facilitating precise determination of Cepheid relative distances. Remaining statistical errors in Cepheid relative distance determination, primarily associated with Cepheid and SNe Ia brightness zero points, are sufficiently small for competitiveness, 2.3% (or about 1.65 km/s/Mpc) using the R22 data release. The Cepheid-related systematic errors are significantly smaller than the statistical uncertainty, rendering the accuracy adequate to identify 3σ tension with the Planck H_0 value. 

Once the decision is made to forego the local anchor points, there is no inherent requirement to shift to the NIR. As demonstrated in Section <ref>, we established that the systematic uncertainty associated with optical photometry is considerably smaller than the statistical uncertainty. This implies the potential for achieving greater accuracy compared to NIR photometry, primarily because the optical background is approximately an order of magnitude lower than that in the NIR, owing to factors such as higher resolution, smaller pixels, and reduced flux from red giants (R22). The observed low sensitivity to extinction correction stems from the minimal extinction experienced by Cepheids within our sample (Section <ref>). However, in practice, the quality of photometry in optical bands does not exhibit substantial improvement over NIR photometry, and the reliability of the optical photometry sample does not match that of the NIR photometry sample, as demonstrated in Section <ref>.

The primary drawback of our approach lies in the pronounced reliance of H_0 on the distance to NGC 4258, D_4258, as expressed by the equation:

    dH_0/H_0≈-fdD_425/D_4258,
 
where f≈1, in contrast to f≈1/3 with the inclusion of three anchors (NGC 4258, LMC, and MW). This heightened dependency renders our method more susceptible to systematic uncertainties in D_4258. Nonetheless, given the resilience of our method against Cepheid-related systematic errors and the absence of discernible signs of SNe Ia-related systematic errors in detailed examination  <cit.>, our analysis suggests two potential scenarios: either standard cosmology requires revision or there exists a systematic error in the current determination of the distance to NGC 4258, D_4258≈7.57±0.24 Mpc. To fully alleviate the Hubble tension, a distance of D_4258≈8.15 Mpc is necessary, and considering the prevailing uncertainties, a distance (with a central value) exceeding D_4258≳7.8 Mpc is needed to reduce the tension below 2σ. Moreover, the scenario of systematic error in the distance to NGC 4258 to resolve the Hubble tension would also necessitate an unrelated systematic error in the comparison of distant host Cepheids with those much closer to explain the H_0 measurement of R22. However, solid evidence supporting this assertion has not been presented so far. 

Our investigation underscores the importance of conducting a meticulous analysis of optical observations of extragalactic Cepheids to enable the application of our method with an optical Cepheid sample. Looking ahead, observations with JWST hold promise for further reducing optical crowding corrections <cit.>. Moreover, the heightened sensitivity of JWST may facilitate the measurement of Cepheids in additional megamaser galaxies with precise geometrical distances, such as UGC 3789 <cit.>, thereby enabling the extension of our method with supplementary anchors.



§ ACKNOWLEDGEMENTS

We thank members of the SH0ES collaboration: Adam Riess, Dan Scolnic, Lucas Macri, Stefano Casertano, and Wenlong Yuan for extensive correspondences and discussions that significantly improved the manuscript. We thank Andy Gould, Eli Waxman, Boaz Katz, Barak Zackay, Subo Dong, and Eran Ofek for useful discussions. DK is supported by a research grant from The Abramson Family Center for Young Scientists, and by the Minerva Stiftung. 



§ DATA AVAILABILITY


All data used in this study are publicly available through other publications. 













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§ THE PERIOD-COLOR RELATION



This appendix utilizes the catalog provided by <cit.> to investigate the intrinsic period-color (P-C) relationship of Cepheids falling within the range of 1 < log P < 1.72 across various colors, as pertinent to the analysis discussed in Section <ref>. We adopt the methodology outlined by <cit.>. Initially, we compute (B-V)_0 by subtracting E(B-V)[E(B-V) values in the catalog are sourced from <cit.>.] from (B-V), as illustrated in the upper left panel of Figure <ref> (following the exclusion of outliers VZ-Pup and ER-Aur). We fit (B-V)_0 against log P linearly and observe a relatively high reduced χ^2, suggesting that the scatter around the fit is intrinsic, so we calibrate an intrinsic scatter of approximately 0.07 mag to obtain a unity for the reduced χ^2. Upon incorporating the calibrated intrinsic scatter, the resulting fit is (B-V)_0=(0.387±0.048)+(0.331±0.059) mag (depicted by black lines, with a scatter of 0.10 mag). In the bottom left panel, we depict the deviation of each Cepheid from the best-fit line, Δ(B-V), against the estimated E(B-V). No discernible trend is observed (the slope is -0.010 ± 0.027, indicated by black lines), indicating the absence of a systematic scaling error in the determined E(B-V) values. Notably, our derived intrinsic (B-V)_0 color closely resembles the findings of <cit.>, albeit with updated E(B-V) values sourced from <cit.> for 56 Cepheids and a limitation of Cepheid periods to 1 < log P < 1.72.



We continue to adhere to the methodology outlined by <cit.> for the determination of (V-I)_0. By calibrating E(V-I) = 1.33E(B-V), and subsequently analyzing (V-I) in a manner akin to that of (B-V), we find no discernible dependence of Δ(V-I) on E(V-I) (depicted in the middle panels of Figure <ref>, following the exclusion of outlier V0396-Cyg). Our best-fit result is (V-I)_0=(0.225±0.044)+(0.517±0.054) mag (represented by black lines, with a scatter of 0.11 mag). Additionally, the ratio between E(V-I) and E(B-V), as well as the obtained best fit, closely resemble the findings of <cit.>.

Finally, we calibrate E(F555W-F814W) = 1.42E(B-V) (as depicted in the right panels of Figure <ref>), resulting in (F555W-F814W)_0=(0.156±0.083)+(0.750±0.102) mag (illustrated by black lines, with a scatter of 0.08 mag).






