A Reanalysis of the Latest SH0ES Data for H₀: Effects of New Degrees of Freedom on the Hubble Tension

Abstract

We reanalyze in a simple and comprehensive manner the recently released SH0ES data for the determination of $H_0$. We focus on testing the homogeneity of the Cepheid+SnIa sample and the robustness of the results in the presence of new degrees of freedom in the modeling of Cepheids and SnIa. We thus focus on the four modeling parameters of the analysis: the fiducial luminosity of SnIa $M_B$ and Cepheids $M_W$ and the two parameters ($b_W$ and $Z_W$) standardizing Cepheid luminosities with period and metallicity. After reproducing the SH0ES baseline model results, we allow for a transition of the value of any one of these parameters at a given distance $D_c$ or cosmic time $t_c$, thus adding a single degree of freedom in the analysis. When the SnIa absolute magnitude $M_B$ is allowed to have a transition at $D_c\simeq 50$ Mpc (about 160 Myrs ago), the best-fit value of the Hubble parameter drops from $H_0=73.04\pm 1.04$ km s${}^{-1}$ Mpc${}^{-1}$ to $H_0=67.32\pm 4.64$ km s${}^{-1}$ Mpc${}^{-1}$ in full consistency with the Planck value. Additionally, the best-fit SnIa absolute magnitude $M_B^{>}$ for $D>D_c$ drops to the Planck inverse distance ladder value $M_B^{>}=-19.43\pm 0.15$, while the low distance best fit $M_B^{<}$ parameter remains close to the original distance ladder calibrated value $M_B^{<}=-19.25\pm 0.03$. Similar hints for a transition behavior is found for the other three main parameters of the analysis ($b_W$, $M_W$ and $Z_W$) at the same critical distance $D_c\simeq 50$ Mpc, even though in that case, the best-fit value of $H_0$ is not significantly affected. When the inverse distance ladder constraint on $M_B^{>}$ is included in the analysis, the uncertainties for $H_0$ reduce dramatically ($H_0=68.2\pm 0.8$ km s${}^{-1}$ Mpc${}^{-1}$), and the $M_B$ transition model is strongly preferred over the baseline SH0ES model ($\Delta {\chi }^2\simeq -15$, $\Delta AIC\simeq -13$) according to the AIC and BIC model selection criteria.

1. Introduction

1.1. The Current Status of the Hubble Tension and Its Four Assumptions

Measurements of the Hubble constant using observations of type Ia supernovae (SnIa) with Cepheid calibrators by the SH0ES team has led to a best-fit value $H_0^{R21}=73.04\pm 1.04$ km s${}^{-1}$ Mpc${}^{-1}$ [1] (hereafter R21). This highly precise but not necessarily accurate measurement is consistent with a wide range of other less precise local measurements of $H_0$ using alternative SnIa calibrators [2,3,4,5], gravitational lensing [6,7,8,9], gravitational waves [10,11,12,13,14], gamma-ray bursts as standardizable candles [15,16,17,18,19], quasars as distant standard candles [20], type II supernovae [21,22], $\gamma -$ray attenuation [23], etc. (for recent reviews, see Refs. [24,25]). This measurement is based on two simple assumptions:

• There are no significant systematic errors in the measurements of the properties (period, and metallicity) and luminosities of Cepheid calibrators and SnIa.
• The physical laws involved and the calibration properties of Cepheids and SnIa in all the rungs of the distance ladder are well understood and modeled properly.

This measurement, however, is at $5\sigma$ tension (Hubble tension) with the corresponding measurement from Planck observations of the CMB angular power spectrum (early time inverse distance ladder measurement) $H_0^{P18}=67.36\pm 0.54$ km s${}^{-1}$ Mpc${}^{-1}$ [26] (see also Refs. [24,25,27,28,29,30,31,32,33,34] for relevant recent reviews). This inverse distance ladder measurement is also based on two basic assumptions:

• The scale of the sound horizon at the last scattering surface is the one calculated in the context of the standard cosmological model with the known degrees of freedom (cold dark matter, baryons and radiation) and thus, it is a reliable distance calibrator.
• The evolution of the Hubble free expansion rate $E\left(z\right)\equiv H\left(z\right)/H_0$ from the time of recombination (redshift $z=z_{rec}$) until the present time ($z=0$) is the one predicted by the standard $\Lambda$CDM model as defined by the best-fit Planck parameters (Planck18$/\Lambda$CDM) [26,35].

A wide range of approaches have been implemented in efforts to explain this Hubble tension (for reviews, see Refs. [24,25,27,34,36,37,38,39]). These approaches introduce new degrees of freedom that violate at least one of the above four assumptions and may be classified in accordance with the particular assumption that they are designed to violate.

Thus, early time sound horizon models introduce new degrees of freedom at the time just before recombination (e.g., early dark energy [36,40,41,42,43,44,45,46,47,48,49,50,51,52], radiation [53,54,55,56] or modified gravity [57,58,59,60,61,62]) to change the expansion rate at that time and thus decrease the sound horizon scale $r_s$ (early time distance calibrator) to increase $H_0$, which is degenerate with $r_s$, to a value consistent with local measurements.

The mechanism proposed by these models attempts to decrease the scale of the sound horizon at recombination, which can be calculated as

$$r_s={\int }_0^{t_{*}}\frac{c_s\left(a\right)}{a\left(t\right)}dt={\int }_{z_{*}}^{\infty }\frac{c_s\left(z\right)}{H(z;{\rho }_b,{\rho }_{\gamma },{\rho }_c)}dz={\int }_0^{a_{*}}\frac{c_s\left(a\right)}{a^{2}H(a;{\rho }_b,{\rho }_{\gamma },{\rho }_c)}da$$

(1)

where the recombination redshift $z_{*}$ corresponds to time $t_{*}$; ${\rho }_b$, ${\rho }_c$ and ${\rho }_{\gamma }$ denote the densities for baryon, cold dark matter and radiation (photons), respectively; and $c_s$ is the sound speed in the photon–baryon fluid. The angular scale of the sound horizon is measured by the peak locations of the CMB perturbations angular power spectrum and may be expressed in terms of $r_s$ as

$${\theta }_s=\frac{r_s}{d_A}=\frac{H_0{r}_s}{c{\int }_0^{z_{*}}\frac{d{z}^{\prime }}{E\left(z^{\prime }\right)}}$$

(2)

where $d_A$ is the comoving angular diameter distance to last scattering (at redshift $z\approx 1100$) and $E\left(z\right)$ is the dimensionless normalized Hubble parameter, which for a flat $\Lambda$CDM model is given by

$$E\left(z\right)\equiv \frac{H\left(z\right)}{H_0}={\left[{\Omega }_{0m}{(1+z)}^3+(1-{\Omega }_{0m})\right]}^{1/2}$$

(3)

Equation (2) indicates that there is a degeneracy between $r_s$, $H_0$ and $E\left(z\right)$, given the measured value of ${\theta }_s$. A decrease in $r_s$ would lead to an increase in the predicted value of $H_0$ (early time models), and a late time deformation of $E\left(z\right)$ could lead to an increase in the denominator of Equation (2), leading also to an increase in $H_0$ (late time models).

Early dark energy models have the problem of predicting the stronger growth of perturbations than implied by dynamical probes, such as redshift space distortion (RSD) and weak lensing (WL) data and thus may worsen the ${\Omega }_m$−${\sigma }_8$ growth tension [63,64,65,66,67,68,69,70,71,72] and reduce consistency with growth data and with other cosmological probes and conjectures [38,39,47,48,49,72,73,74,75,76]. Thus, a compelling and full resolution of the Hubble tension may require multiple (or other) modifications beyond the scale of the sound horizon predicted by $\Lambda$CDM cosmology. Even though these models are severely constrained by various cosmological observables, they currently constitute the most widely studied class of models [42,43,44,50,77,78].

Late time $H\left(z\right)$ deformation models introduce new degrees of freedom (e.g., modified gravity [79,80,81,82], dynamical late dark energy [83,84,85,86,87,88,89,90,91,92,93,94,95,96], or interacting dark energy with matter [85,97,98,99,100,101]) to deform $E\left(z\right)$ at redshifts $z\sim O\left(1\right)$ so that the present time value of $H(z=0)=H_0$ increases and becomes consistent with the local measurements. This class of models is even more severely constrained [84,102,103,104,105,106,107,108,109,110,111,112] by other cosmological observables (SnIa, BAO and growth of perturbations probes), which tightly constrain any deformation [35] from the Planck18/$\Lambda$CDM shape of $E\left(z\right)\equiv H\left(z\right)/H_0$.

The third approach to the resolution of the Hubble tension is based on a search for possible unaccounted systematic effects, including possible issues in modeling the Cepheid data, such as non-standard dust-induced color correction [113], the impact of outliers [114,115,116], blending effects, SnIa color properties [117], robustness of the constancy of the SnIa absolute magnitude in the Hubble flow [67,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134], etc. There is currently a debate about the importance of these potential systematic effects [135,136,137,138]. Additionally, the possibility for redshift evolution of the Hubble constant was studied by Ref. [139]. In [139], the Hubble tension was analyzed with a binned analysis on the Pantheon sample of SNe Ia through a multidimensional MCMC analysis. Finally, the need for new standardizable candles with redshift values far beyond the SnIa ($3\lesssim z\lesssim 9$) was studied (see Refs. [15,16,17,18,19] for gamma-ray bursts and Refs. [140,141] for quasars).

A fourth approach related to the previous one is based on a possible change of the physical laws (e.g., a gravitational transition [142,143]) during the past $200Myrs$ ($z\lesssim 0.01$) when the light of the Cepheid calibrator hosts was emitted [143,144,145,146,147,148,149,150,151,152]. In this context, new degrees of freedom should be allowed for the modeling of either Cepheid calibrators and/or SnIa to allow for the possibility of this physics change. If these degrees of freedom are shown not to be excited by the data, then this approach would also be severely constrained. It is possible, however, that nontrivial values of these new parameters are favored by the data, while, at the same time, the best-fit value of $H_0$ shifts to a value consistent with the inverse distance ladder measurements of $H_0$ using the sound horizon at recombination as the calibrator. In this case [146], this class of models would be favored, especially in view of the severe constraints that have been imposed on the other three approaches.

The possible new degrees of freedom that could be allowed in the Cepheid+SnIa modeling analysis are clearly infinite, but the actual choices to be implemented in a generalized analysis may be guided by three principles: simplicity, physical motivation, and improvement of the quality of fit to the data.

In a recent analysis [146], using a previous release of the SH0ES data [153,154,155], we showed that a physically motivated new degree of freedom in the Cepheid calibrator analysis allowing for a transition in one of the Cepheid modeling parameters $R_W$ or $M_W$ is mildly favored by the data and can lead to a reduced best fit value of $H_0$. Here, we extend that analysis in a detailed and comprehensive manner to a wider range of transition degrees of freedom using the latest publicly available SH0ES data described in R21.

1.2. The New SH0ES Cepheid+SnIa Data

The new Cepheid+SnIa data release and analysis from the SH0ES collaboration in R21 includes a significant increase of the sample of SnIa calibrators from 19 in Ref. [153] to 42. These SnIa reside in 37 hosts observed between 1980 and 2021 in a redshift range $0.0015\lesssim z<0.011$ (see

Table 1. A comparison of the latest SH0ES data release (R21) with previous data updates.

SH0ES	Cepheid + SnIa	Cepheids	Calibrator	Hubble Flow
Year/Ref.	Host Galaxies		SnIa	SnIa
		MW            15
		LMC a         785
2016	19	N4258         139	19	217
R16 [153]	$z<0.01$	M31           372	$z<0.01$	$0.0233<z<0.15$
		Total         1311
		In SnIa hosts     975
		Total All      2286
		MW            15
		LMC b     785 + 70
2019	19	N4258          139	19	217
R19 [154]	$z<0.01$	M31           372	$z<0.01$	$0.0233<z<0.15$
		Total         1381
		In SnIa hosts     975
		Total All      2356
		MW            75
		LMC b    785 + 70
2020	19	N4258         139	19	217
R20 [155]	$z<0.01$	M31           372	$z<0.01$	$0.0233<z<0.15$
		Total         1441
		In SnIa hosts     975
		Total All       2416
		LMC b     270 + 69
		SMC a         143
2021	37	N4258         443	42	277
R21 [1]	$0.0015\lesssim z<0.011$	M31              55	$0.0015\lesssim z<0.011$	$0.023<z<0.15$
		Total            980
		In SnIa hosts    2150	(77 lightcurve meas.)
		Total All      3130

NOTE: (a) From the ground. (b) From the ground + HST.

for a more detailed comparison of the latest SH0ES data release with previous updates). These SnIa are calibrated using Cepheids in the SnIa host galaxies. In turn, Cepheid luminosities are calibrated using geometric methods in calibrator nearby galaxies (anchors). These anchor galaxies include the megamaser host NGC42581, the Milky Way (MW) where distances are measured with several parallaxes, and the Large Magellanic Cloud (LMC) where distances are measured via detached eclipsing binaries [154] as well as two supporting anchor galaxies ($M31$ [157] and the Small Magellanic Cloud (SMC). These supporting galaxies are pure Cepheid hosts and do not host SnIa but host large and well measured Cepheid samples. However, geometric measurements of their distances are not so reliable and thus are not directly used in the analysis2. The calibrated SnIa in the Hubble flow ($z\gtrsim 0.01$) are used to measure $H_0$ due to to their high precision (5% in distance per source) and high luminosity which allows deep reach and thus reduces the impact of local velocity flows.

The new SH0ES data release includes a factor of 3 increase in the sample of Cepheids within NGC4258. In total it has 2150 Cepheids in SnIa hosts3, 980 Cepheids in anchors or supporting galaxies4, 42 SnIa (with total 77 lightcurve dataset measurements) in 37 Cepheid+SnIa hosts with redshifts $0.0015\lesssim z<0.011$ and 277 SnIa in the Hubble flow in the redshift range $0.023<z<0.15$. In addition 8 anchor based constraints (with uncertainties included) constrain the following Cepheid modeling parameters: $M_W$ (the Cepheid absolute magnitude zeropoint), $b_W$ (the slope of the Cepheid Period-Luminosity P-L relation), $Z_W$ (the slope of the Cepheid Metallicity-Luminosity M-L relation), a zeropoint parameter $zp$ used to refine the Cepheid P-L relation by describing the difference between the ground and HST zeropoints in LMC Cepheids (zp is set to 0 for HST observations), the distance moduli of the anchors NGC4258 and LMC and a dummy parameter we call X which has been included in the R21 data release and is set to 0 with uncertainty ${10}^{-9}$5.

The parameters that fit with these data include the four modeling parameters $M_W$, $b_W$, $Z_W$, $M_B$ (the SnIa absolute magnitude), the 37 distance moduli of SnIa/Cepheid hosts, the distance moduli to the 2 anchors (NGC4258, LMC) and to the supporting Cepheid host M31, the zeropoint $zp$ of the Cepheid P-L relation in the LMC ground observations, the Hubble parameter, and the dummy parameter mentioned above (tightly constrained to 0). This is a total of 47 parameters (46 if the dummy parameter X is ignored).

In addition to these parameters, there are other modeling parameters such as the color and shape correction slopes of SnIa (usually denoted as $\beta$ and $\alpha$) as well as the Wesenheit dust extinction parameter $R_W$, which were incorporated in the released SnIa and Cepheid apparent magnitudes and thus cannot be used as independent parameters in the analysis, in contrast to the previous data release.

The provided R21 Cepheid Wesenheit dust-corrected dereddened apparent magnitudes $m_H^W$ are connected with the Wesenheit dust extinction parameter $R_W$ as [158] (see also Refs. [153,154])

$$m_H^W\equiv m_H-R_W(V-I)$$

(4)

where $m_H$ is the observed apparent magnitude in the near-infrared H (F160W) band, and V (F555W) and I (F814W) are the optical mean apparent magnitudes in the corresponding bands. The empirical parameter $R_W$ is also called ’the reddening-free “Wesenheit” color ratio’ and is different from $R_H$, which can be derived from a dust law (e.g., the Fitzpatrick law [159]). The parameter $R_W$ corrects for both dust and intrinsic variations applied to observed blackbody colors $V-I$.

The provided in R21 SnIa apparent magnitudes $m_B^0$, standardized using light curve color c and shape $x_1$ corrections, are defined as

$$m_B^0\equiv m_B-\alpha x_1-\beta c=\mu +M_B$$

(5)

where $m_B$ is the peak apparent magnitude, $\mu$ is the SnIa distance modulus, while the B-band absolute magnitude, $M_B$, and correction coefficients $\alpha$ and $\beta$ are fit directly using the SnIa data. The latest SH0ES data release provides the measured values of $m_H^W$ and $m_B^0$ for Cepheid and SnIa, respectively, which are also used in the corresponding analysis, while the parameters $R_W$, $\alpha$ and $\beta$ were fit previously and independently by the SH0ES team to provide the values of $m_H^W$ and $m_B^0$.

1.3. The Prospect of New Degrees of Freedom in the SH0ES Data Analysis

The homogeneity of the SH0ES data with respect to the parameters $R_W$, $\alpha$ and $\beta$ were analyzed in previous studies with some interesting results. In particular, using the data of the previous SH0ES data release [153,154,155], it was shown [146] (see also [113] for a relevant study) that if the parameter $R_W$ is allowed to vary among Cepheid and SnIa hosts, then the fit quality is significantly improved, and the best-fit value of $H_0$ is lowered to a level consistent with the inverse distance ladder best fit. In addition, a more recent analysis allowed the parameter $\beta$ to have a different value in Hubble flow SnIa ($\beta ={\beta }_{HF}$ for $z>0.02$) compared to calibrating SnIa ($\beta ={\beta }_{cal}$ for $z<0.01$). A reanalysis allowing for this new degree of freedom indicated a tension between the the two best-fit values (${\beta }_{HF}$ and ${\beta }_{cal}$) at a level of up to $3.8\sigma$ [117].

Motivated by these hints for inhomogeneities in the SH0ES data, in what follows, we introduce new degrees of freedom in the analysis that are designed to probe the origin of these inhomogeneities. We thus accept three of the four above-mentioned assumptions that have led to the Hubble tension and test the validity of the fourth assumption. In particular, we keep the following assumptions:

• There are no significant systematics in the SH0ES data, and thus they are reliable.
• The CMB sound horizon scale used as a calibrator in the inverse distance ladder approach is correctly obtained in the standard model using the known particles.
• The Hubble expansion history from the time of recombination up to $z=0.01$ (or even $z=0$) used in the inverse distance ladder measurement of $H_0$ is provided correctly by the standard Planck18$/\Lambda$CDM cosmological model.

As discussed above, there are several studies in the literature that support the validity of these assumptions (e.g., [45,104]). If these assumptions are valid, then the most probable source of the Hubble tension is the violation of the fourth assumption stated above, namely ‘the physical laws involved and the calibration properties of Cepheids and SnIa in all the rungs of the distance ladder are well understood and modeled properly’.

If this assumption is violated, then the modeling of Cepheids+SnIa should be modified to take into account possible changes of physics by introducing new degrees of freedom that were suppressed in the original (baseline) SH0ES analysis. In the context of this approach, if these degrees of freedom are properly introduced in the analysis, then the best-fit value of $H_0$ will become consistent with the corresponding inverse distance ladder value of $H_0=67.36\pm 0.54$ km s${}^{-1}$ Mpc${}^{-1}$.

In an effort to pursue this approach for the resolution of the Hubble tension, we address the following questions:

• How can new degrees of freedom (new parameters) be included in the SH0ES data analysis for the determination of $H_0$?
• What are the new degrees of freedom that can expose internal tensions and inhomogeneities in the Cepheid/SnIa data?
• What new degrees of freedom can lead to a best-fit value of $H_0$ that is consistent with Planck?

The main goal of the present analysis is to address these questions. The new degree of freedom we allow and investigate is a transition of any one of the four Cepheid/SnIa modeling parameters at a specific distance $D_c$ or equivalently (in the context of the cosmological principle) at a given cosmic time $t_c$ such that $t_0-t_c=D_c/c$, where $t_0$ is the present cosmic time (age of the universe). In the context of this new degree of freedom, we reanalyze the SH0ES data to find how the quality of fit to the data and the best-fit value of $H_0$ change when the new degree of freedom is excited. The possible introduction of new constraints included in the analysis is also considered.

The structure of this paper is the following: in the next Section 2, we describe the standard analysis of the SH0ES Cepheid+SnIa data in a detailed and comprehensive manner, stressing some details of the released dataset that are not described in R21. We also describe some tension between the values of the best-fit Cepheid modeling parameters $b_W$ and $Z_W$ obtained in anchor or pure Cepheid host galaxies and the corresponding mean values obtained in SnIa host galaxies. In Section 3, we present our generalized analysis with new degrees of freedom that allow a transition of the main modeling parameters at specific distances (cosmic times of radiation emission). We also investigate the effect of the inverse distance ladder constraint on $M_B$ [149,160,161] for both the baseline SH0ES analysis and for our analysis involving the $M_B$ transition degree of freedom. Finally, in Section 4, we conclude and discuss the implications of our results and the possible extensions of our analysis.

2. The New SH0ES Data and Their Standard Analysis: Hints for Intrinsic Tensions

2.1. The Original Baseline SH0ES Analysis: A Comprehensive Presentation

The main equations used to model the Cepheid SnIa measured apparent magnitudes with parameters that include $H_0$ are described as follows:

• The equation that connects the measured Wesenheit magnitude of the jth Cepheid in the ith galaxy with the host distance moduli ${\mu }_i$ and the modeling parameters $M_W$, $b_W$ and $Z_W$ is of the following form6

  $$m_{H,i,j}^W={\mu }_i+M_H^W+b_W{\left[P\right]}_{i,j}+Z_W{[O/H]}_{i,j}$$

  (6)

  where ${\mu }_i$ is the inferred distance modulus to the galaxy, $M_H^W$ is the zeropoint Cepheid absolute magnitude of a period $P=10d$ Cepheid (d for days), and $b_W$−$Z_W$ are the slope parameters that represent the dependence of magnitude on both period and metallicity. The $[O/H]$ is a measure of the metallicity of the Cepheid. The usual bracket shorthand notation for the metallicity $[O/H]$ represents the Cepheid metal abundance compared to that of the Sun

  $$[O/H]\equiv log(O/H)-log{(O/H)}_{\odot }=\Delta log(O/H)$$

  (7)
  Here, O and H are the number of oxygen and hydrogen atoms per unit of volume, respectively. The unit often used for metallicity is the dex (decimal exponent) defined as $ndex\equiv {10}^n$. Additionally, the bracket shorthand notation for the period $\left[P\right]$ is used as (P in units of days)

  $$\left[P\right]\equiv logP-1$$

  (8)
• The color- and shape-corrected SnIa B-band peak magnitude in the ith host is connected with the distance modulus ${\mu }_i$ of the ith host and with the SnIa absolute magnitude $M_B$ as shown in Equation (5), i.e.,

  $$m_{B,i}^0={\mu }_i+M_B$$

  (9)
  The distance modulus is connected with the luminosity distance $d_L$ in $Mpc$ as

  $$\mu =5log(d_L/Mpc)+25$$

  (10)

  where in a flat universe

  $$d_L\left(z\right)=c(1+z){\int }_0^z\frac{d{z}^{\prime }}{H\left(z^{\prime }\right)}=c{H}_0^{-1}(1+z){\int }_0^z\frac{H_{0}d{z}^{\prime }}{H\left(z^{\prime }\right)}\equiv H_0^{-1}{D}_L\left(z\right)$$

  (11)

  where $D_L\left(z\right)$ is the Hubble free luminosity distance, which is independent of $H_0$.
• Using Equations (9)–(11), it is easy to show that that $H_0$ is connected with the SnIa absolute magnitude and the Hubble free luminosity distance as

  $$5log{H}_0=M_B+5log{D}_L\left(z\right)-m_B^0\left(z\right)+25$$

  (12)
  In the context of the cosmographic expansion of $H\left(z\right)$ valid for $z<<1$, we have

  $$log{D}_L{\left(z\right)}_c\simeq log\left[cz\left(1+\frac{1}{2}(1-q_0)z-\frac{1}{6}(1-q_0-3q_0^2+j_0)z^2+\mathcal{O}\left(z^3\right)\right)\right]$$

  (13)

  where $q_0\equiv -\frac{1}{H_0^2}\frac{d^{2}a\left(t\right)}{d{t}^2}{|}_{t=t_0}$ and $j_0\equiv \frac{1}{H_0^3}\frac{d^{3}a\left(t\right)}{d{t}^3}{|}_{t=t_0}$ are the deceleration and jerk parameters, respectively. Thus, Equations (12) and (13) lead to the equation that connects $H_0$ with the SnIa absolute magnitude $M_B$, which may be expressed as

  $$5log{H}_0=M_B+5log{D}_L\left(z\right)-m_B^0\left(z\right)+25\equiv M_B+5a_B+25$$

  (14)

  where we have introduced the parameter $a_B\equiv log{D}_L\left(z\right)-0.2m_B^0\left(z\right)$ as defined in the SH0ES analysis [153].

Thus, the basic modeling equations used in the SH0ES analysis for the measurement of $H_0$ are Equations (6), (9) and (14). In these equations, the input data are the measured apparent magnitudes (luminosities) of Cepheids $m_{H,i,j}^W$ and the SnIa apparent magnitudes $m_{B,i}^0$ (in Cepheid+SnIa hosts and in the Hubble flow). The parameters to be fit using a maximum likelihood method are the distance moduli ${\mu }_i$ (of the anchors and supporting hosts, the Cepheid+SnIa hosts and Hubble flow SnIa), the four modeling parameters ($M_H^W$, $b_W$, $Z_W$ and $M_B$), the Hubble constant $H_0$, the zeropoint $zp$ of the Cepheid P-L relation in the LMC ground measurements and the dummy parameter X. This is a total of 47 parameters. The actual data were released by the SH0ES team as a .fits file in the form of a column vector Y with 3492 entries, which include 8 constraints on the parameters obtained from measurements in anchor galaxies where the distance moduli are measured directly with geometric methods.

The entries of the provided Y data column vector do not include the pure measured apparent magnitudes. Instead, its entries are residuals defined by subtracting specific quantities. In particular, we have the following:

• The Cepheid Wesenheit magnitudes are presented as residuals with respect to a fiducial P-L term as

  $${\overline{m}}_{H,i,j}^W\equiv m_{H,i,j}^W-b_W^0\left[P\right]$$

  (15)

  where $b_W^0=-3.286$ is a fiducial Cepheid P-L slope. As a result of this definition, the derived best-fit slope is actually a residual slope $\Delta b_W\equiv b_W-b_W^0$.
• The residual Cepheid Wesenheit magnitudes of the Cepheids in the anchors N4258, LMC and the supporting pure Cepheid host SMC (non SnIa hosts), are presented after subtracting a corresponding fiducial distance modulus obtained with geometric methods7.
• The SnIa standardized apparent magnitudes in the Hubble flow are presented as residuals after subtracting the Hubble free luminosity distance with cosmographic expansion $5log{D}_L{\left(z\right)}_c+25$ (see Equation (13)).

Thus, the released data vector Y has the following form

The eight external anchor constraints on the parameters that appear in this vector are the following:

$$\begin{array}{ccc}\hfill M_H^W& =& -5.803\pm 0.082\hfill \\ \hfill M_H^W& =& -5.903\pm 0.025\hfill \\ \hfill Z_W& =& -0.21\pm 0.12\hfill \\ \hfill X& =& 0\pm 0.00003\hfill \\ \hfill \Delta zp& =& 0\pm 0.1\hfill \\ \hfill \Delta b_W& =& 0\pm 10\hfill \\ \hfill \Delta {\mu }_{N4258}& =& 0\pm 0.03\hfill \\ \hfill \Delta {\mu }_{LMC}& =& 0\pm 0.026\hfill \end{array}$$

(16)

The parameters to be fit using the Y vector data may also be expressed as a vector q with 47 entries of the following form

q = $\left.\left(\begin{array}{c}{\mu }_1\\ \dots \\ {\mu }_{37}\\ \Delta {\mu }_{N4258}\\ M_H^W\\ \Delta {\mu }_{LMC}\\ {\mu }_{M31}\\ \Delta b_W\\ M_B\\ Z_W\\ X\\ \Delta zp\\ 5log{H}_0\end{array}\right)\right\}$

47  parameters

Using the column vectors Y and q, Equations (6), (9) and (14) and the constraints stated above can be expressed in matrix multiplication form as

$$\mathbf{Y}=\mathbf{Lq}$$

(17)

with $\mathbf{Y}$, the matrix of measurements (data vector), $\mathbf{q}$, the matrix of parameters and $\mathbf{L}$, a model (or design) matrix which has 3492 rows corresponding to the entries of the $\mathbf{Y}$ data vector and 47 columns corresponding to the entries of the parameter vector $\mathbf{q}$. The model matrix $\mathbf{L}$ also includes some data (Cepheid period and metallicity), and in the context of this baseline, the modeling of data has the form

The system (17) has 3492 equations and 47 unknown parameter values. Thus, it is overdetermined and at best, it can be used in the context of the maximum likelihood analysis to find the best-fit parameter values that have maximum likelihood and thus minimum ${\chi }^2$. For the definition of ${\chi }^2$, the measurement error matrix (covariance matrix) $\mathbf{C}$ is needed and provided in the data release as a square matrix with dimension $3492\times 3492$8. In Appendix A, we present the schematic form of the matrix $\mathbf{C}$, which also includes the standard uncertainties of the constraints as diagonal elements. Using the covariance matrix that quantifies the data uncertainties and their correlation, the ${\chi }^2$ statistic may be constructed as

$${\chi }^2={(\mathbf{Y}-\mathbf{Lq})}^{\mathbf{T}}{\mathbf{C}}^{-\mathbf{1}}(\mathbf{Y}-\mathbf{Lq})$$

(18)

The numerical minimization of ${\chi }^2$ in the presence of 47 parameters that need to be fit would be very computationally demanding, even with the use of Markov chain Monte Carlo (MCMC) methods. Fortunately, the linear form of the system (17) allows the analytical minimization of ${\chi }^2$ and the simultaneous analytic evaluation of the uncertainty of each parameter. In Appendix B, we show that the analytic minimization of ${\chi }^2$ of Equation (18) leads to the best-fit parameter maximum likelihood vector9

$${\mathbf{q}}_{\mathbf{best}}={\left({\mathbf{L}}^{\mathbf{T}}{\mathbf{C}}^{-\mathbf{1}}\mathbf{L}\right)}^{-\mathbf{1}}{\mathbf{L}}^{\mathbf{T}}{\mathbf{C}}^{-\mathbf{1}}\mathbf{Y}$$

(19)

The $1\sigma$ standard errors for the parameters in ${\mathbf{q}}_{\mathbf{best}}$ are obtained as the square roots of the 47 diagonal elements of the transformed error matrix

$$\Sigma ={\left({\mathbf{L}}^{\mathbf{T}}{\mathbf{C}}^{-\mathbf{1}}\mathbf{L}\right)}^{-\mathbf{1}}$$

(20)

For example the best fit of the parameter10 $5log{H}_0$ is obtained as the 47th entry of the best fit parameter vector ${\mathbf{q}}_{\mathbf{best}}$ and the corresponding $1\sigma$ standard error is the $\sqrt{{\Sigma }_{\mathbf{47},\mathbf{47}}}$ element of the error matrix. Using Equations (19) and (20) and the latest released data of the SH0ES team presented in R21 we find full agreement with all values of the best fit parameters. For example for $H_0$ we find (after implementing error propagation) $H_0=73.04\pm 1.04$ km s${}^{-1}$ Mpc${}^{-1}$ fully in agreement with the published result of R21.

In

Table 2. Best-fit parameter values in the absence and in the presence of the inverse distance ladder constraint for baseline model.

Parameter	Best Fit Value	$\mathit{\sigma }$	Best Fit Value a	$\mathit{\sigma }$a
${\mu }_{M101}$	29.16	0.04	29.20	0.04
${\mu }_{M1337}$	32.92	0.08	32.97	0.08
${\mu }_{N0691}$	32.82	0.09	32.87	0.09
${\mu }_{N1015}$	32.62	0.069	32.67	0.06
${\mu }_{N0105}$	34.49	0.12	34.56	0.12
${\mu }_{N1309}$	32.51	0.05	32.56	0.05
${\mu }_{N1365}$	31.33	0.05	31.37	0.05
${\mu }_{N1448}$	31.3	0.04	31.33	0.03
${\mu }_{N1559}$	31.46	0.05	31.51	0.05
${\mu }_{N2442}$	31.47	0.05	31.51	0.05
${\mu }_{N2525}$	32.01	0.06	32.08	0.06
${\mu }_{N2608}$	32.63	0.11	32.69	0.11
${\mu }_{N3021}$	32.39	0.1	32.45	0.1
${\mu }_{N3147}$	33.09	0.09	33.16	0.08
${\mu }_{N3254}$	32.4	0.06	32.46	0.05
${\mu }_{N3370}$	32.14	0.05	32.19	0.04
${\mu }_{N3447}$	31.94	0.03	31.98	0.03
${\mu }_{N3583}$	32.79	0.06	32.84	0.06
${\mu }_{N3972}$	31.71	0.07	31.76	0.07
${\mu }_{N3982}$	31.64	0.06	31.69	0.05
${\mu }_{N4038}$	31.63	0.08	31.69	0.08
${\mu }_{N4424}$	30.82	0.11	30.87	0.11
${\mu }_{N4536}$	30.84	0.05	30.87	0.05
${\mu }_{N4639}$	31.79	0.07	31.83	0.07
${\mu }_{N4680}$	32.55	0.15	32.61	0.15
${\mu }_{N5468}$	33.19	0.05	33.25	0.05
${\mu }_{N5584}$	31.87	0.05	31.91	0.04
${\mu }_{N5643}$	30.51	0.04	30.56	0.04
${\mu }_{N5728}$	32.92	0.1	32.98	0.1
${\mu }_{N5861}$	32.21	0.08	32.26	0.07
${\mu }_{N5917}$	32.34	0.08	32.4	0.08
${\mu }_{N7250}$	31.61	0.1	31.65	0.1
${\mu }_{N7329}$	33.27	0.07	33.33	0.07
${\mu }_{N7541}$	32.58	0.08	32.64	0.08
${\mu }_{N7678}$	33.27	0.08	33.33	0.08
${\mu }_{N0976}$	33.54	0.08	33.61	0.08
${\mu }_{U9391}$	32.82	0.05	32.86	0.05
$\Delta {\mu }_{N4258}$	−0.01	0.02	0.01	0.02
$M_H^W$	−5.89	0.02	−5.92	0.02
$\Delta {\mu }_{LMC}$	0.01	0.02	0.03	0.02
${\mu }_{M31}$	24.37	0.07	24.4	0.07
$\Delta b_W$	−0.013	0.015	−0.026	0.015
$M_B$	−19.25	0.03	−19.33	0.02
$Z_W$	−0.22	0.05	−0.22	0.05
X	0.	0.	0.	0.
$\Delta zp$	−0.07	0.01	−0.07	0.01
$5log{H}_0$	9.32	0.03	9.24	0.02
$H_0$	73.04	1.04	70.5	0.7

NOTE: (a) With constraint M_B = −19.401 ± 0.027 included in the data vector Y and in the model matrix L with included uncertainty in the extended covariance matrix C.

, we show the best-fit values and uncertainties for all 47 parameters of the vector $\mathbf{q}$, including the four Cepheid modeling parameters ($b_W=b_W^0+\Delta b_W\equiv -3.286+\Delta b_W$, $M_H^W$, $Z_W$ and $M_B$) along with the best-fit value of $H_0$ for the baseline SH0ES model. The corresponding best-fit values and uncertainties with an additional constraint on $M_B$ from the inverse distance ladder are also included and discussed in detail in the next section. The agreement with the corresponding results of R21 (left three columns) is excellent.

Before closing this subsection, we review a few points that are not mentioned in R21 but are useful for the reproduction of the results and the analysis:

• The number of entries of the parameter vector $\mathbf{q}$ is 47 and not 46 as implied in R21 due to the extra dummy parameter X, which is included in the released data $\mathbf{Y}$, $\mathbf{L}$ and $\mathbf{C}$ but not mentioned in R21.
• The entry referred as $b_W$ in R21 in the parameter vector $\mathbf{q}$ should be $\Delta b_W$ because it is actually the residual $b_W$ ($\Delta b_W\equiv b_W-b_W^0$) as stated above and not the original slope of the P-L relation (its best-fit value is $\Delta b_W=-0.014\pm 0.015$).
• The number of constraints shown in the definition of the $\mathbf{Y}$ vector in R21 is five, while in the actual released data $\mathbf{Y}$, $\mathbf{L}$ and $\mathbf{C}$, we found the eight constraints defined above.

2.2. Homogeneity of the Cepheid Modeling Parameters

Before generalizing the model matrix $\mathbf{L}$ with new degrees of freedom and/or constraints, it is interesting to test the self-consistency of the assumed universal modeling parameters $b_W$ and $Z_W$ of the Cepheids. These parameters can be obtained separately for each one of the $i=1,...,40$ Cepheid hosts11 by fitting linear relations.

In particular, in order to obtain the best-fit P-L slope $b_{W,i}$ of the $ith$ Cepheid host, we fit the $m_{H,i,j}^W-log{P}_{i,j}$ data (where $P_{i,j}$ is the period in units of days of the $jth$ Cepheid in the $ith$ host) with a linear relation of the form

$$m_{H,i,j}^W=s_i+b_{W,i}log{P}_{i,j}$$

(21)

with parameters to be fit in each host $s_i-b_{W,i}$ (intercept and slope). These equations may be expressed as a matrix equation of the form

$${\mathbf{Y}}_{\mathbf{i}}={\mathbf{A}}_{\mathbf{i}}{\mathbf{X}}_{\mathbf{i}}$$

(22)

with ${\mathbf{Y}}_{\mathbf{i}}$ as the vector of measurements, ${\mathbf{X}}_{\mathbf{i}}$ as the vector of parameters and ${\mathbf{A}}_{\mathbf{i}}$ as the model (or design) matrix. These are defined as

$$\begin{array}{ccccc}\hfill {\mathbf{Y}}_{\mathbf{i}}=\left(\begin{array}{c}{m}_{H,i,1}^W\\ m_{H,i,2}^W\\ ⋮\\ m_{H,i,N}^W\end{array}\right)& ,& {\mathbf{X}}_{\mathbf{i}}=\left(\begin{array}{c}{s}_i\\ b_{W,i}\end{array}\right)\hfill & ,& {\mathbf{A}}_{\mathbf{i}}=\left(\begin{array}{cc}1& \log P_{i,1}\\ 1& \log P_{i,2}\\ ⋮& ⋮\\ 1& \log P_{i,N}\end{array}\right)\\ & & & \end{array}$$

(23)

The analytically (along the lines of the previous subsection and of Appendix B) obtained minimum of

$${\chi }_i^2={({\mathbf{Y}}_{\mathbf{i}}-{\mathbf{A}}_{\mathbf{i}}{\mathbf{X}}_{\mathbf{i}})}^{\mathbf{T}}{{\mathbf{C}}_{\mathbf{i}}}^{-\mathbf{1}}({\mathbf{Y}}_{\mathbf{i}}-{\mathbf{A}}_{\mathbf{i}}{\mathbf{X}}_{\mathbf{i}})$$

(24)

with respect to the slope $b_{W,i}$ and intercept $s_i$ leads to the best-fit values and standard errors for these parameters. For all Cepheids, we adopt the $\mathrm{N}\times \mathrm{N}$ covariance matrix ${\mathbf{C}}_{\mathbf{i}}$ of standard errors of the magnitude measurements from R21.

Thus, the analytic minimization of ${\chi }^2$ of Equation (24) leads to the best-fit parameter maximum likelihood vector

$${\mathbf{X}}_{\mathbf{i},\mathbf{best}}={\left({{\mathbf{A}}_{\mathbf{I}}}^{\mathbf{T}}{{\mathbf{C}}_{\mathbf{i}}}^{-\mathbf{1}}{\mathbf{X}}_{\mathbf{i}}\right)}^{-\mathbf{1}}{{\mathbf{A}}_{\mathbf{i}}}^{\mathbf{T}}{{\mathbf{C}}_{\mathbf{i}}}^{-\mathbf{1}}{\mathbf{Y}}_{\mathbf{i}}$$

(25)

The $1\sigma$ standard errors for $b_{W,i}$ slope and intercept in ${\mathbf{X}}_{\mathbf{i},\mathbf{best}}$ are obtained as the square roots of the two diagonal elements of the error matrix

$${\Sigma }_{\mathbf{i}}={\left({{\mathbf{A}}_{\mathbf{i}}}^{\mathbf{T}}{{\mathbf{C}}_{\mathbf{i}}}^{-\mathbf{1}}{\mathbf{A}}_{\mathbf{i}}\right)}^{-\mathbf{1}}$$

(26)

A similar analysis was implemented for the other Cepheid modeling parameter, the metallicity slopes $Z_{W,i}$. In this case, the linear fit considered was of the form

$$m_{H,i,j}^W=s_i+Z_{W,i}{[O/H]}_{i,j}$$

(27)

The best-fit values of the slopes $b_{W,i}$ and $Z_{W,i}$ for each of the 40 Cepheid hosts are shown in Figure 1 and Figure 2 in terms of the host distance, respectively. The actual Cepheid data in each host with the best-fit $log{P}_{i,j}-m_{H,i,j}^W$ and ${[O/H]}_{i,j}-m_{H,i,j}^W$ straight lines are shown in Figure A2 and Figure A3 in Appendix C for each of the 40 Cepheid hosts i. The $b_{W,i}$ slopes shown in Figure A4 for each host in sequence of increasing uncertainties, is in excellent agreement with the corresponding Figure 10 shown in R2112. The corresponding numerical values of the best fit $b_{W,i}$ and $Z_{W,i}$ are shown in

Table A1. The best-fit individual slopes $b_W$ and $Z_W$ with their standard errors $\sigma$ and scatter uncertainties ${\sigma }_{scat}$. The scatter uncertainty ensures that the model of a universal best-fit slope becomes an acceptable model with ${\chi }_{min}^2/dof\simeq 1$, as described in Section 2.2. The number of hosts in the Table is $N=42$, including the Milky Way (MW). For LMC and SMC, the metallicities are not provided individually for each Cepheid in R21, and thus no $Z_W$ slope could be estimated in our analysis.

Galaxy	${\mathit{D}}^{\mathit{a}}$	${\mathit{b}}_{\mathit{W}}$	$\mathit{\sigma }$	${\mathit{\sigma }}_{\mathbf{scat}}$	${\mathit{Z}}_{\mathit{W}}$	$\mathit{\sigma }$	${\mathit{\sigma }}_{\mathbf{scat}}$
	[Mpc]	[mag/dex]	[mag/dex]	[mag/dex]	[mag/dex]	[mag/dex]	[mag/dex]
M101	6.71	−2.99	0.14	0.18	−1.14	0.86	3.2
M1337	38.53	−4.	0.85	0.18	−19.44	6.27	3.2
N0691	35.4	−2.64	0.57	0.18	−0.73	2.54	3.2
N1015	35.6	−3.63	0.43	0.18	2.57	1.3	3.2
N0105	73.8	−5.13	2.13	0.18	6.21	3.8	3.2
N1309	29.4	−3.23	0.44	0.18	−0.77	0.59	3.2
N1365	22.8	−2.9	0.37	0.18	−1.45	0.39	3.2
N1448	16.3	−3.21	0.12	0.18	−1.3	0.39	3.2
N1559	19.3	−3.27	0.18	0.18	−2.55	1.62	3.2
N2442	20.1	−3.05	0.2	0.18	3.94	0.58	3.2
N2525	27.2	−3.2	0.37	0.18	−1.75	2.74	3.2
N2608	35.4	−3.24	0.73	0.18	1.82	2.33	3.2
N3021	26.6	−3.3	0.89	0.18	−1.03	2.85	3.2
N3147	42.5	−4.72	0.73	0.18	−11.43	9.13	3.2
N3254	24.4	−3.05	0.31	0.18	−0.22	0.57	3.2
N3370	24	−3.55	0.23	0.18	−1.46	0.4	3.2
N3447	20.8	−3.48	0.13	0.18	−0.62	0.66	3.2
N3583	34.4	−2.57	0.31	0.18	−0.06	0.62	3.2
N3972	15.1	−3.58	0.32	0.18	4.9	1.58	3.2
N3982	19	−2.47	0.36	0.18	0.46	0.47	3.2
N4038	29.3	−2.63	0.47	0.18	0.54	2.07	3.2
N4424	16.4	1.01	1.26	0.18	0.29	4.33	3.2
N4536	31.9	−3.25	0.19	0.18	−2.82	0.56	3.2
N4639	19.8	−3.08	0.51	0.18	−2.12	0.64	3.2
N4680	42.1	−3.12	1.22	0.18	1.48	3.27	3.2
N5468	46.3	−2.62	0.35	0.18	1.12	0.74	3.2
N5584	28	−3.26	0.16	0.18	0.4	0.36	3.2
N5643	20.7	−3.1	0.12	0.18	2.9	1.17	3.2
N5728	44.9	−5.09	1.34	0.18	−15.19	5.71	3.2
N5861	30.3	−2.5	0.45	0.18	1.65	0.72	3.2
N5917	31	−4.54	1.17	0.18	−17.82	6.56	3.2
N7250	12.8	−2.75	0.39	0.18	−8.33	4.44	3.2
N7329	46.8	−3.04	0.76	0.18	6.74	1.86	3.2
N7541	34.4	−3.54	0.71	0.18	−0.04	1.08	3.2
N7678	46.6	−2.73	1.03	0.18	2.16	2.27	3.2
N0976	60.5	−1.79	1.24	0.18	3.71	2.73	3.2
U9391	29.4	−3.33	0.43	0.18	−4.2	1.51	3.2
N4258	7.4	−3.32	0.06	0.18	−7.67	0.31	3.2
M31	0.86	−3.29	0.07	0.18	−4.85	0.35	3.2
LMC ${}^b$	0.05	−3.31	0.017	0.18	−	−	3.2
SMC ${}^b$	0.06	−3.31	0.017	0.18	−	−	3.2
MW ${}^c$	0	−3.26	0.05	0.18	−0.2	0.12	3.2

Note: (a) Distances from https://ned.ipac.caltech.edu/, (accessed on 23 August 2022). (b) No individual Cepheid metallicities were provided for LMC and SMC in R21 and thus we could not estimate the slope Z_W for these hosts. (c) For the Milky Way (MW), we directly used the values provided in R21 obtained from Gaia EDR3+HST.

.

Some inhomogeneities are visible in the distance distribution of both the individual Cepheid slopes in Figure 1 and Figure 2 as well as in Figure A4. For example, in Figure 1 and Figure A4, there is a consistent trend of most non-anchor hosts having an absolute best-fit slope $b_W$ that is smaller than the corresponding best-fit $b_W$ of the anchor hosts (above the dotted line). Similarly, in Figure 2, there seems to be a trend of most $Z_{W,i}$ absolute slopes being larger than the corresponding slope of the Milky Way (MW) (below the dotted line).

A careful inspection of Figure 1 and Figure 2 indicates that the scatter is significantly larger than the standard uncertainties of the slopes. Indeed, a ${\chi }^2$ fit to a universal slope for $Z_W$ of the form

$${\chi }^2\left(Z_W\right)=\sum _{i=1}^N\frac{{(Z_{W,i}-Z_W)}^2}{{\sigma }_{Z_{W,i}}^2}$$

(28)

leads to a minimum ${\chi }_{min}^2$ per degree of freedom ($dof$)13 $\frac{{\chi }_{ZW,min}^2}{dof}=22$ (with a best-fit $Z_{W,bf}\simeq -1$) while $\frac{{\chi }_{min}^2}{dof}=O\left(1\right)$ would be expected for an acceptable fit. Additionally, for $b_W$, we find $\frac{{\chi }_{bW,min}^2}{dof}=1.55$ ($b_{W,bf}=-3.3$), which is more reasonable but still larger than 1, indicating a relatively poor quality of fit to a universal slope.

There are two possible causes for this poor quality of fit to universal slopes: either many of the uncertainties of the individual host slopes were underestimated, or the universal slope model is not appropriate. In view of the fact that the uncertainties of the Cepheid periods and metallicities are not included in the ${\chi }^2$ fit because they were not provided by R21 released data, we make the working hypothesis that the uncertainties are underestimated, and thus we add a scatter error adjusted so that $\frac{{\chi }_{bW,min}^2}{dof}\simeq \frac{{\chi }_{ZW,min}^2}{dof}\simeq 1$. Thus for ${\chi }^2\left(Z_W\right)$, we have

$${\chi }^2\left(Z_W\right)=\sum _{i=1}^N\frac{{(Z_{W,i}-Z_W)}^2}{{\sigma }_{Z_{W,i}}^2+{\sigma }_{Z,scat}^2}$$

(29)

For $\frac{{\chi }_{ZW,min}^2}{dof}\simeq 1$, we must set ${\sigma }_{Z,scat}\simeq 3.2$, which is significantly larger than most uncertainties of individual host $Z_W$ slopes. Similarly, for $\frac{{\chi }_{bW,min}^2}{dof}\simeq 1$ we must set ${\sigma }_{b,scat}\simeq 0.18$, which is comparable to or smaller than most uncertainties of individual host $b_W$ slopes as shown in Table A1.

In order to quantify possible inhomogeneities of Cepheid hosts $b_{W,i}$ and $Z_{W,i}$ slopes, we split each sample of slopes in two bins: a low-distance bin with Cepheid host distances D smaller than a critical distance $D_c$, and a high-distance bin with $D>D_c$. Given $D_c$, for each bin, we find the best-fit slope and its standard $1\sigma$ error using the maximum likelihood method. For example, for a low distance $b_W$ bin, we minimize

$${\chi }^2\left(b_W^{<}\right)=\sum _{i=1}^{N^{<}}\frac{{(b_{W,i}-b_W^{<})}^2}{{\sigma }_{b_{W,i}}^2+{\sigma }_{scat}^{2<}}$$

(30)

where $N^{<}$ is the number of hosts in the low-distance bin and ${\sigma }_{scat}$ is the additional uncertainty (scatter error) chosen in each bin so that ${\chi }^2\simeq 1$ so that the fit becomes consistent with a constant $b_W$ in each bin. Thus, we find the best-fit low-distance bin slope $b_W^{<}$ and its standard error and similarly for the high distance bin $D>D_c$, $b_W^{>}$ and for the slopes $Z_W^{<}$, $Z_W^{>}$. Thus for each $D_c$, we obtain two best-fit slopes (one for each bin) and their standard errors. The level of consistency between the two binned slopes at each $D_c$ determines the level of homogeneity and self consistency of the full sample. The results for the best-fit binned slopes for $b_W$ and $Z_W$ are shown in Figure 3 and Figure 4, respectively, as functions of the dividing critical distance $D_c$.

Interestingly, for a range of $D_c$, there is a mild tension between the best-fit values of the high- and low-distance bins, which reaches levels of $2-3\sigma$, especially for the metallicity slopes $Z_W$. For both $b_W$ and $Z_W$, the absolute value of the difference between the high- and low-distance bin slopes is maximized for $D_c>47$ Mpc. For the case of $Z_W$, this difference is significant statistically, as it exceeds the level of $2\sigma$. The level of statistical consistency between high- and low-distance bins for both slopes $b_W$ and $Z_W$ is shown in Figure 5. In the range of $D_c$ between 40 Mpc and 50 Mpc and also between 10 Mpc and 20 Mpc, it is shown that in the case of $Z_W$, the $\sigma$-distance

$${\sigma }_d\equiv \frac{|Z_W^{>}-Z_W^{<}|}{\sqrt{{\sigma }_{Z_W^{>}}^2+{\sigma }_{Z_W^{<}}^2}}$$

(31)

between the best-fit binned $Z_W$ metallicity slopes can reach a level beyond $2\sigma$ (see Figure 4 and Figure 5).

An additional test of possible intrinsic tensions of the Cepheid properties in the SH0ES Cepheid sample is obtained by comparing the probability distributions of the Cepheid period and metallicity for the full sample of the 3130 Cepheid with high- and low-distance subsamples. In Figure 6 and Figure 7, we show histograms of the probability distributions of the Cepheid period and metallicity for the whole Cepheid sample and for the high- ($D>D_c$) and low-distance ($D<D_c$) subsamples for $D_c=50$ Mpc. The two subsamples for each observable are clearly inconsistent with each other and with the full sample. This is demonstrated visually and also through the Kolmogorov–Smirnov consistency test, which quantifies the inconsistency and gives a p-value very close to 0 for the three sample pairs. However, as communicated to us by members of the SH0ES team (private communication), this inconsistency can be justified by observational selection effects and does not necessarily indicate a physics change at $D_c=50$ Mpc. For example, bright Cepheids have longer periods and they are more easily observed at high distances. Thus, it is expected that there will be higher period (brighter) Cepheids observed at higher distances. Other variables, such as the time span of the observations, also play a role. For more distant galaxies, there is a trend to allow a longer window of observations so that longer period Cepheids can be found. Additionally, the star formation history of the galaxies dictates if one will have very long-period Cepheids which come from massive, short-lived stars. However, even though the observed inconsistency in the Cepheid properties probability distributions may be explained using observational biases and anticipated galactic properties, it may also be a hint of interesting new physics and/or systematic effects.

In view of the above level of mild inhomogeneities identified is the SH0ES data; it would be interesting to extend the SH0ES modeling of the Cepheids and SnIa with new degrees of freedom that can model the data, taking into account these inhomogeneities. This is the goal of the next section.

3. Generalized Analysis: New Degrees of Freedom Allowing for a Physics Transition

An obvious generalization of the SH0ES analysis described in Section 2.1 which models the Cepheid-SnIa luminosities using four main parameters is to allow for a transition of any one of these parameters at a particular distance or equivalently cosmic time when radiation is emitted. Such a transition could be either a result of a sudden change in a physics constant (e.g., the gravitational constant) during the last 200 Myrs in the context, e.g., of a first-order phase transition or a result of the presence of an unknown systematic.

Another modification of the standard SH0ES analysis in R21 would be to extend the number of constraints imposed on the data vector $\mathbf{Y}$ taking into account other cosmological data, such as the inverse distance ladder estimate of $M_B$ [149,160,161]. Both of these generalizations will be implemented in the present section.

3.1. Allowing for a Transition of a Cepheid Calibration Parameter

A transition of one of the Cepheid calibration parameters can be implemented by replacing one of the four modeling parameters in the $\mathbf{q}$ vector by two corresponding parameters: one for high and one for low distances (recent cosmic times). Thus, in this approach, the number of parameters and entries in the parameter vector $\mathbf{q}$ increases by one, from 47 to 48. Since one of the entries of $\mathbf{q}$ is replaced by two entries, the corresponding column of the modeling matrix $\mathbf{L}$ should be replaced by two columns. The high distance parameter column should be similar to the original column it replaced, but with zeros in the entries, corresponding to low-distance data (or constraints), and the reverse should happen for the low-distance parameter column. This process is demonstrated in the following schematic diagram, where the $q_j$ parameter is replaced by the two parameters $q_j$ (or $q_j^{<}$) and $q_{j+1}$ (or $q_j^{>}$), and the j column of $\mathbf{L}$ is replaced by the j and $j+1$ corresponding columns, which have zeros in the low- or high-distance data (or constraint) entries.

$$\begin{array}{ccccc}\hfill {\mathbf{L}}_{(\mathbf{3492}\times \mathbf{47})}=\left(\begin{array}{ccc}\dots & L_{1,j}& \dots \\ \dots & L_{2,j}& \dots \\ \dots & L_{3,j}& \dots \\ \dots & \dots & \dots \\ \dots & \dots & \dots \\ \dots & L_{3491,j}& \dots \\ \dots & L_{3492,j}& \dots \end{array}\right)& \left.\begin{array}{cc}& \\ & \\ & \\ & \\ & \\ & \\ & \end{array}\right\}& \begin{array}{c}{D}_{Y_1}<D_c\\ D_{Y_2}>D_c\\ D_{Y_3}>D_c\\ \dots \\ \dots \\ D_{Y_{3491}}<D_c\\ D_{Y_{3492}}>D_c\end{array}\hfill & \Rightarrow & {\mathbf{L}}_{(\mathbf{3492}\times \mathbf{48})}=\left(\begin{array}{cccc}\dots & L_{1,j}& 0& \dots \\ \dots & 0& L_{2,j+1}& \dots \\ \dots & 0& L_{3,j+1}& \dots \\ \dots & \dots & \dots & \dots \\ \dots & \dots & \dots & \dots \\ \dots & L_{3491,j}& 0& \dots \\ \dots & 0& L_{3492,j+1}& \dots \end{array}\right)\end{array}$$

(32)

$$\begin{array}{c}\hfill {\mathbf{q}}_{(\mathbf{47}\times \mathbf{1})}=\left(\begin{array}{c}\dots \\ \dots \\ q_j\\ \dots \\ \dots \end{array}\right)\to {\mathbf{q}}_{(\mathbf{48}\times \mathbf{1})}=\left(\begin{array}{c}\dots \\ \dots \\ q_j\\ q_{j+1}\\ \dots \\ \dots \end{array}\right)\end{array}$$

(33)

In this manner, if the parameter $b_W$ was to be split to $b_W^{>}$ and $b_W^{<}$ for example, Equation (6) would be replaced by

$$m_{H,i,j}^W\left(D\right)={\mu }_i+M_H^W+b_W^{>}\Theta (D-D_c){\left[P\right]}_{i,j}+b_W^{<}\Theta (D_c-D){\left[P\right]}_{i,j}+Z_W{[O/H]}_{i,j}$$

(34)

and similarly for splittings of each of the other three parameters, $M_H^W$, $Z_W$ and $M_B$. In (34), D is a distance that may be assigned to every entry of the data vector $\mathbf{Y}$ (Cepheids, SnIa and constraints). Notice that the splitting of any parameter to high- and low-distance versions does not affect the form of the $\mathbf{Y}$ data vector and the covariance matrix $\mathbf{C}$ of the data. In order to properly place the 0 entries of each one of the new columns, a distance must be assigned to every entry of $\mathbf{Y}$. We specified this distance for each host using the literature resources or the best-fit distance moduli of each host. These distances along with other useful properties of the Cepheids used in our analysis are shown in

Table 3. Properties of the Cepheid hosts considered in the analysis.

Galaxy	SnIa	Ranking in	Ranking in	Ranking in	Da	Number of Fitted	Initial Point	Final Point
		Figure A4	Table 3 in R21	Data Vector Y	[Mpc]	Cepheids	in Vector Y	in Vector Y
M101	2011fe	8	1	1	6.71	259	1	259
M1337	2006D	34	2	2	38.53	15	260	274
N0691	2005W	28	4	3	35.4	28	275	302
N1015	2009ig	24	6	4	35.6	18	303	320
N0105	2007A	42	3	5	73.8	8	321	328
N1309	2002fk	25	7	6	29.4	53	329	381
N1365	2012fr	20	8	7	22.8	46	382	427
N1448	2001el,2021pit	7	9	8	16.3	73	428	500
N1559	2005df	11	10	9	19.3	110	501	610
N2442	2015F	13	11	10	20.1	177	611	787
N2525	2018gv	21	12	11	27.2	73	788	860
N2608	2001bg	31	13	12	35.4	22	861	882
N3021	1995al	35	14	13	26.6	16	883	898
N3147	1997bq,2008fv,2021hpr	32	15	14	42.5	27	899	925
N3254	2019np	16	16	15	24.4	48	926	973
N3370	1994ae	14	17	16	24	73	974	1046
N3447	2012ht	9	18	17	20.8	101	1047	1147
N3583	2015so	15	19	18	34.4	54	1148	1201
N3972	2011by	17	20	19	15.1	52	1202	1253
N3982	1998aq	19	21	20	19	27	1254	1280
N4038	2007sr	29	22	21	29.3	29	1281	1309
N4424	2012cg	40	23	22	16.4	9	1310	1318
N4536	1981B	12	24	23	31.9	40	1319	1358
N4639	1990N	27	25	24	19.8	30	1359	1388
N4680	1997bp	38	26	25	42.1	11	1389	1399
N5468	1999cp,2002cr	18	27	26	46.3	93	1400	1492
N5584	2007af	10	28	27	28	165	1493	1657
N5643	2013aa,2017cbv	6	29	28	20.7	251	1658	1908
N5728	2009Y	41	30	29	44.9	20	1909	1928
N5861	2017erp	26	31	30	30.3	41	1929	1969
N5917	2005cf	37	32	31	31	14	1970	1983
N7250	2013dy	22	33	32	12.8	21	1984	2004
N7329	2006bh	33	34	33	46.8	31	2005	2035
N7541	1998dh	30	35	34	34.4	33	2036	2068
N7678	2002dp	36	36	35	46.6	16	2069	2084
N0976	1999dq	39	5	36	60.5	33	2085	2117
U9391	2003du	23	37	37	29.4	33	2118	2150
Total						2150	1	2150
N4258	Anchor	4	38	38	7.4	443	2151	2593
M31	Supporting	5	39	39	0.86	55	2594	2648
LMC b	Anchor	2	40	40	0.05	270	2649	2918
SMC b	Supporting	3	41	41	0.06	143	2919	3061
LMC c	Anchor	2	40	42	0.05	69	3062	3130
Total						980	2151	3130
Total All						3130	1	3130

Note: (a) Distances from NASA/IPAC Extragalactic Database (https://ned.ipac.caltech.edu/, (accessed on 8 June 2022)). (b) From the ground. (c) From HST.

.

We thus considered four generalizations of the baseline SH0ES analysis, each one corresponding to a high–low-distance split of each of the four modeling parameters. For each generalization, we obtained the best-fit values and uncertainties of all 48 parameters of the generalized parameter vector $\mathbf{q}$ for several values of the critical distance $D_c\in [1,60Mpc]$, which defined in each case the high–low-distance data bins. The best-fit values with uncertainties for the high–low D split parameter (green and blue points) and for $H_0$ (red points), for each generalization considered, is shown in Figure 8, Figure 9, Figure 10 and Figure 11 in terms of $D_c$. Dotted lines correspond to the SH0ES R21 $H_0$ best fit and to the Planck18/$\Lambda$CDM best fit for $H_0$. The following comments can be made based on the results of Figure 8, Figure 9, Figure 10 and Figure 11.

• When the SnIa absolute magnitude $M_B$ is allowed to change at $D_c$ ($M_B$ transition) and for $D_c>47Mpc$, the best-fit value of $H_0$ drops spontaneously to the best-fit Planck18/$\Lambda$CDM value, albeit with larger uncertainty $67.33\pm 4.65$ (see Figure 8 and second row of

  Table 4. A model comparison (baseline vs transitions) and best-fit parameter values in the absence (top five rows) and in the presence (last two rows) of the inverse distance ladder constraint. Notice that in the presence of the inverse distance ladder constraint and the transition degree of freedom, the best-fit value of $H_0$ is identical to the inverse distance ladder result of the completed SDSS-IV extended BAO survey [35].

  Model	${\mathit{\chi }}_{\mathbf{min}}^2$	${\mathit{\chi }}_{\mathbf{red}}^2$a	$\mathit{\Delta }\mathbf{AIC}$	$\mathit{\Delta }\mathbf{BIC}$	${\mathit{H}}_0$	${\mathit{M}}_{\mathit{B}}$	${\mathit{M}}_{\mathit{H}}^{\mathit{W}}$	$\Delta {\mathit{b}}_{\mathit{W}}$	${\mathit{Z}}_{\mathit{W}}$
  					km s${}^{-1}$ Mpc${}^{-1}$	[mag]	[mag]	[mag/dex]	[mag/dex]
  Baseline	3552.76	1.031	0	0	$73.043\pm 1.007$	$-19.253\pm 0.029$	$-5.894\pm 0.018$	$-0.013\pm 0.015$	$-0.217\pm 0.045$
  Transition b $M_B$	3551.31	1.031	0.55	6.71	$67.326\pm 4.647$	$-19.250\pm 0.029$	$-5.894\pm 0.018$	$-0.013\pm 0.015$	$-0.217\pm 0.045$
  						$-19.430\pm 0.150$
  						$1.2\sigma$
  Transition b $M_H^W$	3551.31	1.031	0.55	6.71	$73.162\pm 1.014$	$-19.250\pm 0.029$	$-5.894\pm 0.018$	$-0.013\pm 0.015$	$-0.217\pm 0.045$
  							$-5.713\pm 0.151$
  							$1.2\sigma$
  Transition b $Z_W$	3549.99	1.030	$-0.77$	5.39	$72.981\pm 1.007$	$-19.255\pm 0.029$	$-5.894\pm 0.018$	$-0.014\pm 0.015$	$-0.217\pm 0.045$
  									$2.588\pm 1.686$
  									$1.7\sigma$
  Transition b $b_W$	3550.86	1.030	0.10	6.26	$73.173\pm 1.013$	$-19.249\pm 0.029$	$-5.894\pm 0.018$	$-0.013\pm 0.015$	$-0.217\pm 0.045$
  								$0.315\pm 0.239$
  								$1.4\sigma$
  Baseline+Constraint c	3566.78	1.035	0	0	$70.457\pm 0.696$	$-19.332\pm 0.020$	$-5.920\pm 0.017$	$-0.026\pm 0.015$	$-0.220\pm 0.045$
  Transition b,c $M_B$ + Constraint	3551.34	1.031	−13.44	−7.27	$68.202\pm 0.879$	$-19.249\pm 0.029$	$-5.893\pm 0.018$	$-0.013\pm 0.015$	$-0.217\pm 0.045$
  						$-19.402\pm 0.027$
  						$3.9\sigma$

  NOTE: (a) ${\chi }_{\mathrm{red}}^2={\chi }_{\min }^2/dof$, where dof = N − M is typically the number of degrees of freedom (with N the number of datapoints used in the fit and M the number of free parameters) for each model. (b) At critical distance D_c ≃ 50 Mpc. (c) With constraint M_B = −19.401 ± 0.027.

  ). This remarkable result appears with no prior or other information from the inverse distance ladder results. Clearly, there are increased uncertainties of the best-fit parameter values for this range of $D_c$ because the most distant usable Cepheid hosts are at distances $46.8$ Mpc (N7329), $60.5$ Mpc (N0976) and $73.8$ Mpc (N0105) and only two of them are at distance beyond 47 Mpc. These hosts (N0975 and N0105) have a total of 41 Cepheids and 4 SnIa. Due to the large uncertainties involved, there is a neutral model preference for this transition degree of freedom at $D_c>47$ Mpc (the small drop of ${\chi }^2$ by $\Delta {\chi }^2\simeq -1.5$ is balanced by the additional parameter of the model). This, however, changes dramatically if the inverse distance ladder constraint on $M_B$ is included in the vector $\mathbf{Y}$ as discussed below.
• For all four modeling parameters considered, there is a sharp increase in the absolute difference between the high–low-distance best-fit parameter values for $D_c\gtrsim 47$ Mpc. The statistical significance of this split, however, is low due to the relatively small number of available Cepheids at $D>47$ Mpc.
• The best-fit value of $H_0$ changes significantly when the SnIa absolute magnitude $M_B$ is allowed to make a transition at $D_c>47$ Mpc but is not significantly affected if the three Cepheid modeling parameters $M_W$, $b_W$ and $Z_W$ are allowed to make a transition at any distance. This is probably due to the large uncertainties involved and due to the fact that $H_0$ is only indirectly connected with the three Cepheid modeling parameters.
• Each one of the transition degrees of freedom mildly reduces ${\chi }^2$, thus improving the fit to the data, but these transition models are not strongly preferred by the model selection criteria, which penalize the extra parameter implied by these models. This is demonstrated in Figure 12 and Figure 13, which show the values of $\Delta {\chi }_{min}^2$, $\Delta AIC$ and $\Delta BIC$ of the four transition models with respect to the baseline SH0ES model for various $D_c$ transition distances. As discussed below, this changes dramatically if the inverse distance ladder constraint on $M_B$ is introduced in the analysis.

3.2. The Effect of the Inverse Distance Ladder Constraint

In view of the spontaneous transition of the SnIa absolute magnitude $M_B$ that appears to lead to a Hubble tension resolution, while being mildly favored by the SH0ES data ($\Delta {\chi }^2\simeq -1.5$) without any inverse distance ladder information included in the fit, the following question arises:

How does the level of preference for the $M_B$ transition model (compared to the SH0ES baseline model) change if an additional constraint is included in the analysis obtained from the inverse distance ladder best fit for $M_B$?

The inverse distance ladder constraint on $M_B$ is [149,160,161]

$$M_B^{P18}=-19.401\pm 0.027$$

(35)

In order to address this question, we modify the analysis by adding one more constraint to the $\mathbf{Y}$ data vector: after entry 3215, which corresponds to the 8th constraint, we add the value −19.401 for the $M_B$ constraint. We also add a line to the model matrix $\mathbf{L}$ after line 3215 with all entries set to zero except the entry at column 43 corresponding to the parameter $M_B^{<}$. A column after column 43 is added in $\mathbf{L}$ to accommodate the high distance parameter $M_B^{>}$ as described above. The new constraint is assigned a large distance (larger than the distance of the most distant SnIa of the sample) so that it only affects the high-distance parameter $M_B^{>}$ (the entry at line 3216 column 43 of $\mathbf{L}$ is set to 0 for all $D_c$, while the entry at column 44 of the same line is set to 1 for all $D_c$). To accommodate the corresponding uncertainty of the new constraint, we also add a line at the covariance matrix after line 3215 with a single nonzero entry at the diagonal equal to ${\sigma }_{MB}^2=0.{027}^2=0.000729$. Thus, after the implementation of the constraint in the $M_B$ transition model, the $\mathbf{Y}$ vector has 3493 entries, the $\mathbf{L}$ model matrix has dimensions $3493\times 48$, the $\mathbf{q}$ parameter vector has 48 entries, and the covariance matrix $\mathbf{C}$ matrix has dimensions $3493\times 3493$. In a similar way, we may implement the constraint (35) in the SH0ES model without allowing for the additional transition degree of freedom and implement model selection criteria to compare the baseline with the transition model in the presence of the inverse distance ladder constraint.

The new constraints on $H_0$ and on the parameters $M_B^{<}$, $M_B^{>}$ emerging from this modified modeling analysis are shown in Figure 14 in terms of $D_c$. The corresponding quality of fit expressed via the value of ${\chi }_{min}^2$ and model selection [162,163] expressed via the AIC [164] and BIC criteria is shown in Figure 15. The definitions and properties of the AIC and BIC criteria are described in Appendix D.

Clearly, the transition degree of freedom at $D_c\simeq 50$ Mpc which is mildly preferred by the data, even in the absence of the inverse distance ladder constraint, is strongly preferred by the data in the presence of the constraint while the $\sigma$-distance between the low-distance best-fit parameter $M_B^{<}$ and the high distance $M_B^{>}$ reaches a level close to $4\sigma$. These results are described in more detail in Table 4. The full list of the best-fit values of all the parameters of the vector $\mathbf{q}$ for the SH0ES baseline model with and without the inverse distance ladder constraint in the data vector $\mathbf{Y}$ is shown in Table 2 along with the corresponding uncertainties.

The following comments can be made on the results shown in Table 4.

• The $M_B$ transition degree of freedom resolves the $H_0$ tension both in the absence of the inverse distance ladder constraint (second raw of Table 4) and in the presence of it (last row of Table 4).
• In the presence of the inverse distance ladder constraint, the model with the $M_B$ transition degree of freedom at $D_c=50$ Mpc is strongly preferred over the baseline $SH0ES$ model as indicated by the model selection criteria, despite the additional parameter it involves (comparison of the last two rows of Table 4).
• The transition degree of freedom, when allowed in each of the other three modeling parameters, does not lead to a spontaneous resolution of the Hubble tension since the best-fit value of $H_0$ is not significantly affected. However, it does induce an increased absolute difference between the best-fit values of the high-distance and low-distance parameters, which, however, is not statistically significant due to the large uncertainties of the bin with $D>D_c\simeq 50$ Mpc (see also Figure 9, Figure 10 and Figure 11).

The above comments indicate that interesting physical and or systematic effects may be taking place at distances at or beyond 50 Mpc in the SH0ES data, and therefore more and better quality Cepheid/SnIa data are needed at these distances to clarify the issue. This point is further enhanced by the recent study of Ref. [117], indicating that SnIa in the Hubble flow (at distances $D>90$ Mpc) appear to have different color calibration properties than SnIa in Cepheid hosts (at distances $D<75$ Mpc).

The hints for a transition in the SnIa absolute luminosity and magnitude $M_B$ are also demonstrated in Figure 16, where we show the mean14 SnIa absolute magnitude $M_{Bi}$ for each Cepheid+SnIa host i is obtained from the equation

$$M_{Bi}=m_{Bi}^0-{\mu }_i$$

(36)

where $m_{Bi}^0$ is the measured apparent magnitude of the SnIa, and ${\mu }_i$ are the best-fit distance host distance moduli obtained using the SH0ES baseline model (left panel of Figure 16) and the $M_B$ transition model (right panel), which allows (but does not enforce) an $M_B$ transition at $D_c=50$ Mpc. Notice that when the $M_B$ transition degree of freedom is allowed in the analysis, the best-fit values of $M_{Bi}$ for the more distant hosts N0976 ($D=60.5$ Mpc) and N0105 ($D=73.8$ Mpc) spontaneously drop to the inverse distance ladder calibrated value range.

The inverse distance ladder calibrated values of the absolute magnitudes $M_{Bi}$ of SnIa in the Hubble flow are obtained by assuming $H_0=H_0^{P18}=67.36\pm 0.54$ km s${}^{-1}$ Mpc${}^{-1}$ and using the following equation

$$M_{Bi}=m\left(z_{Bi}^0\right)+5{log}_{10}\left[H_0^{P18}·Mpc/c\right]-5{log}_{10}\left[D_L\left(z_i\right)\right]-25$$

(37)

where $D_L\left(z_i\right)$ is the Hubble free luminosity distance in the context of Planck18/$\Lambda$CDM and $m\left(z_{Bi}^0\right)$ are the binned corrected SnIa apparent magnitudes of the Pantheon sample. The corresponding binned Cepheid+SnIa host values of $M_B$ obtained assuming the baseline Sh0ES model (red points) and the $M_B$ transition model ($D_c=50$ Mpc, green points) are shown in Figure 17 along with the inverse distance ladder calibrated binned $M_B$ of the Hubble flow SnIa of the Pantheon dataset (blue points). When the transition dof is allowed, the data excite it, and a hint for a transition appears (the green data point of the transition model is clearly below the red point corresponding to the constant $M_B$ SH0ES baseline model), even though the statistical significance of the indicated transition is low due to the small number of Cepheids (41) included in the last bin with $D\in [50$ Mpc,75 Mpc].

4. Conclusions

We described a general framework for the generalization of the Cepheid+SnIa modeling in the new SH0ES data. Such a modeling generalization approach is motivated by the increased statistical significance of the Hubble tension and may hint toward new degrees of freedom favored by the data. Such degrees of freedom may be attributed to either new physics or to unknown systematics hidden in the data.

In the present analysis, we focused on a particular type of new modeling degree of freedom, allowing for the transition of any one of the four Cepheid/SnIa modeling parameters at a distance $D_c$. However, our analysis can be easily extended to different degrees of freedom with physical motivation. Examples include possible modeling parameter dependence on properties other than distance (e.g., dust extinction and color and stretch calibration properties). In addition, other degrees of freedom that could be excited by the data could involve the modeling parameters that have been absorbed in the R21 SH0ES data, such as the Cepheid dust extinction parameter $R_W$ and the color and stretch SnIa calibration parameters $\beta$ and $\alpha$. These degrees of freedom may also be probed and excited by the fit to the data if allowed by the modeling [117].

We demonstrated that our proposed transition degrees of freedom are mildly excited by the SH0ES data, and in the case of the SnIa absolute magnitude $M_B$ transition degree of freedom at $D_c\simeq 50$ Mpc, the best-fit value of $H_0$ shifts spontaneously to a value almost identical to the Planck18/$\Lambda$CDM best-fit value, thus annihilating the Hubble tension. However, the high-distance bin in this case involves only 41 Cepheids and 4 SnIa, and therefore, the best-fit value of $M_B^{>}$, which effectively also fixes $H_0$, involves significant uncertainties.

In the presence of the inverse distance ladder constraint on the high-distance bin parameter $M_B^{>}$ of the $M_B$ transition model and on the single $M_B$ parameter of the SH0ES model, the uncertainties reduce dramatically. The Hubble tension is fully resolved only in the presence of the $M_B$ transition degree of freedom at $D_c\simeq 50$ Mpc. This transition model is also strongly selected over the baseline SH0ES model with the constraint, by both model selection criteria AIC and BIC, despite the penalty imposed by AIC and especially BIC on the additional parameter involved in the new modeling. This behavior along with other recent studies [117] hints toward the need of more detailed Cepheid+SnIa calibrating data at distances $D\gtrsim 50$ Mpc, i.e., at the high end of rung two on the distance ladder.

Our comprehensively described generalized modeling approach opens a new direction in the understanding of the Hubble tension and may be extended by the introduction of a wide range of new degrees of freedom in the SH0ES data analysis and also by the introduction of new constraints motivated by other cosmological data. Such extensions could, for example, involve more distance bins in the distance dependence of the four main modeling parameters. In that case, the relevant column of the modeling matrix $\mathbf{L}$ would have to be replaced by more than two columns (one for each distance bin). Such bins could also be defined not in terms of distance, but in terms of other Cepheid/SnIa properties (e.g., Cepheid metallicity or period and/or SnIa light curve color or stretch). In addition, other parameters beyond the four basic modeling parameters may be considered, including the dust extinction parameter $R_W$ and/or the SnIa light curve color and stretch parameters. Additionally, similar modeling generalizations may be implemented on different distance calibrator data used for the measurement of $H_0$, such as the TRGB data [165].

Physical motivation is an important factor for the evaluation of any new modeling degree of freedom, especially if it is favored by the data. A transition of the SnIa luminosity at a particular recent cosmic time could be induced by a sudden change of the value of a fundamental physics constant, e.g., the gravitational constant in the context of a recent first-order phase transition [166,167,168] to a new vacuum of a scalar–tensor theory, or in the context of a generalization of the symmetron screening mechanism [143]. A similar first-order transition was implemented in early dark energy models [169], attempting to change the last scattering sound horizon scale without affecting other well-constrained cosmological observables. Thus, even though no relevant detailed analysis has been performed so far, there are physical mechanisms that could potentially induce the SnIa luminosity transition degree of freedom.

The emerging new puzzles challenging our standard models may soon pave the way to exciting discoveries of new physical laws. The path to these discoveries goes through the deep and objective understanding of the true information that is hidden in the cosmological data. The present analysis may be a step in that direction.