# The Unsettled Number: Hubble’s Tension

^{1}

^{2}

^{3}

^{4}

^{5}

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## Abstract

**:**

_{0}, called the Hubble constant. Once again, different observational techniques bring about different results, causing new “Hubble tension”. In the present work, we review the historical roots of the Hubble constant from the beginning of the twentieth century, when modern cosmology originated, to the present. We develop the arguments that gave rise to the importance of measuring the expansion of the Universe and its discovery, and we describe the different pioneering works attempting to measure it. There has been a long dispute on this matter, even in the present epoch, which is marked by high-tech instrumentation and, therefore, in smaller uncertainties in the relevant parameters. It is, again, currently necessary to conduct a careful and critical revision of the different methods before one invokes new physics to solve the so-called Hubble tension.

## 1. Introduction

_{0}). This is a fixed number used as a unit of measurement to describe this expansion. The importance of knowing this value with good certainty lies in the fact that this knowledge provides a measurement scale of the present universe through the Hubble Sphere r

_{HS}= c/H

_{0}and its time scale through Hubble time t

_{H}= 1/H

_{0}~14 Gyrs. It also serves to estimate the critical density of the universe, ρ

_{critical}= 3 H

_{0}

^{2}/8 π G, which, in general relativity, creates flat space geometry. Since H

_{0}is our least well-determined cosmological parameter, things are often expressed in terms of h = H

_{0}/(100 km/s/Mpc), or h

^{2}, in this case, so that the result can be scaled properly when accounting for final errors and results. Other cosmological parameters are intimately related to these quantities. Thus, the precise determination of H

_{0}could reveal missing pieces in our current understanding of physics. The present review provides a historical description of the origin, measurement, and present status of the Hubble constant.

_{0}has emerged, involving measurements from the two routes of its assessment. In defiance of all expectations, estimates of H

_{0}from early route assessments do not agree with those of the late route. This discrepancy is known as the Hubble tension. This tension might be nothing more than a measurement error. However, it has stubbornly prevailed in spite of the fact that all the recent measurements have increased their precision.

_{0}to be about 625 km/s/Mpc.

_{0}-Tension”. This was a problem involving the age of the Earth based on early-20th radioactive dating as compared to the age of the universe, inferred from the value of the Hubble constant. It turned out that the Earth was found to be older than the Universe. Today, we know that the problem had its origin in the then-accepted value for H

_{0}, which was far from today’s estimates.

_{0}at very large distances, since gravitational interactions among our neighboring galaxies may be causing some of them to move much more quickly or slowly than the rest of the universe.

_{0}) whose measurement provides values for the Hubble constant and the average density of matter in the universe. This method requires an accurate determination of the apparent luminosity of an object as a function of its redshift z. Then, we move on to the use of gravitational lensing to measure the Hubble constant.

_{0}, Section 27, Section 28 and Section 29 describe the lensing, masers, and the Sunyaev–Zeldovich Effect, respectively. Then, we move to an “early” method: The cosmic microwave background radiation (CMB) method, which is closely related to baryon acoustic oscillations (BAO) H

_{0}determination (Section 30). In Section 31, we briefly mention the standard siren method, which is a promising and different technique with which to determine the Hubble constant. In Section 32, we explain how shadows of supermassive black holes can be used to determine H

_{0}, and in Section 33, we explain how fast radio bursts can also be used to constrain the Hubble constant.

_{0}value. The results of “late” route measurements for H

_{0}all converge to a value within a very narrow range of uncertainty. The same can be said about the “early” route, but its values converge to a different H

_{0}. Thus, the results of both (“early” and “late”) sets of measurements do not coincide with each other within the error bars. We conclude this work by presenting some of the views and outlooks on this crisis, which have arisen because of the Hubble tension.

**Part I. Early historical roots.**

## 2. The Great Debate of the 1920s

## 3. The First Candle

_{v}> = a + b log

_{10}P, where P is the period of the luminous oscillation in days and a and b are constants to be determined. This pair of constants is univocally determined by a single point that astronomers call the zero point in their jargon. The zero point is defined as the absolute magnitude of a hypothetical Cepheid with P = 1 day.

_{10}P. He also carried out a statistical parallax analysis on 13 Cepheids previously reported by Leavitt for which proper motions were available. Surprisingly, his paper had a misprint, as he reported a distance to SMC Cepheids of only 3000 light-years [9]. This result was well below the value of the actual distance, and perhaps this was the reason why his publication did not receive much attention. Some years later, Harold Shapley and Henry Norris Russell realized the error that had appeared in Hertzsprung’s article and corrected the misprinted distance to 30,000 light years, see footnote 2 p. 434 in [10]. An interesting fact in Hertzsprung’s work is that he ignored the interstellar absorption effects that cause dimming of light during its passage through the interstellar medium and alter the magnitudes of real stars.

## 4. The End of the Great Debate

^{−1}toward the Earth [23]. The fact that Slipher’s discovery came from the premises of Lowell’s observatory caused a stir of doubt regarding its soundness. Nevertheless, by 1914, Slipher had already collected data from 15 nebulae, of which 13 were receding and 2 were moving towards Earth [24].

## 5. Velocity–Distance Early Searches

_{m}), where D

_{m}represents the angular diameter (scales like 1/r) in arc minutes of the observed nebula. Clearly, this was incorrect, but his relation followed the right tendency for V as it increases with distance r.

_{obs}/λ

_{em}) − 1, where λ

_{obs}is the observed and λ

_{em}is the emitted wavelength). Silberstein applied his formula to a list of stellar clusters as well as to the small and large Magellan clouds. The application of this law to this mix of objects (some approaching and others receding) gave him a value for the radius of curvature of the universe of R of about 10

^{8}lyr. As expected, it did not take long for Silberstein’s work to be criticized by various astronomers, including Arthur Eddington [35].

## 6. Hubble’s Entrance

## 7. The Emergence of Cosmological Relativity

“As to relativity, I must confess that I would rather have a subject in which there would be a half dozen members of the Academy competent enough to understand at least a few words of what the speakers were saying if we had a symposium upon it. I pray to God that the progress of science will send relativity to some region of space beyond the fourth dimension, from whence it may never return to plague us”.[40]

## 8. Friedmann

^{2}/c, where the value R, i.e., the radius of curvature, is constant. To recover Einstein’s and De Sitter’s models, he set $M=1$ for Einstein’s and $M=\mathrm{cos}{(x}_{1}$) for De Sitter’s, respectively. He demonstrated that these are the only two solutions for static universes. Then, he moved on to the non-static cases.

_{4}) and the matter density $\rho $ = $\rho $ (x

_{4}), both of which were allowed to vary with time according to Einstein’s equations:

_{4}) is determined by $\lambda $ and $\rho $. The energy–momentum tensor reduces to

## 9. Lemaître

“Dear Professor Eddington, I just read the February No., of the Observatory and your suggestion of investigating of non-statical intermediary solutions between those of Einstein and de Sitter. I made these investigations two years ago. I consider a Universe of curvature constant in space but increasing in time. And I emphasize the existence of a solution in which the motion of the nebulae is always a receding one from time minus infinity to plus infinity”.

## 10. Lemaître’s Expanding Universe

^{3}, so energy conservation becomes$\text{}d\left(V\rho \right)+pdV=0$. This means that the work done by radiation pressure plus the variation in total energy add up to zero. Then, he explains why pressure coming from the mass loss of stars that is converted or transformed into energy does not contribute to the pressure. Also, he recovers Einstein’s model by setting the condition of the constancy of the universe’s radius while setting $\rho $ = 0, as well as retrieving De Sitter’s solution.

_{1}at σ

_{1}will arrive at σ

_{2}with a wavelength

_{1}is subtracted from both sides, and the equation is rearranged:

_{0}(known as the Hubble constant today) to be about 625 km/s/Mpc.

## 11. The Earliest H_{0}-Tension

“... both incline to the opinion, however, that if the red-shift is not due to recessional motion, its explanation will probably involve some quite new physical principles [... and] use of a static Einstein model of the Universe, combined with the assumption that the photons emitted by a nebula lose energy on their journey to the observer by some unknown effect, which is linear with distance, and which leads to a decrease in frequency, without appreciable transverse deflection”.

## 12. Cepheids

## 13. Baade and Thackeray, a Good Fresh Breeze

“…Whatever the final outcome we would know where we stand in the view of a most vexing question. Both Hubble and I hope that Shapley’s tendency to consider the Magellanic Clouds as his personal property will not deter you from attacking this problem. He has monopolized the Clouds all too long and it is high time that the barbed wire fences and the warning signs “Keep out. This means you!” are taken down. Monopolies in science are intolerable and should never be respected. Moreover lately Shapley has worked his gold mine only if he needed money for booze (some stuff for publication). The whole situation has become intolerable and a good fresh breeze is most desirable…”.[84]

## 14. The Changing Value of H_{0}

_{0}in 1927 to the mid-twentieth century measurements. This has already been carried out many times, often by the original researchers or their close collaborators (see, e.g., Fernie 1969 [81]). Our intention here is to show the downward trend experienced over time during the second quarter of the last century for the values of the Hubble constant. This trend was not due to an arbitrary value-adjustment of H

_{0}from the observers, but to efforts to improve observations, “revise” bias and confusions made by others on observed objects, and to consider the sources of errors that were previously overlooked or ignored (shown in Table 1). As we have already seen in the previous section, by the mid-twentieth century, Shapley’s calibration had been found to be erroneous and was substituted by Baade and Thackeray’s calibration.

_{0}estimates experienced during the second quarter of the twentieth century.

## 15. The Need to See Further

_{0}. This obstacle was that, even though the Cepheid stars were bright (on average, about 10

^{4}brighter than the sun) their luminosity was not sufficient for them to be observed them at that time beyond a few dozen parsecs. On this scale, this meant that the universe could not be assumed to be isotropic or homogeneous.

**Part I**

**I. Building and adding rungs to the cosmic ladder**

^{+19}

_{−15}km/s/Mpc), luminosity classification of galaxies (98

^{+19}

_{−17}km/s/Mpc), brightest globular clusters in galaxies (72

^{+15}

_{−12}km/s/Mpc), mass-to-light ratios (105

^{+33}

_{−26}km/s/Mpc), third-brightest cluster galaxy (126

^{+48}

_{−35}km/s/Mpc), supernovae and the extra-galactic distance scale (123

^{+47}

_{-34}km/s/Mpc), surface brightness and diameter of galaxies (89

^{+46}

_{−30}km/s/Mpc), brightest stars in galaxies (95

^{+15}

_{−12}km/s/Mpc), and regional variations of the Hubble constant (not specified). Van den Bergh computed their mean value, obtaining 95

^{+15}

_{−12}km/s/Mpc.

## 16. Planetary-Nebula Luminosity Functions

_{¤}) towards the end of their evolution. Planetary nebulae is an inaccurate term for these objects because they are unrelated to planets or exoplanets. PNe are important tools used to estimate distances, despite the brief lifespans of their terminal phases (~30,000–50,000 y).

## 17. The Tully–Fisher Relation (TFR)

_{0}using TRF in 1977 was 80 km/s/Mpc.

^{4}, if the spirals possessed some properties. In 1983, Aaronson suggested an improvement to the TFR method by measuring luminosities in the infrared (K band) radiation rather than the optical band to reduce scattering effects [108].

## 18. The Faber–Jackson Relation (F-J)

^{n}). Originally, Faber and Jackson found that n ≈ 4. But later, it was found that the value of n depends on the range of galaxy luminosity. For low-luminosity elliptical galaxies, the F-J relation is fitted with a value of n ≈ 2. This value was found by a team led by Roger Davies [112]. A value of n ≈ 5 for luminous elliptical galaxies was reported by Paul L. Schechter [113]. These findings have been confirmed observationally by many authors. For a comprehensive reference list in the literature, see, for instance, Markovic and Guzman [114].

## 19. Fundamental Plane, the D_{n}-σ Relation

_{e}) and the average surface brightness (I

_{e}) within that radius. They found a relationship connecting the three parameters: R

_{e}= k σ

^{1.36}I

_{e}

^{−0.85}, where k is a constant.

_{e}and σ are measured to obtain an effective radius R

_{e}. This effective radius is used as a standard rod. Once this rod’s length has been obtained, by directly measuring its angular size, it becomes easy to determine the distance from the observer to the galaxy through small-angle approximation.

_{n}, which represents the diameter of a central circular region of an elliptical galaxy within which the total average surface brightness is some particular value (20.75 magnitude per square second of arc). The correlation can be expressed as:

## 20. Surface Brightness Fluctuation Method (SBF)

^{−2}).

## 21. Tip of the Red Giant Branch (TRGB)

^{+5}

_{−5}km/s/Mpc, while the TRGB group reported 73

^{+6}

_{−6}(statistical)

^{+8}

_{−8}(systematic) km/s/Mpc [127]. As Wendy Freedman put it in a short review in 1998 [128], the results seemed to reach an uncertainty of 10% for the Hubble constant when it was measured using different methods, leaving in the past a factor of two that astronomers had used throughout the previous two decades.

_{0}via measurement of the TRGB in nearby galaxies. Their results will be outlined below. The most recent TRGB result was 72.94 ± 1.98 km/s/Mpc, with an additional uncertainty due to algorithm choices of 0.83 km/s/Mpc [129].

## 22. Global Cluster Luminosity Function (GCLF)

_{0}.

_{0}is the peak or turnover point TO (i.e., the magnitude at which most GC are found), σ is the width of the distribution, and A is the normalization constant. The distribution is then characterized by only two parameters, σ and TO. The latter is the standard candle, and the distance measurement is relative to the Milky Way GCLF TO value or to that of M31. The GCLF is a secondary distance indicator, since the absolute distances to either our galaxy or M31 globular clusters must be known. This method has been used to estimate distances and, hence, the Hubble constant as well.

## 23. Cepheid Calibration Development in the Twentieth Century

## 24. Type Ia Supernovae (SN Ia)

_{☉}) accretes mass from its companion star until it becomes denser, and a thermonuclear blast occurs. The explosion blows the dwarf star completely apart, spewing out material at remarkably high speeds (10

^{4}km/s). The glow of this expanding fireball takes a few weeks to reach its peak brightness, and afterwards, its luminosity declines over a period of months in a normal way. Figure 10 shows a typical light curve of an SN Ia.

_{0}, the deceleration parameter, had always been part of the aims of both the HZT and SCP programs. Both projects found in 1998 that high-redshift SN appeared to be 10% to 15% more distant than expected, which meant that the deceleration parameter q

_{0}was negative (q

_{1998}~ −0.7518). In simple words, the universe’s expansion was not slowing down, but rather accelerating, and there was a prevailing feeling that a cosmological constant was at work again in the field equations. The interpretation of this as a new acceleration was actually “decided” by vote at a scientific congress held in May 1998, the month during which Alan Riess of the HZT headed the first publication on the subject [161], and a short while later, a similar publication was authored by the SCP group [162]. In 2011, Saul Perlmutter, Brian Schmidt, and Adam Riess received the Nobel Prize in Physics for this discovery.

## 25. Mira Variables

_{0}of around 73 $\pm $ 4 km/s/Mpc. The intention is to find Mira variables in a wide range of local SN Ia hosts in order to better determine the uncertainties.

## 26. The Deceleration Parameter q_{0}

_{i}compared to a critical density, ρ

_{crit}. The ratio of the average density of matter compared to the critical density is called the density parameter, Ω

_{Μ}= ρ

_{i}/ρ

_{crit}. This critical density is the value where the gravitational attraction of matter in the universe causes space to become geometrically flat. As an alternative, a universe could have open hyperbolic geometry if its density were below this critical value, and if it were above, the universe would be characterized by closed spherical geometry.

_{L}is the luminosity distance, defined as the inverse square law of an object of luminosity, L, and observed flux, f:

## 27. Gravitational Lenses

_{0}value have been based on using the cosmic ladder to obtain a final estimate of absolute distance to the object being observed. Each rung of the ladder requires gauging a particular distance indicator, and each is subject to inaccuracies that contribute, in the end, to increasing the final uncertainty. Direct and more precise methods to estimate absolute distances, such as orbital parallax, are limited to nearby objects, and are difficult to apply to objects outside our own galaxy.

_{0}from the time delays between gravitationally lensed images. They also pointed out that this ambivalence can be addressed with a measurement of the lensing galaxy's velocity dispersion. The velocity dispersion constrains the lensing galaxy's mass, which breaks the degeneracy and leads to a point value for H

_{0}[180].

_{0}of 37 ± 14 km/s/Mpc. Another independent estimate was made of the same quasar by Roberts et al. [182], and a value of 46 ± 14 (42 ± 14) km/s/Mpc was given for H

_{0}. A third estimate was made by G. Rhee [183] that yielded 50 ± 17 km/s/Mpc.

_{0}= 64 ± 13 km/s/Mpc. They argued that their estimate was of comparable quality to those based on more conventional techniques at that time. In the same paper, the authors pointed out the advantages of the gravitational lens method. Briefly, it is a geometrical method based on the well-understood physics of general relativity in the weak-field limit. It yields a direct, single-step method for measuring H

_{0}and thus avoids error propagation along the “distance ladder”, which is no more secure than its weakest rung. It measures distances to cosmologically faraway distant objects, thus precluding the possibility of confusing a local with a global expansion rate. It gives an independent measurement of H

_{0}in two or more lensed systems with different sources and lens redshifts. This provides an internal consistency check of the obtained H

_{0}value, i.e., if a small number of time delay measurements all give the same H

_{0}, this value can be regarded as correct with considerable confidence.

_{0}from lensing that had been accused of being a black-box, can now be put on solid grounds” [187].

## 28. Megamasers

_{2}O maser produced inside vast clouds of water molecules located in the vicinities of galaxy centers and pumped by infrared radiation. These masers, as we shall see, facilitated a more precise distance assessment of the NGC4258 galaxy.

_{2}O maser emissions, with each spike corresponding to a lump of H

_{2}O masering material. These spikes came from masers with radial velocities equal to the average velocity of the galaxy, but they also found spikes from masers with higher and lower radial velocities. The simplest explanation is that there is a disk of H

_{2}O gas orbiting a central mass in NGC4258. The high and low velocity spikes originated from masers at the edges of the disk. From the dynamical characteristics of the Keplerian motion of the disk, the group deduced that there was a black hole (BH) at the center of rotation, with a mass of at least 3.6 × 10

^{7}M

_{☉}. The discovery made the news. Incidentally, at nearly that time, there were independent investigations by Reinhard Genzel and Andrea Ghez regarding the existence of a BH at the center of our Milky Way that, years later, resulted in Nobel prizes for this latter pair of scientists.

_{2}O masers in the disk of masing clouds surrounding the nucleus of the NGC4258 (M106) galaxy [189].

_{2}O masers in NGC 4258, measurements continued to be refined over the years. Table 3 shows the values of the distance to the NGC 4258 galaxy, with uncertainty estimates. The reader will notice that, currently, the margin of error is small. This, as we shall later comment, has also resulted in a dramatic reduction in uncertainty surrounding the Hubble constant.

_{2}O masers orbiting around the BH using larger data sets. Table 3 shows these newer estimates of distance to NGC 4258, with the last three reports showing only very small changes in the estimated distance, but with successive improvements in the uncertainty.

_{0}= 73.9 ± 3.0 km/s/Mpc [194] based on measurements of five megamaser galaxies in the Hubble flow. Their measurements are consistent with the high end of H

_{0}.

## 29. The Sunyaev–Zeldovich Effect

## 30. H_{0} Measurements with CMB Probes and BAO

_{0}= 67.4 ± 0.5 km/s/Mpc [214].

## 31. Gravity Waves Standard Sirens

_{0}is through gravity waves emitted from the coalescence of neutron stars at the final stages of binary systems. In principle, coalescence of black holes emits gravity waves as well, but does not provide redshifts nor precise distances. An electromagnetic counterpart is needed, and that is why one employs neutron stars. This method is considered to be applicable to every neutron coalescence. The name “standard siren” was picked to distinguish it from the known standard candles [218]. This method was proposed long ago by Bernard Schutz [219], but its realization materialized just few years ago. The method relies on the physics behind the collapse of the binary, and it is free of the systematics of local distance scales. This technique provides luminosity distances, and because they are well modeled during its coalescence phase, their uncertainties could be potentially small. On 17 August 2017, LIGO/Virgo collaboration detected a pulse of gravitational waves, called GW170817, associated with the merging of two neutron stars in NGC 4993, an elliptical galaxy in the constellation Hydra located 43 Mpc from us. The collaboration paper [220] reported a Hubble constant of 70

^{+12}

_{−8}km/s/Mpc. With this, we are entering into a new era of gravitational wave multi-messenger astronomy. The associated uncertainties will be greatly reduced in the coming years as more events of this type are measured.

_{0}can be derived purely from the gravitational waves of neutron star–black hole mergers. This new method provides an estimate of H

_{0}spanning the redshift range of z < 0.25, with the sensitivity of current gravity waves and without the need for any afterglow detection. The authors employed the inherently tight neutron star mass function together with the merge’s waveform amplitude and frequency to estimate the distance and redshift, respectively, thereby obtaining H

_{0}statistically. Their first estimate is H

_{0}= 86

^{+55}

_{−46}km/s/Mpc for the secure neutron star–black hole events GW190426 and GW200115. One expects that with ten more such events, one may reach a precision of δH

_{0}/H

_{0}≲ 20%.

## 32. Black Hole Shadows

_{0}. Technically, however, the problem is intricate, since light paths must be computed under the influence of the local action of gravity and, at the same time, in an expanding background. This technique is applicable to the late universe for small redshifts (z $\le $ 0.1), for large black hole masses ($\ge $ 10

^{9}solar masses), and potentially also for much larger redshifts in the coming years. Even though the angular diameter distance decreases for higher redshifts, the angular size of a shadow is expected to increase for high redshifts [224]. This scope will allow us to study angular diameter distances as functions of time. The current precision of H0 is on the order of 10% according to data such as those from the Einstein Horizon Telescope [225], or a small (few) percentage when considering probes in the near future [226,227,228]. A note of caution has been raised on this technique regarding the difficulty of obtaining reliable estimates of a black hole’s mass; achieving the required angular resolution; and having sufficient knowledge of high redshift accretion dynamics, especially challenging high redshift measurements [229]. Other possible effects may come from alternative models to ΛCDM in which pressure singularities may exist, which change the shadows of black holes and, therefore H

_{0}. This could eventually resolve the Hubble tension [230,231].

## 33. Fast Radio Bursts

_{0}. One approach to computing H

_{0}is to assume that the FRB energetics do not depend on redshift, being a kind of standard candle, and the sensitivity to H

_{0}is given through the signal-to-noise ratio of the pulse. It appears unlikely that they are standard candles unless there is a correction factor, since repeater pulses are far from identical from one pulse to the next. The other approach considers the cosmic contribution to the FRB dispersion measure, whose average depends directly on both the Hubble constant and the baryon density parameter; thus, synergies with CMB and nucleosynthesis results could help to constrain H

_{0}. In the study of C.W. James et al. [232], a detailed methodology and bias estimates are provided. Using a sample of 9 FRBs, Steffen Hagstotz et al. [233] found a Hubble constant of 62.3 $\pm $ 9.1 kms/Mpc/s, whereas Qin Wu et al. [234] used 18 localized FRBs and found an H

_{0}with a smaller uncertainty of 68.81

^{+4.99}

_{−4}.

_{33}kms/Mpc/s. Finally, using a sample of 16 localized and 60 unlocalized FRBs, C.W. James et al. [235] found a best fit of 73

^{+12}

_{−8}kms/Mpc/s. This new approach is certainly developing, and will surely provide more stringent constraints on H

_{0}in the near future from the CHIME catalogs [236], among others.

## 34. The Current Situation and Final Remarks

_{0}were grouped into one of two separate intervals. In the first set of the two, dubbed the “long” timescale, the value of H

_{0}was situated in the interval between 40 to 60 km/s/Mpc, while in the second set, labeled the “short” timescale, H

_{0}was located between 80 to 100 km/s/Mpc (i.e., a large value of the Hubble constant and a “short” timescale for the cosmological expansion). This indicated a “factor-of-two” difference!

_{0}of around 72 to ca. 10% accuracy [237]. This value was in the middle of the short and long timescales.

_{0}of ~73 ± 1 km/s/Mpc; these results were confirmed using the latest James Webb Space Telescope (JWST) measurements by Adam Riess et al. [238]. Other methods, such as strong lensing techniques, also obtain a Hubble constant in this interval. For instance, the H0LICOW collaboration reported H

_{0}= 73.3

^{+1.7}

_{−1}.

_{8}km/s/Mpc at 2.4% precision using the light from six multiply-imaged quasar systems [239]; although different results have been obtained since the modeling of the lens, mass distribution is an important systematic. These methods use late-time physics. On the contrary, early-universe physics at last scattering are anchored to the sound horizon, yielding CMB and BAO results that indicate smaller values of H

_{0}, around 67 km/s/Mpc. The uncertainties reported in these papers show a discrepancy of around four sigma, or larger in some cases. In Figure 12, we show the curves of the ΛCDM model that predict SH0ES data together with clustering data from the BOSS collaboration, which measured the Hubble expansion rate as a function of redshift. The differences are clear.

_{0}around 70 $\pm $ 2, as Figure 13 shows. These results ameliorate the discrepancy, and eventually may point to a solution within the ΛCDM model. A likely way to resolve the apparent discrepancy is for at least two of the methods to have additional systematic errors that are underestimated. The big question is which are those two or three out of three?

_{0}measurements is shown in Figure 14. It is clear that early physics determinations were smaller than those of late physics, except for those measurements from the TRGB obtained by CCHP. See further details in reference [242], and this reference [243] added during galley proofing corrections that accounts for the different modern techniques to measure H

_{0}.

_{0}is not a fundamental physical constant. It provides a scale for the present-day universe, as it is a reference. There is even some evidence that H

_{0}might have a decreasing trend when computed using data at higher redshifts (see reference [239] and others [248,249]). Now, its different estimations resulting from diverse standards based on different anchors may be indicative of some new physics, although this also might simply be a difference due to different systematic errors in the various techniques. We believe that time, as well as additional and improved observations, will settle the argument, just as occurred with the Great Shapley–Curtis debate.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | For thousands of years people had referred to the “fixed stars”. In astronomy, the fixed stars (Latin: stellae fixae) are the luminary points, mainly stars, that appear not to move relative to one another against the darkness of the night sky in the background. This is in contrast to those lights visible to naked eye, namely planets and comets, that appear to move slowly among those “fixed” stars. So, we had thousands of years prejudice for a static Universe and a human lifetime natural time scale while the stars move on time scales of millions of years or more. |

2 | What Einstein did not imagine is that the cosmological constant that he would later regret having introduced, and called his “biggest blunder” would later represent the dark energy that is an essential ingredient of the modern ΛCDM model. |

3 | All initial cosmological models assumed spherical (actually homogeneous and isotropic) symmetry in order to make cosmology a tractable issue. In the first half century this was an assumption but later radio surveys and most importantly the isotropy of the Cosmic Background Radiation have justified and improved this assumption. The CMB limits are uniformity to the part in 100,000 level or better. This is sufficient to find the first order solutions and treat the rest with perturbation theory. |

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**Figure 1.**Period–luminosity curves and best fits.

**Left**: Abscissas in days are equal to periods, ordinates corresponding to star magnitudes at their maxima and minima.

**Right**: abscissas are equal to logarithms of the periods. Figure taken from ref. [8].

**Figure 2.**Knut Lundmark’s 1924 [36] relation between relative distances and observed radial velocities of spiral nebulae (The scale unit is the distance to the Andromeda “nebula”).

**Figure 3.**Galactic redshift vs. distance, plotted by Hubble and Humason (1931) [39]; the rectangle in the lower left corner encloses data points plotted in 1929.

**Figure 4.**Approximate regions of type I, type II, and variable RR Lyrae populations that are expected to appear.

**Figure 5.**Apparent overlap between regions of type I and type II Cepheids. Arrows indicate that type I were misidentified as type II Cepheids. Also shown are RR Lyrae.

**Figure 6.**Luminosity–period curve of Cepheid variation. The various symbols designate variables from seven different systems. Credit: figure taken from Shapley’s 1918 paper [79].

**Figure 8.**One of two 2D grids with Earth and the Virgo cluster on the x-axis. Redshift contours are plotted for a Virgocentric flow. Note that a pure Hubble flow would be concentric (from Tonry and Davis 1981) [92].

**Figure 9.**The planetary luminosity function for M31 from Merrett et al. [103]. The sharp cut-off at the bright end of the [O III] PNLF is characteristic of all galaxies. Note the good fit of the standard PNLF.

**Figure 11.**Kowal’s redshift–magnitude relation. The crosses represent the average velocity and average magnitude of supernovae in the Virgo and Coma clusters (Kowal 1968 [150]).

**Figure 12.**Hubble expansion rate, as observed in the present (2023) and earlier (vs. redshift), along with the best-fitted cosmological model predictions from CMB observations (the lower curve). The upper curve is the model used for fitting the “nearby” supernova data, which extended out to z ~1. The actual data fit fairly tightly on this curve. Note that by quoting as the Hubble Constant H

_{0}one suppresses the model dependence of the nearby observations. Doing so allows us to see that the model would need to have “new physics” to bring these into agreement. If we assume some form of continuity in the expansion of the universe, then “new physics” has to jump between the light-based standard candles and the size-based standard candles (cosmic rulers). Credit: Figure was modified by us after a plot from reference [214]. The blue point corresponds to the measurement in in ref. [240].

**Figure 13.**Determination of Hubble constant using different techniques in the last two decades, with shadows showing the associated uncertainties. Credit: Wendy Freedman, reference [241].

**Figure 14.**The spread of the Hubble constant measurements performed in recent years, showing small values for CMB and clustering physics (

**upper**panel) and large values for different rugs used in late measurements (

**middle**panel). Also, different combinations of methods are shown, as well as their discrepancies with the early-physics results (

**bottom**panel). Credit: Vivien Bonvin and Martin Millon [244].

Year | Author | Value (km/s/Mpc) | Method | Ref. |
---|---|---|---|---|

1927 | Lemaître | 600 | Used data of Hubble and Strömberg | [86] |

1929 | Hubble | 500 | Cepheids; Shapley’s calibration P−L | [38] |

1931 | Hubble and Humason | 526 ± 10% | Cepheids; Shapley’s calibration P−L | [39] |

1946 | Mineur | 320 | Interstellar absorption corrections making a major recalibration to Shapley’s scale. Used zero point M_{0} = −1.54, see Baade (1956) | [87,88] [83] |

1951 | Behr | 240 | He heuristically pre-discovered the Scott effect and made the corresponding corrections to observations by J. Stebbins and A. E. Whitford (Astrophys J. 108 413 (1948), Mt Wilson Contr. 753, using Shapley’s calibration | [89] |

1952 | Baade and Thackeray | 280 ± 30 | RR Lyrae stars; corrected Shapley’s calibration (see next section) | [84,90] |

1956 | Humason, Mayall, and Sandage | 180 ± 20 | Comparison of photographic magnitudes for 576 galaxies to their redshifts; their calibration set the brightest galaxies in clusters equal to M31 luminosity | [91] |

**Table 2.**Leading calibrations of the Cepheid period in the twentieth century. Luminosity relation (<Mv> = −a − b log10 P, P (days)).

Year | a | b | <Mv> | Author |
---|---|---|---|---|

1913 | 0.60 | 2.10 | −2.70 | Hertzsprung [134] |

1918 | 0.72 | 2.10 | −2.82 | Shapley [135] |

1961 | 1.67 | 2.54 | −4.21 | Kraft [136] |

1968 | 1.43 | 2.80 | −4.23 | Sandage and Tammann [137] |

1987 | 1.35 | 2.78 | −4.13 | Feast and Walker [138] |

1991 | 1.40 | 2.76 | −4.16 | Madore and Freedman [139] |

1997 | 1.38 | 2.77 | −4.15 | Tanvir [140] |

1997 | 1.43 | 2.81 | −4.24 | Feast and Catchpole [141] |

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**MDPI and ACS Style**

Cervantes-Cota, J.L.; Galindo-Uribarri, S.; Smoot, G.F.
The Unsettled Number: Hubble’s Tension. *Universe* **2023**, *9*, 501.
https://doi.org/10.3390/universe9120501

**AMA Style**

Cervantes-Cota JL, Galindo-Uribarri S, Smoot GF.
The Unsettled Number: Hubble’s Tension. *Universe*. 2023; 9(12):501.
https://doi.org/10.3390/universe9120501

**Chicago/Turabian Style**

Cervantes-Cota, Jorge L., Salvador Galindo-Uribarri, and George F. Smoot.
2023. "The Unsettled Number: Hubble’s Tension" *Universe* 9, no. 12: 501.
https://doi.org/10.3390/universe9120501