Proc. Natl. Acad. Sci. USAVol. 90, pp. 4806-4810, June 1993Colloquium Paper

This paper was presented at a colloquium entitkd "Physical Cosmology," organized by a committee chaired by DavidN. Schramm, held March 27 and 28, 1992, at the National Academy of Sciences, Irvine, CA.

The Hubble constantR. BRENT TULLYInstitute for Astronomy, University of Hawaii, Honolulu, HI 96822

ABSTRACT Five methods of estimating distances havedemonstrated internal reproducibility at the level of5-20% rmsaccuracy. The best of these are the cepheid (and RR Lyrae),planetary nebulae, and surface-brightness fluctuation tech-niques. Luminosity-line width and Dg-a methods are lessaccurate for an individual case but can be applied to largenumbers of galaxies. The agreement is excellent between thesefive procedures. It is determined that Hubble constant Ho = 90± 10 km-s'1Mpc- [1 parsec (pc) = 3.09 x 1016 m]. It is difficultto reconcile this value with the preferred world model even in thelow-density case. The standard model with Q1 = 1 may beexcluded unless there is something totally misunderstood aboutthe foundation of the distance scale or the ages of stars.

The Hubble constant, Ho, is a measure of the current scale ofthe universe compared with the expansion rate: the charac-teristic recessional velocity between two objects equals theHubble constant times the distance separating the objects.The inverse of this constant is taken to be on order of the ageof the universe because it gives the look-back time to whenall matter in expansion can be traced back to the same point,assuming no modifying forces.There is a common perception that the value of the Hubble

constant is only poorly known, confined to the range 50 < Ho< 100 km s-l Mpc-1 [1 parsec (pc) = 3.09 x 1016 m]. It willbe argued here that this perception is incorrect-that there isnow a concordance ofgood methods with low internal scatterand excellent external consistency that lead to Ho = 90 + 10km-s-l Mpc-1. The inverse characteristic age for the uni-verse is Ho1 = 11 + 1 gigayears (Gyr).

This value is different from the generally accepted age ofthe oldest stars in our Galaxy of tG = 16 + 1.5 Gyr. The abovenumbers lead to a product HotG = 1.5 + 0.2, where theuncertainties here and above are standard deviations. Thepreferred model of the universe posits that the only cosmo-logical constraint on the motions of particles since the "BigBang" explosion is retardation due to gravity, and for thepopular range of densities of 0.1-1.0 times the critical densityfor closure, then 0.9 < Hoto < 0.67, where the age of theuniverse is to ' tG. The current observations rule out thepreferred model at any reasonable density at the 99% confi-dence level and clearly discriminate against the most popularflat-space variant.The Hubble constant controversy has a long history.

Recent reviews that give results consistent with those givenhere are provided by Huchra (1) and Jacoby et al. (2). Asomewhat contrary view is provided by Tammann (3).

Good Distance Estimator Methods

There are now five high-quality techniques with the additionof two excellent methods in the last couple of years. Here are

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brief words on each, ordered by the chronological appear-ance of the methodology.Cepheid and RR Lyrae Variables. Stars in a variety of

luminosity-temperature regimes are unstable and fluctuate inbrightness or color. Population I cepheids are interesting asdistance estimators because there is a tight correlation be-tween their luminosities and pulsation periods. Also thesestars are bright. They have been studied since the pioneeringwork by Leavitt (4) in 1908, and by now there is quite detailedinformation about their properties (5, 6). There has beenenough confidence in this methodology to advertise that theHubble space telescope will observe cepheids and solve thedistance scale problem. The dispersion in the near-infrared,time-averaged versions of the period-luminosity relationshipis only 0.15 magnitude (mag), so with enough stars andinformation about reddening, it should be possible to measurethe relative distance of a galaxy to better than 5%. By"relative" I mean the distance with respect to the fiducialdistance of the fundamental calibrators-in this case, theLarge Magellanic Cloud (LMC). However, it is difficult toidentify and obtain the necessary photometry on cepheidsthat lie in complex regions in galaxies. After eight decades ofeffort, there has only been success in using the method ongalaxies within 4 Mpc (7-9), with the heroic and marginalexception of M101 (10).RR Lyrae pulsating stars also have well-defined luminos-

ities, which make good "standard candles." They are somuch fainter than cepheids that they have been studied inonly a few galaxies within the Local Group (11, 12). Still, RRLyrae are important because they are old stars and providea foundation complementary to the cepheids for the distancesto the LMC and other key calibrators of other techniques.

It can be mentioned in passing that the distance to the LMCcan be determined by yet other methods with good accuracybut limited range of applicability. The concordance of thedetermination based on the light echo ring ofsupernova 1987a(13) is particularly noteworthy, since this geometric mea-surement is free of intermediate steps. There are also othermethods that exploit the known color-luminosity propertiesof groups of stars, such as the characteristics of the red giantbranch tip. The LMC distance should be established todaywithin an uncertainty of 5%.

Luminosity-Line Width Relations for Spirals. In 1922, Opik(14) used the virial theorem and an assumed mass-to-lightratio to estimate the distance of M31 with remarkable accu-racy. In the 1960-1970s the related "indicative mass"method was in vogue (15, 16). A mass estimate was calculatedfrom the product of the photometric radius times the squareof the neutral hydrogen line profile width. In 1977, Tully andFisher (17) showed that a simpler two-parameter relationshipexists between the HI line width and either luminosity orradius alone.

Abbreviations: LMC, Large Magellanic Cloud; Mpc, megaparsec;mag, magnitude.

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Proc. Natl. Acad. Sci. USA 90 (1993) 4807

A great deal of the early work with this method was doneby Aaronson and coworkers (18-20) with 1.6-gm (H band)magnitudes. This system is sensitive to old star populationsthat might couple with mass and not be sensitive to obscu-ration. Nowadays, the preference is probably I band (21, 22).Global magnitudes can be determined with Charge-Coupled-Device imagers and the contrast against the night sky is morefavorable. Optical rotation curve measurements provide analternative to global HI profiles and allow the technique to beused to distances beyond the sensitivity range of radiotelescopes (23-25). With good data, the rms dispersion in therelative distance estimate of a single galaxy is 15%. Ifenoughgalaxies in a cluster are observed, the ensemble distanceshould be known to better than 5%. The zero point has beenaccurately established based on nearby systems with cepheiddistance determinations (26, 27). With six local calibrators,the zero point is established to an uncertainty of about 7%.The luminosity-line width technique can be applied in a

relatively straightforward manner to large numbers of galax-ies. A major industry is emerging with the current interest inthe study of large-scale galaxy flows. Distance estimates arebecoming available for some 2000 individual galaxies.Three-Parameter Relations for Ellipticals. In gas-poor gal-

axies it is difficult to observe a global measure of the internalmotions, and the most accessible substitute is the velocitybroadening of stellar absorption lines in the galactic cores.Curiously, while the preferred distance-estimator techniquefor gas-rich systems has evolved from a three-parameter to atwo-parameter relationship, as just discussed, in the case ofelliptical systems the situation is the reverse. Faber andJackson (28) discussed a relationship between an ellipticalgalaxy's luminosity and its core velocity dispersion, butsubsequently it was found (29, 30) that there is a tighterfundamental plane that requires information about luminos-ity, internal motions, and dimension. In fact, the empiricalrelationship between these parameters can be reformulatedto look very much like the virial law.

This three-parameter correlation is often abbreviatedDn-oa, where Dn is the diameter of an aperture at a specifiedsurface brightness (hence, combines dimension and luminos-ity information) and or is the root-mean-square velocitydispersion. The method can provide relative distances withan rms uncertainty of 20%. Unfortunately, the zero-pointcalibration presents a difficulty because there are no large,nearby ellipticals to act as local calibrators. A calibration hasbeen tried based on the bulge component of spirals (31), butit is susceptible to large systematic error. The Dno-c methodcan be used to measure relative distances to many hundredsofgalaxies in scores ofgroups and clusters, where the fiducialzero point is set by some nearby entity like the Virgo Cluster.There is intriguing, as yet unraveled, information about the

process of galaxy formation in the differences between theseempirical descriptions of elliptical and spiral systems. For

ellipticals, the "fundamental plane" is skewed in the spacedescribed by luminosity, kinematics, and surface brightness.For spirals, rotation motions are coupled directly to totalluminosity with little scatter.

Planetary Nebulae. At a late stage in stellar evolution, massis ejected and then ionized by the illumination of the hotcentral star. It is being demonstrated that there is a remark-ably abrupt upper limit to the luminosity of the line emissionfrom the ionized shell. The signature of this cutoff luminosityprovides the distance estimator. Ciardullo, Jacoby, and co-workers (32-34) have studied the phenomenon in a diverseset ofenvironments and find little scatter among host galaxieswith a range of luminosities, metallicities, and morphologicaltype. If on the order of 102 planetary nebulae over a 1-maginterval below the brightest objects can be observed, then thedistance to an individual galaxy can be determined with arelative rms uncertainty of -5%. Operationally, the methodworks best on ellipticals because then there is minimalconfusion from HII complexes and dust.The zero-point calibration of the technique is provided by

an assumed distance to M31, based on cepheid and RR Lyraeinformation (35). Consistency with the cepheid scale is dem-onstrated with the relative distances between this galaxy andboth the LMC and M81 (36, 37). There are results availableon 16 galaxies within 20 Mpc.

Surface Brightness Fluctuations. The brightest stars inuneventful elliptical galaxies are red giants. Nearby systemswould have a grainy appearance because of incipient reso-

lution. The \/Nv statistics of stars within a pixel result in asmoother image at greater distances. Hence, a measure ofmottledness is a measure of proximity. Translating thissimple concept into distance estimates with low internalscatter is the impressive accomplishment of Tonry and co-workers (38-40).As with the planetary nebulae method, the most convincing

evidence of the internal accuracy of the procedure comesfrom the comparison of distance estimates to galaxies in thesame cluster. There is an apparent color term coupled withsystem luminosity. Once corrected, the rms deviations are5%. The zero point is provided by M32, the close companionof M31. This situation is not entirely satisfactory becauseM32 is considerably less luminous than any more distantelliptical galaxies on the program. However, remarkably, themethodology is now being applied with apparent success togalaxies as far away as 4000 km s-1.

Comparison of the Five Good Methods

Table 1 provides a comparison of measured distances bydifferent techniques both to individual galaxies and to groupsof galaxies. Some cases that are noted are calibrators forcertain of the methods, which precludes a comparison.Within the groups there are multiple observations (numbers

Table 1. Consistency in distance moduli between five good methodsPlanetary

Object Cepheids nebulae SBF* T-Ft D, rms, %LMC 18.50 18.44 - 2M31 24.34 Calib Calib Calib - -M81 27.59 27.72 Calib - 4Leo 30.02 (2) 29.84 (1) 30.00 (2) 29.85 (3) 5N1023 29.97 (2) 29.77 (5) 7Virgo 30.84 (6) 30.89 (12) 30.78 (58) Calib 3Fornax 30.85 (13) 30.84 (2) 30.95 (11) 3Eridanus - 31.10 (5) 31.23 (5) 31.68 (5) 15Coma 34.44 (30) 34.50 (33) 2Data are distance moduli = 5 (log distance in Mpc) +25; numbers in parentheses are the number of observations (n).

*SBF, surface brightness fluctuations.tT-F, Tully-Fisher luminosity line width; Calib, calibrator.

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in brackets), and the distance moduli are averages. Uncer-tainties are reduced roughly as VN.The last column in Table 1 is of greatest interest. The entry

here is the rms agreement in distance between whateverestimates are available for a given galaxy or group. Except inone case (a deviant Dn-o- distance to the Eridanus group),agreements are <10%! These results are consistent with theclaimed internal accuracies of each of the five methods andprovide evidence that each method is free of any substantialsystematic external error.

Other Methods

There is some evidence that supernovae of both types I andII may provide good distance estimates. Type Ia supernovaemay have the small intrinsic luminosity dispersion of a goodstandard candle. These objects are thought to involve thecollapse of stars that acquire barely enough mass to exceedthe Chandrasekhar limit, and this well-defined onset condi-tion might result in well-defined luminosity characteristics.There are theoretical energy-budget arguments that constrainthe brightness of these objects (41, 42). Alternatively, themethod could be given an empirical calibration, but then theprocedure is wedded to the conventional distance ladder.Type II supernovae have a wide dispersion in luminosities,but individual cases can be given a Baade-Wesselink treat-ment; the kinematic expansion and luminosity evolution ofthe photosphere can be modeled to give a distance (43, 44).At least some workers in this field evaluate the uncertaintiesin distance estimates to type II supernovae to be very large(45).Both the energy-budget analyses oftype Ia supernovae and

the atmosphere models for type II supernovae give highluminosities, hence large distances and low Ho values com-pared with the five methods described previously. The con-currence of these results from two lines of physical modelingprovides, according to this author, the only serious evidencefor a low value of Ho.The companion communication by Kirshner (3) discusses

the supernovae issues. The conclusions ofour two papers areat the limits of incompatibility, so whom should one believe?The line is clearly drawn between the pedestrian empiricalapproach and the attractiveness (seductiveness?) of a coher-ent physical description. Recall that Opik (14) had a remark-ably good distance to M31 with his physical virial theoremargument. Still, the derivative luminosity-line width andD"-ormethods are messy enough in detail that they manifestlyrequire empirical tweaks. The fact that the supernovae meth-ods give respectable distance estimates provides consider-able comfort that the models are on track. However, hasevery subtlety that could give rise to a systematic effect beenconsidered? The empirical evidence suggests not.There are a lot of other distance estimator methods of

uncertain value. For example, modeling the separate-pathtiming delays in the fluctuations of gravitationally lensedquasi stellar objects is intriguing and could mature into animportant methodology. At present there is only one case(46).

It is not appropriate to consider the uncertain techniques inan attempt to produce a weighted derivation for Ho. Theprobability is too great that the uncertain methods giveanswers that center around the "expectation" value of thepreferred world model (Hoto s 1) rather than whatever thereal value might be. Hence, any averaged estimate of Hoinvolving dubious techniques will be shifted away from thereal value toward the expectation value.

Resolution of Two Nagging Issues

The good news is that there are now several independentdistance estimators of high quality. Until only a couple ofyears ago, there was excessive dependence on the luminos-ity-line width method because the cepheid method lacksrange and Dr-or lacks a reliable local calibration. There weredoubts expressed about the luminosity-line width procedure[possible type dependencies (47), large dispersions in somesamples (48), environmental effects (49), Malmquist biases(50)] that in extreme cases seemed to some of us to bemisguided. Fig. 1 illustrates the status of the I-band lumi-nosity-line width calibration. The relationship is so tight thatthe scatter could be expected to arise from only observationaluncertainties and cluster depth effects. There is not muchroom for type or surface-brightness dependencies, and noth-ing of significance is seen. The new planetary nebulae andsurface-brightness fluctuations methods are providing com-fort that we really have understood what is going on in somecontentious areas. With the confidence provided by this newsupport, two battles that have played a nonnegligible role inthe Ho controversy will be reviewed.The Local Velocity Anomaly Versus Malmquist Bias.

Nearby galaxies give low values for the Hubble parameter,and estimates of Ho increase with distance (50, 51). Thiseffect is expected if there is Malmquist bias. However, theeffect could be physical: the local velocity anomaly. Theposition of this author (52, 53) is that (i) some studies trulysuffer Malmquist bias, but the effect has been overestimatedbecause of pessimism about the scatter in the luminosity-linewidth relations; (ii) with proper techniques the bias can beneutralized (by determining distances based on the relation-ship derived by a regression on the distance-independentvariable with a complete sample), hence statistically unbi-ased distances can be obtained to individual galaxies in thefield; (iii) the region of lowest Ho values coincides with thestructures of our local neighborhood-the Coma-SculptorCloud and its Leo Spur-and the effect can be reasonablyexplained by the self-gravity of this region; and (iv) in theearly days of the Hubble constant controversy, a lot of theevidence for a low Ho was coming from these nearby galaxiesthat have velocities that are systematically retarded fromHubble flow.

-24

- 22

-20 X -

-18- xA<

-16

1.8 2.2 2.6 3.0log WR

FIG. 1. I-band calibration ofthe luminosity-line width relation. x,Members of the Ursa Major Cluster; * and *, local calibrators withcepheid distances; the straight line, regression with minimization inline widths, first fit to Ursa Major data, then shifted to best fit withlocal calibrators, and then iterated one time with fit to all data. MI,absolute I-band magnitude; W'R, neutral hydrogen profile line widthdeprojected to edge-on orientation.

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Backside Contamination in the Virgo Cluster Versus aBreakdown of the Luminosity-Line Width Methodology. Ifone considers the brightest galaxies superimposed upon theVirgo Cluster, one finds that the luminosity-line width rela-tion has low dispersion. However, as fainter galaxies areadded, the dispersion increases and these additional galaxiessystematically shift the apparent modulus of the cluster tolarger distances (54, 55). The scatter becomes much largerthan that seen with the local calibrators or for the Ursa MajorCluster. This effect could signal an environmental breakdownof the luminosity-line width methodology, in which case itwould be difficult to know when the procedure could beapplied. However, the effect could equally be due to confu-sion from a significant number of objects in the line-of-sightof the Virgo Cluster but in the background. The position ofthis author (ref. 22; M. J. Pierce and R.B.T., unpublisheddata) is that (i) use of techniques susceptible to distance bias(and also marginal magnitudes) confuses the issue, whereasunbiased techniques do not generate the artifact of an appar-ent increase in distance modulus going to fainter galaxies; (ii)the galaxies that are determined to be in the background withthe luminosity-line width method are clustered on the planeof the sky, and there is a tight intermediate backgroundconcentration 50% beyond the cluster (de Vaucouleurs'Virgo W', part of his S' cloud) and a couple of furtherbackground concentrations at twice the cluster distance(Virgo Wand M plus a smatter of objects at the same distancelying within a narrow band parallel to the supergalacticplane); (iii) these concentrations have about the expectedvelocities for their positions according to basic Virgocentricinfall models; (iv) a couple of elliptical galaxies are placed bythe surface brightness fluctuation method into the back-groundW and W' structures in fine agreement; and (v) hence,contrary to providing an example of the breakdown of thedistance estimator relationships, the evidence suggests thatthere is no dangerous environmental effect tainting the meth-odology in this cluster situation.

Fig. 2 is offered in an attempt to summarize a messysituation. The squares within the 70 circle, both filled andopen, are considered to be legitimate cluster members. Thesquares outside this circle are part of the Southern Extensionand are presumed to be falling into the cluster. It is the largecircles that deserve particular attention. These cases fall in anextremely tight window: 20 < distance < 26 Mpc and 600 <velocity < 1200 km*s- and they are almost all congregatedat 5 o'clock and 5° from M87 (the two exceptions are probablydeviant cluster members). This clearly distinguished group-ing is called Virgo W'. Almost all of the systems that are lefthave velocities greater than 1200 km s-1 and estimated dis-tances greater than 26 Mpc. The elliptical NGC4261 is in theVirgo W Cluster which, for the most part, projects onto theSouthern Extension rather than the actual Virgo Cluster.However, there are other contaminants projected onto thecluster from roughly this same distance in the background.

Conclusion: de Vaucouleurs Was Right

Gdrard de Vaucouleurs (56-58) looked at the evidence avail-able and derived a value of Ho that is consistent withinreasonable errors with the current estimate. However, thesituation has certainly changed over the intervening decade!Although the attack on the preferred world model was alwaysserious, it was not strong enough until recently to be com-pelling. But now (i) the step to the LMC is known to 5%(agreement between population I and population II high-quality indicators and confirmed by the geometric measure-ment of the supernova 1987a light-echo shell); (ii) the stepfrom the LMC to local calibrators is known to 5% (concur-rence of cepheid, RR Lyrae, and planetary nebulae dis-tances); (iii) the step from the local calibrators to the Virgo

20 _

15

C_0

_ 1000

5-

0

13h

Region around Virgo Cluster

1 2h40m 1 2h20m

Right Ascension12h

FIG. 2. Location of all galaxies with well-measured distances inthe Virgo Cluster region. The cluster proper is confined to the 70radius circle centered on M87. Galaxies with measured distancesgreater than 20 Mpc are indicated two ways. Those with measureddistances of 20-26 Mpc and systemic velocities of 600-1200 km.si1are indicated by larger round symbols [the elliptical NGC4365 is thefilled circle, and the rest are gas-rich (open circles)]. Otherwise,galaxies measured to be beyond 20 Mpc are marked by crosses or,in the case of the sole elliptical NGC4261, the symbol is a 6-point star.

Cluster and neighborhood is known to 5% (concurrence ofluminosity-line width, planetary nebulae, and surface bright-ness fluctuation methods; the only concern is disagreementwith theoretical supernova models); and (iv) the step from theLocal Supercluster to large scales is known to 5% (concur-rence of spiral and elliptical indicators in enough clusters tohave confidence that peculiar velocities are probably not afactor).Adding these uncertainties in quadrature gives an overall

uncertainty of 10o (sense of a standard deviation). A valueof Ho1 = 11 + 1 Gyr does not totally exclude the preferredmodel, but alternatives should be considered. The preferredmodel with Ql = 1 can probably be rejected.

My involvement in distance scale work over the past few years hasbeen done in close collaboration with Mike Pierce. This research issupported by a grant from the National Science Foundation.

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