Using The Baryonic Tully-Fisher Relation to Measure H o
UUsing The Baryonic Tully-Fisher Relation to Measure H o James Schombert ,Stacy McGaugh and Federico Lelli Institute for Fundamental Science, University of Oregon, Eugene, OR 97403 Department of Astronomy, Case Western Reserve University, Cleveland, OH 44106 School of Physics and Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff,CF24 3AA, UK
ABSTRACT
We explore the use of the baryonic Tully-Fisher relation (bTFR) as a new distanceindicator. Advances in near-IR imaging and stellar population models, plus preciserotation curves, have reduced the scatter in the bTFR such that distance is the dominantsource of uncertainty. Using 50 galaxies with accurate distances from Cepheids or tipmagnitude of the red giant branch, we calibrate the bTFR on a scale independent of H o . We then apply this calibrated bTFR to 95 independent galaxies from the SPARCsample, using CosmicFlows-3 velocities, to deduce the local value of H o . We find H o =75 . ± . ± . − Mpc − .
1. Introduction
The classic Tully-Fisher (TF) relation links the rotation velocity of a disk galaxy to its stellarmass and/or luminosity in a given photometric band. Because the observed rotation velocities donot depend on galaxy distance D , while stellar luminosities depend on D , the TF relation is a keydistance indicator that has played a crucial and historical role in constraining the value of H o (seeTully & Fisher 1977, Sakai et al. M (cid:12) , when dwarf galaxies in groups and in the field environmentbecome progressively more gas rich. By replacing the stellar mass with the total baryonic mass(stars plus gas, M b ), one recovers a single linear relation: the so-called baryonic Tully-Fisher relation(bTFR, Freeman 1999; McGaugh et al. et al. et al. M b (McGaugh 2012, Iorio etal. a r X i v : . [ a s t r o - ph . C O ] J u l V ) and baryon mass ( M b ) in the form of M b = AV x that, unlike most astrophysical correlations, does not show any room for a third parameter such ascharacteristic radius or surface brightness (Lelli et al. x becomesthe slope of the relationship, and log A becomes the zeropoint. As V is distance independent,errors in distance only reflect into M b (going as D ). While the scatter in the bTFR can be usedto constrain galaxy formation models in a ΛCDM cosmology (Dutton 2012), it also presents aunique opportunity to test the consistency of the distance scale zero-point (i.e., H o ) with redshift-independent calibrators of the bTFR.A new local test of H o has become critical, for over the past decade there have been growingdiscrepancies in the determination of H o by different methods. On one hand, the Cepheid (C)calibration to type-Ia supernovae (SN) yields H o = 73 . ± . − Mpc − (Riess et al. H o = 69 . ± . − Mpc − (Freedman et al. H o of 67 . ± . H o value of 67 . ± . et al. σ difference from theCepheid-calibrated SN distance scale (Verde, Treu & Riess 2019).In this paper, we present a redshift-independent calibration of the bTFR using galaxies withCepheid and TRGB distances (see Sorce et al. Spitzer
Photometry andAccurate Rotation Curves (SPARC) sample and of Ponomareva et al. (2018). This baseline bTFRis then compared to the larger SPARC data set with new CosmicFlows-3 velocities (Tully et al. H o values.
2. Data2.1. The SPARC galaxy database
The SPARC data set consists of HI rotation curves (RC) accumulated over the last threedecades of radio interferometry combined with deep near-IR photometry from the
Spitzer µ mIRAC camera (Lelli, McGaugh & Schombert 2016b). This provides the community an importantcombination of extended HI rotation curves (mapping the galaxy gravitational potential out tolarge radii) plus near-IR surface photometry to map the stellar component (see also Zaritsky et al. to 10 M (cid:12) ), surface brightness(3 to 1000 L (cid:12) pc − ) and rotation velocity ( V f from 20 to 300 km sec − ). The SPARC dataset 3 –also contains every Hubble late-type producing a representative sample of disk galaxies from dwarfirregulars to massive spirals with large bulges. The details of the sample are listed in Lelli, McGaugh& Schombert (2016b) and the resulting science outlined in McGaugh, Lelli & Schombert (2016).The SPARC sample and analysis with respect to the bTFR are presented in Lelli, McGaugh &Schombert (2016a). The analysis presented herein follows that paper with respect to error analysisplus small additions and corrections to the data as outlined in Lelli et al. (2019).For sample selection, we have isolated a subset of 125 galaxies that follow the quality criterionoutlined in Lelli et al. (2019) plus two additional galaxies with good TRGB distances. That studyexamined 153 objects in the original sample, excluding 22 galaxies with low inclinations ( i < et al. (2019), the average circular velocity along the flat portion of the rotation curve ( V f )results in the tightest correlation between rotation and total baryonic mass (see their Figure 2).This measure of rotation velocity is superior to single-dish measures (such as W P or W M ) orvelocities based on some disk scale length (e.g., V R e ) or peak of the rotation curve. Only thoserotation curves that display a flat outer portion (neither rising or falling) were included in thesample, and it is those values ( V f ), that will form one axis of the bTFR for our analysis.The other axis of the bTFR is total baryonic mass, the sum of all of the observed components,stars and gas ( M b = M ∗ + M g ). Of the two components to the baryonic mass, the stellar componenthas the highest uncertainty as it is determined by measuring a luminosity at a specific wavelengthmultiplied by the appropriate mass-to-light ratio (Υ ∗ ) for that wavelength. The mass-to-light ratiois obtained from stellar population models (e.g., Bell et al. et al. etal. ∗ deduced from optical luminosities are highly sensitive to the galaxy’s star formation rate (SFR)and produce uncertain stellar masses. Values in the near-IR are less sensitive to a galaxy’s starformation history and have the additional advantage of minimizing absorption by dust. For thisanalysis, we use the Υ ∗ values obtained from Schombert, McGaugh & Lelli (2019) of 0.5 for diskregions and 0.7 for bulge regions at the IRAC channel 1 wavelength of 3.6 µ m. We use fits to the Spitzer
B/D ) and applythe appropriate Υ ∗ to the luminosities of those components.The SPARC sample pays extra attention to gas-rich, low surface brightness (LSB) galaxiesthat typically populate the low-mass end of the bTFR. Unlike the luminous TF (Tully et al. V f , also provide detailed information about the gas content. Inparticular, neutral atomic (HI) gas dominates the gas component in typical star-forming galaxiesand, therefore, the gas mass can be deduced directly from the physics of the spin-flip transition ofhydrogen times the cosmic hydrogen fraction plus minor corrections for molecular hydrogen and 4 –heavier elements. While the contribution of ionized gas is considered in numerical simulations(Gnedin 2012), we found no evidence of large amounts of diffuse H α emission or X-ray output inthe gas-rich galaxies of our sample (Schombert et al. Aside from distance errors, errors to the baryon mass have four components: (1) errors in the3.6 photometry, (2) errors in the HI fluxes, (3) uncertainty in the conversion of near-IR luminosityto stellar mass (Υ ∗ ) and (4) uncertainty in the conversion of atomic gas mass into total gas mass.The first two are due to errors in the observations and are outlined in Lelli, McGaugh & Schombert(2016b). These error estimates are substantiated by comparison to 2MASS K -band photometryand other HI studies (Schombert & McGaugh 2014). They are typically of the order of 3% for the3.6 luminosities and 10% for HI fluxes, which translate into a mean error of 0.04 in log M b .The second set of errors are systematic to assumptions in the conversion of the observed valuesto mass values. For example, systematic errors in the modeling of Υ ∗ are explored in Schombert,McGaugh & Lelli (2019) where scenarios with different assumptions on the star formation history orthe stellar mass function produced different relationships between color and Υ ∗ . However, there areonly a limited number of stellar population models that also reproduce the main-sequence diagrams(the correlation of stellar mass versus star formation rate which indicate nearly constant SF over aHubble time for most of the SPARC sample) and the distribution of colors from the UV to near-IR(see Schombert, McGaugh & Lelli 2019). Those models are well approximated by using a singularvalue of 0.5 for Υ ∗ in the disk regions and a value of 0.7 for bulges.Through the use of Bayesian rotation-curve fits with dark matter halos, the plausible galaxy-to-galaxy variations of Υ ∗ can be explored (Li et al. ∗ at 3.6 µ m for the blue colors typical of low-mass galaxies span a range from 0.45 to 0.60.This is in agreement with the possible range of Υ ∗ from stellar population models given the rangein galaxy colors. For early-type galaxies with large bulges, the value of Υ ∗ can range from 0.6 to0.7. Thus, the systematic variation for Υ ∗ is approximately 0.15 for disks and 0.10 for bulges.Likewise, the gas correction term to convert HI values into total gas mass has two componentsof uncertainty, metallicity and the amount of molecular gas in a galaxy. In the past, we have useda conversion factor ( η ) of 1.33 for atomic to total gas mass that corrects for the abundance of Hein low-metallicity systems (high-mass disks have metallicities near solar, but their gas fractionsare small; see McGaugh 2012). We have also ignored molecular gas (primarily H ) owing to thearguments outlined in McGaugh (2012) and observations from McGaugh & Schombert (2015).In this paper, we will make minor corrections for metallicity and molecular gas using scalingrelations from McGaugh, Lelli & Schombert (2020). The gas metallicity causes the gas correctionterm, η , to vary from 1.33 for low-metallicity systems to 1.40 for galaxies with metallicity near 5 –solar (McGaugh 2012). As the gas-rich systems in our study are typically low in stellar mass,their metallicities are expected to be low. In comparison, the high-metallicity galaxies have high η values but their gas component is small. For this study, we allow η to vary with mass/metallicityfollowing the prescriptions in McGaugh, Lelli & Schombert (2020); however, this correction is small(see Table 1).A similar correction is needed for the contribution of molecular gas, which again is measuredto be very low for low-mass galaxies and higher for high-mass systems. From studies of H contentin disk galaxies, we adopt a scaling relation from total stellar mass to M H (McGaugh, Lelli &Schombert 2020). The contribution of H varies from 1% to 8% with a mean of 5% for the SPARCsample. This is at the same level as the variation in η for the combined gas content.To quantify the magnitude of the above variations on the baryon mass, we have listed sixscenarios in Table 1. Here we have made a maximum likelihood fit (Lelli et al. . We fit both the baseline sample of 50 galaxies with Cepheid and TRGBdistances and adequate V f values (this sample is described in greater detail in § H o = 75 km s − Mpc − for illustration). We then alter the prescriptionsused to calculate the baryonic mass in the six different ways listed in Table 1 and recalculate themaximum likelihood fit.The largest uncertainty from the stellar population model assumptions arises from varyingΥ ∗ , although the variations in metallicity or molecules for the gas mass are of the same order ofmagnitude. More importantly, neither corrections to stellar or gas mass have a significant influenceon the deduced slope of the bTFR. All the slopes are well within the errors of the fitted slope ofthe redshift-independent sample of 3.95 ± η due to metallicitychanges the slope of the main sample by only 0.05. Large variations in the slope of the bTFR wouldmake a skewed distribution of residuals from the bTFR, key to deducing an H o for each subsample.In addition, any change in the zeropoint (a shift of the baryon mass to higher or lower valuesdue to systematic changes in Υ ∗ or η ) has no effect on the use of the bTFR as a distance indicatoras those shifts are made to the calibrating galaxies as well as the main sample in identical ways.Because the calibrating Cepheid and TRGB sample also covers the same mass range as the fullsample, any minor changes in the slope would also have a negligible effect on the estimate of H o .We also list in Table 1 the very small uncertainty due to a zeropoint correction to the TRGBdistance scale (46% of the calibrating sample) from the discussion in Freedman et al. (2019). Alarger systematic shift in the TRGB zero-point would imply a major offset between Cepheids andTRGB calibrator galaxies in the bTFR plane, which appears unphysical. This zeropoint error isthe smallest of all of the systematic corrections. The uncertainty in using just TRGB (23 galaxies)or Cepheid (27 galaxies) calibrators is also small. As the Cepheid calibrators are primarily highin mass, this produces a shallower slope to the bTFR ( x = 3 .
70) compared to the entire sample. BayesLineFit is available at http://astroweb.cwru.edu/SPARC/ V f are independent of distance, only constrained by the errors in the HI measurementsthemselves. The uncertainty in V f considers three sources of error: (1) the random error oneach velocity point along the flat part of the rotation curve, quantifying noncircular motions andkinematic asymmetries between the two sides of the disk, (2) the dispersion around the estimateof V f along the rotation curve, quantifying the actual degree of flatness, and (3) the assumedinclination angle, which is generally derived from fitting the HI velocity field. Quantifying thedegree of flatness of the rotation curve and the error of the outer disk inclination is outlined inLelli, McGaugh & Schombert (2016b) and new error calculations are presented in Lelli et al. (2019)with a mean of 8% for the sample. This translates into a mean error of 0.02 in log V f on the x-axisof the bTFR. Distance is the critical parameter for converting 3.6 apparent magnitudes and HI fluxes intostellar and gas masses (as both are dependent on D ) and, thereby, into total baryonic masses( M b ). Within our 125 SPARC galaxies with accurate RCs, there are 30 galaxies with redshift-independent Cepheid or TRGB distances. The Cepheid and TRGB galaxies are listed in Table2 (TRGB distances and errors are from Tully et al. et al. et al. (2018) study, which also has Cepheid or TRGB distances (listed inTable 3). Of the 31 galaxies in the Ponomareva et al. sample that met our inclination and accurateRC criterion, 11 are already in the SPARC sample and we have used the SPARC data. For theremaining 20 galaxies, we have used their published HI fluxes and V f values, but have redeterminedtheir stellar mass values using our own Spitzer ∗ values andgas mass prescriptions. Their quoted errors are similar to the SPARC sample. The final sampleof 50 galaxies is show in Figure 1 and is used to calibrate the bTFR with galaxies having redshift-independent distances (hereafter, the C/TRGB sample). For uniformity, we have only used thedistances from Tully et al. (2019) and Bhardwaj et al. (2016) for both the SPARC and Ponomareva et al. samples.To check the internal consistency in the Cepheid versus TRGB methods, there are 41 galaxies inthe CosmicFlows-3 database with both Cepheid distances and TRGB measurements. Comparisonbetween these Cepheid and TRGB distances indicates an internal dispersion of 2% in distance withno obvious systematics (although there are very few galaxies with C/TRGB distances beyond 10Mpc). As discussed in Freedman et al. (2019), the error in the C/TRGB zeropoint is, at most,0.05 mag (2% in distance), which is consistent with the dispersion between the Cepheid and TRGB 7 –methods from the CosmicFlows-3 dataset.For the C/TRGB sample, observational error dominates the error budget compared to distanceerror in the M b values. The observational error is 0.04 in log M b whereas a 5% uncertainty to theC/TRGB distances contributes 0.05 in log M b ( V f being independent of distance). Thus, theuncertainty in the y-axis of the bTFR is at most 0.06 in log M b . And, if the slope is constant fromsample to sample, the limit to our knowledge of the y-intersect is limited by this error and thesample size. With uncertainties from the previous section in mind, the maximum likelihood fit to theC/TRGB sample is shown as the solid line in Figure 1 (also listed in Table 1). This slope isconsistent, within the errors, to the slope from Lelli et al. (2019, 3.85 versus 3.95 for the C/TRGBsample). For a mean error of 0.06 in log M b and 0.02 in log V f , the expected scatter for a slope of3.95 is 0.050 in log-log space. The perpendicular residuals have a dispersion of 0.048, exactly whatis expected for uncertainty solely from observational error and intrinsic scatter in Υ ∗ . Systematicmodel uncertainties are well mapped, and they are applied equally to all of the galaxies in thesample with only slight differences from low to high-mass due to early-type morphology (i.e., bulgelight) and additional molecular gas for high mass galaxies. Their effects are more significant on theslope of the bTFR (as seen in Table 1), but will be irrelevant because our comparison samples willhave the exact same corrections over the same range in galaxy mass.In our 2016 study, we used a combination of redshift-independent distances, cluster distancesand Hubble flow distances (with an H o = 73 km s − Mpc − ) to calculate the baryonic masses. Sincethat study, the CosmicFlows-3 database has been released (Tully et al. et al. H o . For our analysis we use a subset of the inclination-selected SPARC sampleof 125 galaxies where we have removed the 30 galaxies used for the C/TRGB calibration, leaving95 galaxies with good CosmicFlows-3 or Virgo infall velocities (hereafter the flow SPARC sample).There is a grouping in the flow SPARC dataset of 26 galaxies at 18 Mpc that represents theUrsa Major cluster (Verheijen & Sancisi 2001), where we assumed all the Ursa Major galaxies tobe at the mean cluster distance. In the interim, the CosmicFlows-3 project has assigned a meandistance of 17.2 Mpc for Ursa Major using the luminosity-based TF relation of 35 galaxies. Weadopt this new distance for an H o = 75 km s − Mpc − with the caveat of adjusting this clusterdistance for varying H o values.The slope of the maximum likelihood fit to the flow SPARC sample (using CosmicFlows-3 8 –Fig. 1.— The bTFR diagram for 30 SPARC galaxies (diamond symbols) and 20 Ponomareva et al. (2018)disks (circular symbols) with C/TRGB distances. TRGB galaxies are marked in red, Cepheid galaxies inblue. The baryonic mass is the sum of the
Spitzer V f , is determined directly from HIrotation curves following the techniques outlined in Lelli et al. (2016). A maximum likelihood fit is shownand serves as the baseline slope and zeropoint for comparison to the flow SPARC sample of 95 galaxies. x =3.97 ± H o ’s. In other words, we adopt the same slope of theC/TRGB sample and deduce H o from the varying zeropoint on the flow SPARC sample.Comparing the CosmicFlows-3 distance to the C/TRGB sample (for an H o = 75 km s − Mpc − ) finds a mean of zero and a standard deviation of 20% of the distance. The Virgo infallmodels (Mould et al. et al. (2019). Given that the uncertainty in the baryon mass of the bTFR diagram is nearly twice thatof the velocity axis, plus the distance errors will only map into the baryon mass, we then deducethat the dominant term for uncertainty for the flow SPARC sample is the combined observationalerrors to M b plus distance error. The observational errors contribute 0.04 on log M b where distanceerrors contribute as D for 0.16 in log M b . We assign a combined mean uncertainty of 0.20 in logfor the baryon mass axis. Again, for a mean error of 0.20 in log M b and 0.02 in log V f , the expectedscatter for a slope of 3.95 is 0.085 in log-log space. The perpendicular residuals have a dispersionof 0.067, slightly lower than what is expected for uncertainty solely from observational error. Usinga slope of 3.97 lowered the dispersion by only a small amount to 0.065.To explore the robustness of the derived H o value, we repeat the linear fitting of the calibratingC/TRGB and the flow SPARC sample using three techniques. The first, used in § et al. M b and V f (the Appendix of Lelli et al. H o values from the orthogonal fit as shown in the table as ∆ H o . Eventhough the uncertainties are greater in the vertical direction, we believe the orthogonal fits aremore representative of the data and the non-negligible error in V f . We can assign a systematicerror of ± The bTFR for the flow SPARC galaxies with inclinations > ◦ and flat, outer rotation curvesfor highly accurate V f values. The left panel displays the baryon masses calculated using the CosmicFlows-3 velocities. The solid line is the maximum likelihood fit. The center panel displays the baryon massescalculated for flow velocities based on a Virgo infall model from Mould et al. (2000). There is very littledifference in the bTFR using either CosmicFlows-3 or Mould et al. velocities. The right panel displays theresulting bTFR using only local standard of rest (LSRK) velocities. While similar to the infall distances,the scatter is 15% larger, from which we conclude that LSRK velocities are inadequate to deduce H o . Theresiduals in M b from the C/TRGB fit are shown in the bottom panels. The formal fits are listed in Table 5.
11 –
3. Results
The bTFR for the flow SPARC sample (95 galaxies) is shown in Figure 2 using the CosmicFlows-3 velocities (left panel), the Virgo infall flow model (Mould et al. H o ) will produce a linear shift from the C/TRGBbTFR (upward for larger distances) such that the bTFR can be used as a distance scale indicatormuch like the luminous TF or the Fundamental Plane relation. In other words, one can reproducethe C/TRGB bTFR zeropoint with the appropriate H o to convert redshifts into distance to applyto 3.6 luminosities and HI fluxes to determine the baryon mass. Comparing the residuals alongthe M b axis with the linear fit to the C/TRGB sample becomes a simple t-test. In this case,the empirical correlation of the bTFR is stronger (less scatter) than the Fundamental Plane. Inaddition, errors in the distance apply to each galaxy, rather than a cluster or group uncertainty aswith many traditional applications of the luminous TF relation. So, increasing the sample size hasa notable effect on the scatter versus adding a new galaxy cluster to the luminous TF relation.A formal match from the C/TRGB bTFR to the flow SPARC sample using CosmicFlows-3velocities produces a H o = 75 . ± . − Mpc − using the maximum likelihood orthogonalfitting method. This result is relatively independent of bTFR slope and/or flow model. Slightchanges in the slope (for example, from the 3.9 to 4.1) only widens the dispersion of the residualsfrom the bTFR, and the differences between the different distance samples operate only along the M b axis. The use of a slope of 3.8 or 4.1 (the range from the luminous TF relation) has no effecton the mean normalization and would only increase the error on H o by 0.5%. Likewise, as canbe seen in Figure 2, using the Virgo infall flow model produces a nearly identical bTFR fit to theCosmicFlows-3 bTFR (and an H o = 74 . − Mpc − ).Of the calibrating sample of 50 galaxies, 23 are TRGB galaxies and any zeropoint errorswould enter through errors in the TRGB method. The estimated uncertainty on M T RGB is 0.022(stat) and 0.039 (sys) (primarily through uncertainty to the distance to the LMC, Riess 2019).As a numerical experiment, we allowed the zeropoint for the TRGB galaxies to shift upward and 12 –downward by 0.05 mag then refit the bTFR for the C/TRGB sample. The results are shown inTable 1. Due to the fact that 27 of the 50 galaxies in the C/TRGB sample uses Cepheids as adistance indicator, the effect on the C/TRGB bTFR is small, only ± H o . In a similar fashion, we examined the effect of different fitting methods of the C/TRGBsample. This procedure was discussed in § ± σ of C/TRGB distances with similar dispersions of 20%. We find the CosmicFlows-3 tobe the closest match to the C/TRGB distances followed by the simple Virgo infall then the Virgoplus Great Attractor (Virgo+GA) correction. The Virgo + Great Attractor + Shapley superclusterproduced the same results as the Virgo+GA model. As not applying infall corrections raises thescatter by 15% to 20%, we find, unsurprisingly, that LSRK values do not represent the correct valueof H o . In addition, the slope of the bTFR from orthogonal fits using the CosmicFlows-3 velocitiesis closer to the C/TRGB bTFR slope than using any of the Virgo infall models. For this reason,we adopt the CosmicFlows-3 model and the orthogonal fits for our conclusions.Using the Virgo infall model consistently finds an H o value 0.4 km s − Mpc − lower than thevalue found from CosmicFlows-3 velocities across all three fitting methods. Using the Virgo+GAand Virgo+GA+Shapley infall models consistently finds H o values greater than the baseline by 2.4km s − Mpc − . No correction for peculiar velocities (LSRK) consistently finds H o values 2.5 kms − Mpc − lower than the CosmicFlows-3 values. The standard deviation between the various flowmodels is found to be 1.5 km s − Mpc − , which we adopt as the systematic variation for our flowmodel. As this value dominates the other systematics in the fitting methods or zeropoint errors,we use this value to indicate the expected systematic uncertainty in our measurements. Thus, weadopt a formal fit to the flow SPARC dataset of H o = 75 . ± . ± . − Mpc − .To test the significance level of the orthogonal fit to competing values of H o , we recalculate theflow SPARC sample baryonic masses using distances given by an H o = 67 . − Mpc − . Thisis the value of H o found by the Planck mission (Planck Collaboration et al. H o = 75 . − Mpc − and H o = 67 . − Mpc − sampleswe have, respectively, mean perpendiculars of 0.0 (by definition), − − H o samples. The C/TRGB and H o = 75 . − Mpc − samples are in agreement at a high level, the H o = 67 . − Mpc − sample is rejected at the 99.98% level. In fact, all values of H o below 70.5 are rejected at the 95%level. 13 –Fig. 3.— The bTFR for the flow SPARC galaxies, using CosmicFlows-3 velocities, with a H o = 67 . − Mpc − (left panel) and H o = 75 . − Mpc − (right panel). The solid line is the baseline fit fromFigure 1. Residuals are plotted in the bottom panels. Histograms of the range in M b and residuals are shownon the right (blue for 75.1, red for 67.4). As V f is independent of distance, changing H o has the effect ofincreasing the baryon mass by ∆ H o . H o values less than 70.5 are ruled out at the 95% confidence level.
14 –
4. Conclusions
As outlined in the Introduction, the baryonic Tully-Fisher relation is one of the strongest empirical correlations in extragalactic astronomy. While there exist many other correlations be-tween galaxy characteristics with similar scatter (such as scaling relations between effective radiusand surface brightness in ellipticals; see Schombert 2017), other galaxy correlations often involvecoupled parameters measured by a similar procedure (e.g., photometry). The bTFR involves twodistinct parameters, rotation velocity and baryonic mass, both measured by independent methods(HI observations provide both the HI mass and the rotation velocities, but the former comes fromthe total intensity map and the latter from the velocity map, so they are effectively independentobservables). With the advent of accurate rotation curves, the determination of V f has very littleuncertainty and is mapped to the very edge of the baryonic extent of a galaxy. In addition, thebehavior of V f with respect to past measures of rotation velocity is now well known (see Lelli et al. et al. ∗ (see Zaritsky et al. M b , as applied to as a new distance indicator,is now dominated by the observational error in the HI fluxes and 3.6 photometry, which approaches10% each. This means the baryonic mass, which is obtained without reference to kinematics, isdetermined to a higher degree of accuracy than could be obtained simply from the kinematics.The error on this axis is completely dominated by uncertainty in distance, which in turn makes itan ideal distance scale indicator when calibrated with redshift-independent samples. Our analysisherein results in a high value of H o near 75 km s − Mpc − .It is becoming increasingly obvious that values of H o , determined from empirical astronomicalcorrelations differ significantly from values determined from expectations estimated by cosmologicalmodel fits (e.g., fits to the angular power spectrum of the CMB fluctuations). There is a long historyin observational astronomy of constructing a distance scale to determine the local H o independentof any underlying astrophysics. In fact, the best techniques rely on as little modeling as possible.Where model values are used (for example, Υ ∗ in determining stellar mass from 3.6 luminosities),those values are highly constrained by limits based on knowledge of galaxy colors and past SFRs.However, one cannot ignore that the model fits to the CMB under a ΛCDM cosmology havebeen extremely successful at explaining the details of these observations of the early universe. Forexample, the concordance ΛCDM model is in agreement with measurements of anisotropies in thetemperature and polarization of the CMB (Planck Collaboration et al. et al. H o , but provide an inferred value of 67 . ± . − Mpc − .An important distinction is that observations of the CMB do not measure H o directly, butrather predict what the value of H o should be given a specific model of the expanding universe 15 –with cosmological parameters in a ΛCDM framework. This framework also makes predictions onthe power spectrum, polarization and anisotropies (also expressed through various cosmologicalparameters). The actual measured values of H o , through distance ladder techniques, do not agreewith this deduced value.In this study, we present another empirical method to deduce H o . We calibrate the bTFRusing 50 galaxies with Cepheid and/or TRGB distances and apply this relation to another 95galaxies from the SPARC sample to deduce the value of H o . Due to the nature of the observations,only the baryonic mass axis is sensitive to distance. Leveraging the fit to the redshift independentcalibrators, we find that H o = 75 . − Mpc − ruling out all values below 70 with a 95% degreeof confidence.High values of H o near 75 km s − Mpc − continue to be more appropriate when involvingempirical, observational issues of galaxy distance, while lower values of H o are required for theframework that involves the CMB and the physics of the early universe. These are, in a realsense, separate chains of deductive reasoning. And, while in a rule-driven universe, H o should beconnected from the early universe to today, there exist many explanations that do not require thatto be true (Verde, Treu & Riess 2019). In addition, there is increasing reason to doubt the CDMparadigm on galaxy scales (see McGaugh, Lelli & Schombert 2016; Bullock & Boylan-Kolchin 2017)despite its success on cosmological scalesThe future use of the bTFR as a distance indicator is encouraging due to the fact that theintrinsic bTFR scatter in the C/TRGB sample and the flow SPARC sample are similar, despitethe difference in sample size. This indicates that additional calibrating galaxies will significantlyimprove the slope and zeropoint to the bTFR while additional flow galaxies will severely pushdown both random and systematic errors. In fact, the current dominating systematic uncertainty( ± − Mpc − ) depends on how the 95 flow SPARC galaxies sample specific regions of thenearby universe, where flow velocities can differ by more than the mean global difference fromdifferent flow models (e.g. CosmicFlows-3 versus Virgo Infall). Additional flow galaxies will allowus to sample a larger volume of the nearby universe, where different flow models have, on average,smaller differences. Acknowledgements
We thank Brent Tully and an anonymous referee for comments to improve the text. Softwarefor this project was developed under NASA’s AIRS and ADAP Programs. This work is based in parton observations made with the Spitzer Space Telescope, which is operated by the Jet PropulsionLaboratory, California Institute of Technology under a contract with NASA. Support for this workwas provided by NASA through an award issued by JPL/Caltech. Other aspects of this work weresupported in part by NASA ADAP grant NNX11AF89G and NSF grant AST 0908370. As usual,this research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by 16 –the Jet Propulsion Laboratory, California Institute of Technology, under contract with the NationalAeronautics and Space Administration.
REFERENCES
Alam, S., Zhu, H., Croft, R. A. C., et al. 2017, MNRAS, 470, 2822Aubourg, ´E., Bailey, S., Bautista, J. E., et al. 2015, Phys. Rev. D, 92, 123516Bell, E. F., McIntosh, D. H., Katz, N., et al. 2003, ApJS, 149, 289Bhardwaj, A., Kanbur, S. M., Macri, L. M., et al. 2016, AJ, 151, 88Bradford, J. D., Geha, M. C., & Blanton, M. R. 2015, ApJ, 809, 146Bullock, J. S., & Boylan-Kolchin, M. 2017, ARA&A, 55, 343Cappellari, M., Scott, N., Alatalo, K., et al. 2013, MNRAS, 432, 1709Dutton, A. A. 2012, MNRAS, 424, 3123Freedman, W. L., Madore, B. F., Hatt, D., et al. 2019, ApJ, 882, 34Freeman, K. C. 1999, Ap&SS, 269, 119Gnedin, N. Y. 2012, ApJ, 754, 113Iorio, G., Fraternali, F., Nipoti, C., et al. 2017, MNRAS, 466, 4159Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, ApJ, 816, L14Lelli, F., McGaugh, S. S., & Schombert, J. M. 2016, AJ, 152, 157Lelli, F., McGaugh, S. S., Schombert, J. M., et al. 2019, MNRAS, 484, 3267Li, P., Lelli, F., McGaugh, S., et al. 2020, arXiv e-prints, arXiv:2001.10538McGaugh, S. S., Schombert, J. M., Bothun, G. D., et al. 2000, ApJ, 533, L99McGaugh, S. S. 2005, ApJ, 632, 859McGaugh, S. S. 2012, AJ, 143, 40McGaugh, S. S., & Schombert, J. M. 2015, ApJ, 802, 18McGaugh, S. S., Lelli, F., & Schombert, J. M. 2016, Phys. Rev. Lett., 117, 201101McGaugh, S. S., Lelli, F., & Schombert, J. M. 2020, Research Notes of the American AstronomicalSociety, 4, 45 17 –Meidt, S. E., Schinnerer, E., van de Ven, G., et al. 2014, ApJ, 788, 144Mould, J. R., Huchra, J. P., Freedman, W. L., et al. 2000, ApJ, 529, 786Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13Ponomareva, A. A., Verheijen, M. A. W., Papastergis, E., et al. 2018, MNRAS, 474, 4366Portinari, L., Sommer-Larsen, J., & Tantalo, R. 2004, Dark Matter in Galaxies, 309Qu, Z., & Bregman, J. N. 2019, ApJ, 880, 89Riess, A. G. 2019, Nature Reviews Physics, 2, 10Riess, A. G., Macri, L. M., Hoffmann, S. L., et al. 2016, ApJ, 826, 56Sakai, S., Zaritsky, D., & Kennicutt, R. C. 2000, AJ, 119, 1197Schombert, J., McGaugh, S., & Lelli, F. 2019, MNRAS, 483, 1496Schombert, J. M. 2017, PASA, 34, e016Schombert, J., Maciel, T., & McGaugh, S. 2011, Advances in Astronomy, 2011, 143698Schombert, J., & McGaugh, S. 2014, PASA, 31, e036Shaya, E. J., Tully, R. B., Hoffman, Y., et al. 2017, ApJ, 850, 207Sorce, J. G., Courtois, H. M., Tully, R. B., et al. 2013, ApJ, 765, 94Tully, R. B., & Fisher, J. R. 1977, IAU Colloq. 37: Decalages Vers Le Rouge EtTully, R. B., Courtois, H. M., Dolphin, A. E., et al. 2013, AJ, 146, 86Tully, R. B., Pomar`ede, D., Graziani, R., et al. 2019, ApJ, 880, 24Verde, L., Treu, T., & Riess, A. G. 2019, Nature Astronomy, 3, 891Verheijen, M. A. W. 2001, ApJ, 563, 694Verheijen, M. A. W., & Sancisi, R. 2001, VizieR Online Data Catalog, J/A+A/370/765Zaritsky, D., Courtois, H., Mu˜noz-Mateos, J.-C., et al. 2014, AJ, 147, 134
This preprint was prepared with the AAS L A TEX macros v5.2.
18 –Table 1. Systematic Error Budgetcase log
A x ∆ H o Notesbaseline 1.79 ± ± ∗ ± ± − ∗ =0.4High Υ ∗ ± ± ∗ =0.6No Molecules 1.80 ± ± − ± ± η =1.33High TRGB zeropoint 1.81 ± ± ± ± − − M b = AV f x . 19 –Table 2. SPARC Cepheids/TRGB Calibrating GalaxiesGalaxy D log V f log M b Distance(Mpc) (km s − ) ( M (cid:12) ) MethodD631-7 7.87 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± et al. Cepheids/TRGB Calibrating GalaxiesGalaxy D log V f log M b Distance(Mpc) (km s − ) ( M (cid:12) ) MethodNGC0253 3.56 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± A x ∆ H o Fit1.78 ± ± ± ± − ± ± − x H o flow model3.97 ± ± ± ± ± ± ± ±±