An Improved Distance to NGC 4258 and its Implications for the Hubble Constant
22019 November 18
An Improved Distance to NGC 4258 and its Implications for theHubble Constant
M. J. Reid , D. W. Pesce , & A. G. Riess , ABSTRACT
NGC 4258 is a critical galaxy for establishing the extragalactic distance scaleand estimating the Hubble constant ( H ). Water masers in the nucleus of thegalaxy orbit about its supermassive black hole, and very long baseline interfero-metric observations of their positions, velocities, and accelerations can be mod-eled to give a geometric estimate of the angular-diameter distance to the galaxy.We have improved the technique to obtain model parameter values, reducingboth statistical and systematic uncertainties compared to previous analyses. Wefind the distance to NGC 4258 to be 7 . ± .
082 (stat . ) ± .
076 (sys . ) Mpc.Using this as the sole source of calibration of the Cepheid-SN Ia distance ladderresults in H = 72 . ± . − Mpc − , and in concert with geometric distancesfrom Milky Way parallaxes and detached eclipsing binaries in the LMC we find H = 73 . ± . − Mpc − . The improved distance to NGC 4258 also providesa new calibration of the tip of the red giant branch of M F W = − . ± . H compared tothe LMC-based calibration, because it is measured on the same Hubble SpaceTelescope photometric system and through similarly low extinction as SN Ia hosthalos. The result is H = 71 . ± . − Mpc − , in good agreement with theresult from the Cepheid route, and there is no difference in H when using thesame calibration from NGC 4258 and same SN Ia Hubble diagram intercept tostart and end both distance ladders. Subject headings: (cosmology:) distance scale; (cosmology:) cosmological param-eters; methods: data analysis; stars: variables: Cepheids; galaxies: individual(NGC 4258) Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA Black Hole Initiative at Harvard University, 20 Garden Street, Cambridge, MA 02138, USA Department of Physics and Astronomy, Johns Hopkins University, Baltimore, MD, USA Space Telescope Science Institute, Baltimore, MD, USA a r X i v : . [ a s t r o - ph . GA ] N ov
1. Introduction
The nucleus of NGC 4258 hosts a H O megamaser in a sub-parsec-scale accretion disksurrounding a 4 × M (cid:12) black hole. Very long baseline interferometric (VLBI) mappingand spectral monitoring of the masers yield estimates of angular and linear accelerations ofmasing clouds in their Keplerian orbits about the black hole. Combining these accelerationsyields a very accurate and purely geometric distance to the galaxy. The distance to NGC4258 provides an important calibration for the Cepheid period-luminosity (PL) relation andthe absolute magnitude of the tip of the red giant branch (TRGB). These calibrations, inturn, provide the basis for some of the most accurate estimates of the Hubble constant ( H ).Humphreys et al. (2013) analyzed the very extensive dataset of observations of theH O masers toward NGC 4258 presented by Argon et al. (2007) and Humphreys et al.(2008) and estimated a distance of 7 . ± .
17 (stat . ) ± .
15 (sys . ) Mpc. The fitted dataconsisted of positions in two dimensions, Doppler velocities, and line-of-sight accelerations ofindividual maser features. The statistical (stat.) distance uncertainty was estimated usinga likelihood function that depended, in part, on assumed values for “error floors.” Theseerror floors were added in quadrature to measurement uncertainty in order to account forunknown limitations in the data, including “astrophysical noise.” For example, the 6 , − , H O transition has six hyperfine components, with three dominant components spanning 1.6km s − . When calculating a Doppler velocity one generally assumes that the three dominantcomponents contribute equally to the line profile. However, were one of the outer componentsto dominate the maser amplification, this could shift the assigned line velocity by 0.8 kms − . The heterogeneous nature of the data precludes a simple scaling of data uncertaintiesin order to achieve a post-fit χ ν per degree of freedom of unity. Since there are no strongpriors on the values of the error floors, reasonable variations in these values contribute tothe estimated systematic (sys.) uncertainty. In order to better address these issues, we havereanalyzed the NGC 4258 data using an Markov chain Monte Carlo (MCMC) approach,which includes the error floors as adjustable parameters. Owing to the exquisite quality ofthe dataset, these parameters could be solved for using “flat” priors, with only non-negativerestrictions on the their values. This approach indicated that the position error floors usedby Humphreys et al. (2013) were overly conservative, and that properly accounting forthem reduced the statistical uncertainty in distance, while also removing their contributionto systematic uncertainty. In this Letter, we report a revised distance to NGC 4258 and,correspondingly, estimates of H with reduced uncertainty. 3 –
2. An Improved Distance Estimate for NGC 4258
Over the past 25 yr, the number of VLBI observations used to map the masers inNGC 4258 and measure their accelerations has dramatically increased. Table 1 summarizesthe geometric distance estimates based on modeling the Keplerian orbits of maser featuresabout the galaxy’s supermassive black hole. The distance estimates reported in the firstthree papers listed in the table were based on successively larger data sets and, therefore,are nearly statistically independent. These distance estimates are statistically consistent.The last three papers (i.e., starting with Humphreys et al. (2013)) used the same data set,with the latter two papers improving the analysis approach. These papers report only verysmall changes in the estimated distance, but with successive improvements in the uncertainty.The dynamics of an H O maser cloud in an accretion disk surrounding a supermassiveblack hole can be characterized by four measurements: the eastward and northward offsetsfrom a fiducial position, ( x, y ); its heliocentric Doppler velocity, V ; and its line-of-sightacceleration, A . The relative weightings of these heterogeneous data can affect model fittedparameters. Whereas previously one had the freedom to adjust the individual error floorsfor these data components, we now remove this freedom and incorporate the error floors asparameters that are adjusted automatically with each MCMC trial. This removes potentialbias and “lets the data speak.” Note that in order to allow for adjustable data weights, onemust include the σ pre-factor in the full Gaussian formula, √ π σ e − ∆ / σ , when evaluatingdata uncertainties for the likelihood calculation (e.g., Roe 2015). We have conducted testson mock datasets of megamaser disks, which were generated with different levels of Gaussianrandom noise, and we were able to recover those noise levels. Thus, we are confident thatthis procedure works well.The position error floors previously adopted by Humphreys et al. (2013) were ( σ x , σ y )= ( ± . , ± . . These were based on very conservative estimates of the effectsof potential interferometric delay errors. Allowing the error floors to be model parametersrevealed that the uncertainty of the relative positions measured by VLBI actually approach( ± . , ± . ± D = 7 . ± .
075 (stat . ) Mpc, where theformal statistical uncertainty is now a factor of two smaller than before. The reduced χ ν forthis fit is 1.2 (for 483 degrees of freedom), which is an improvement over the reduced χ ν of1.4 in Humphreys et al. (2013), and we conservatively inflate the statistical component ofdistance uncertainty by √ . ± .
082 Mpc.The MCMC fitting code of Humphreys et al. (2013) employs the Metropolis–Hastingsalgorithm. Modifications to that program were (1) to allow the error floors to be adjustable 4 –parameters, (2) to replace handling of the recessional velocity from a relativistic velocityto the standard (1 + z ) formalism, and (3) to define the warping parameters relative to theaverage maser radius (6.1 mas) instead of at the origin. As an end-to-end check on this code,one of us (DP) has written an independent fitting program, implementing a HamiltonianMCMC approach, and we find essentially identical results from both programs The two-dimensional marginalized probability densities for selected parameters are shown in Fig. 1.Further gains in distance accuracy come from reducing systematic sources of error.Humphreys et al. (2013), in their Table 4, listed the contributions of a number of systematicsto the distance uncertainty. By solving for error floor parameters, their uncertainties arenow incorporated into the marginalized distance estimate, and therefore we remove theircontributions from the systematic error budget. In addition, as done in Riess et al. (2016),we now calculate two orders of magnitude more MCMC trials than in Humphreys et al.(2013), making the fitted parameter values largely insensitive to initial conditions. Finally,since we allow for eccentric orbits for the masing clouds, as well as second-order warpingof the disk, the marginalized distance estimate now includes these uncertainties. The onlyremaining systematic error term in Table 4 of Humphreys et al. that we have not includedin our distance uncertainty is their estimate of the effects of unmodeled spiral structure of ± .
076 Mpc. Thus, we have now reduced the estimated systematic uncertainty by nearly afactor of two. 5 –Table 1. Estimates of Distance to NGC 4258
Reference Distance (Stat., Sys.) Data Comment(Mpc) (Mpc)Miyoshi et al. (1995) 6 . ± . . ± . . ± .
228 (0.167,0.155) 18 VLBI epochsRiess et al. (2016) 7 . ± .
197 (0.170,0.100) 18 VLBI epochs Better MCMC convergenceThis paper 7 . ± .
112 (0.082,0.076) 18 VLBI epochs Improved analysis (see text)Note. — Distance uncertainties are the quadrature sum of the statistical (Stat.) and systematic (Sys.)errors. The distance modulus from this Letter is 29 . ± . D = 7.5762 +0.07610.0742 . . . . . M B H ( M ) M BH = 3.9780 +0.04060.0394 . . . . . x ( m a s ) x = 0.0016 +0.00050.0004 . . . . y ( m a s ) y = 0.0041 +0.00040.0005 . . . . . v , h v ( k m s ) v ,hv = 2.2468 +0.28250.2668 . . . . v , s y s ( k m s ) v ,sys = 0.3122 +0.16860.1810 .
35 7 .
50 7 .
65 7 .
80 7 . D (Mpc) . . . . a ( k m s y r ) .
84 3 .
92 4 .
00 4 .
08 4 . M BH (10 M ) . . . . . x (mas) .
003 0 .
004 0 .
005 0 . y (mas) . . . . . v ,hv (km s ) . . . . v ,sys (km s ) . . . . a (km s yr ) a = 0.4615 +0.04470.0409 Fig. 1.—
Marginalized probability densities for selected parameters: distance ( D ), black hole mass ( M bh ),and error floors for the eastward ( σ x ) and northward ( σ y ) positions, the high ( σ v,hv ) and systemic ( σ v,sys )velocities, and the accelerations ( σ a ). a Distance (Mpc) 7 . ± .
082 7 . ± . M (cid:12) ) 3 . ± .
04 4 . ± . − ) 473 . ± . . ± . b x -position (mas) − . ± . − . ± . b y -position (mas) 0 . ± .
004 0 . ± . c (deg) 87 . ± .
09 86 . ± . st order (deg mas − ) 2 . ± .
07 2 . ± . nd order (deg mas − ) 0 . ± .
018 ...Disk position angle c (deg) 88 . ± .
04 88 . ± . st order (deg mas − ) 2 . ± .
02 2 . ± . nd order (deg mas − ) − . ± . − . ± . . ± .
001 0 . ± . ±
13 294 ± − ) 123 ± ± d σ x eastward offset (mas) 0 . ± . σ y northward offset (mas) 0 . ± . σ v,sys systemic velocities (km s − ) 0 . ± .
20 [1.00] σ v,hv high-vel velocities (km s − ) 2 . ± .
31 [1.00] σ a accelerations (km s − y − ) 0 . ± .
04 [0.30] a Uncertainties are formal statistical estimates, inflated by their respective (cid:112) χ ν . b Positions are measured relative to the maser emission at 510 km s − . The differencebetween the x -position values is largely due to the systematic effect of changing the recessional 8 –velocity from relativistic in Humphreys et al. (2013) to (1 + z ) in this Letter. c Disk inclination and position angle are measured at a radius, r , of 6.1 mas, near the averageradius of the masers. The values from Humphreys et al. (2013) have been adjusted from r = 0to r = 6 . d Brackets for the Humphreys et al. (2013) error floor values indicate that these wereassumed and not solved for. 9 –Our best estimate of the distance to NGC 4258 is 7 . ± .
082 (stat . ) ± .
076 (sys . )Mpc.
3. Estimate of H NGC 4258 has played a central role in the determination of the Hubble constant, becauseits geometric distance has been established to useful and increasingly high precision sinceHerrnstein et al. (1999). The galaxy is near enough to calibrate Cepheid variables (Maoz etal. 1999; Macri et al. 2006; Hoffman 2013), the TRGB (Macri et al. 2006; Mager, Madore& Freedman 2008; Jang & Lee 2017) and Mira variables (Huang et al. 2018) using the
Hubble Space Telescope (HST). These stars in turn are used to calibrate the luminosities ofSNe Ia, which measure the Hubble flow and the Hubble constant.In order to determine the Hubble constant using the improved distance to NGC 4258presented here, we use the Cepheid and SN Ia data and formalism presented in Riess et al.(2016) and revised geometric distances provided in Riess et al. (2019). The distance to NGC4258 has increased modestly from that in Riess et al. (2016) by 0.5%, well within the total ± .
6% error there, or even the ± .
5% total error here, resulting in a small change in H measured using NGC 4258 as the sole, geometric calibrator of Cepheid luminosities. However,there is a larger impact on H measured in conjunction with the other geometric calibrators:Milky Way parallaxes and detached eclipsing binaries (DEBs) in the LMC (Pietrzy´nski etal. 2019). The reason is that the weight of NGC 4258 in the joint solution has increasedsubstantially due to its 40% smaller distance error, and its preferred value for H is 2.7% lowerthan for the other methods. Including uncertainties in the PL relationships and photometriczero-points given in Table 6 of Riess et al. (2019), the net uncertainties in the use of eachanchor for the Cepheid distance ladder are now 2.1%, 1.7% and 1.5% for NGC 4258, MilkyWay parallaxes, and the LMC DEBs, respectively. The values of H and their uncertainties(including systematics) are given in Table 3. Combining estimates from all three anchorsyields a best value for H of 73 . ± . − Mpc − , with the revised distance to NGC 4258reducing H by this combination by 0.7%. The total uncertainty is little changed because theerror is already dominated by the mean of the 19 SN Ia calibrators from Riess et al. (2016)(1.2%), with little impact from the reduction of the error due to the geometric calibration ofCepheids which decreases here from 0.8% to 0.7%. The difference between this late universemeasurement of H and the prediction from Planck and ΛCDM (Planck 2018) of 67 . ± . − Mpc − remains high at 4.2 σ .We can also use the revised distance to NGC 4258 to derive a new calibration of theTRGB on the HST ACS photometric system, which is used to observe the TRGB in the 10 –halos of SN Ia hosts. There are two sets of HST observations with the ACS in F W that have yielded a strong detection of the TRGB in NGC 4258: GO 9477 (PI: Madore, 2.6ks in F W ) and GO 9810 (PI: Greenhill, 8.8 ks in F W ). The GO 9477 observationis of a halo field and has been analyzed by Mager, Madore & Freedman (2008), Madore,Mager & Freedman (2009), and Jang & Lee (2017), with differing definitions of the TRGBmagnitude system (e.g., color transformed in Madore et al. 2009). The recent thoroughanalysis by Jang & Lee find F W =25.36 ± .
03 mag, where a foreground extinction of A F W = 0 . ± .
003 mag was assumed.One expects that there will only be a small amount of extinction of the TRGB in thehalos of galaxies. A statistical value of A I ∼ .
01 mag is indicated from an analysis byM´enard, Kilbinger & Scranton (2010) based on the reddening of background quasars byforeground halos at radii from the host center of 10-20 kpc (M´enard, Scranton et al. 2010).Most importantly for the determination of H is to use a consistent approach to estimatethe TRGB extinction, both where the TRGB is calibrated and where that calibration isapplied, to better reduce systematic errors through their cancellation. In this manner thedetermination of H is relatively independent of whether or not halos have a measurableamount of extinction, and for this reason we default to the convention of assuming no haloextinction.Macri et al. (2006) measured the TRGB in the “Outer field” of NGC 4258 using datafrom GO 9810. This field is primarily from the halo of NGC 4258 and is at a similar separationfrom the nucleus, r ∼
20 kpc, as other TRGB measurements used in Freedman, Madore &Hatt (2019) and where internal extinction is by convention assumed to be negligible. Theobservation is very deep, reaching I ∼
27 and V ∼
28, significantly deeper than the TRGBmagnitude and sufficient to reject all stars in the I -band luminosity function with V − I ≤ I = 25 . ± .
02 mag or transformed using equation (2) inMacri et al. (2006) for the TRGB color of V − I = 1 . F W = 25 . ± .
02 mag. This detection is somewhat stronger inthis data than from GO program 9477, likely due to its greater depth (2.6 ks versus 8.8 ks in F W ) and is reflected in its smaller error (both generated by a bootstrap test). The outerchip of this field (no disk, only halo) gives the same estimated peak to < . σ (L. Macri,private communication). Correcting this by the same amount as the Jang & Lee (2017) resultfor Milky Way extinction yields very good agreement (1 σ ) with the result from Jang & Lee.We take the average of the two and conservatively adopt the larger error (as these errors maybe correlated via edge detection methods and point-spread function fitting packages used)resulting in F W = 25 . ± .
030 mag. Using the distance to NGC 4258 presented here,which translates to µ N = 29 . ± .
032 mag, yields M F W = − . ± .
04 mag for theTRGB. 11 –Although the distance uncertainty is a bit larger for NGC 4258 than for the LMC,systematic errors in the TRGB measurement of H calibrated with NGC 4258 are smallerbecause (i) this calibration is on the same HST photometric system (zero-points, instruments,bandpasses) as TRGB measured in SN Ia hosts, (ii) the extinction is either negligible asassumed in SN Ia host halos or, even if ∼ .
01 mag, it becomes negligible after a consistenttreatment through its cancellation along the distance ladder, and (iii) the metallicity in thehalos of large galaxies is likely to be more similar to each other (i.e., metal poor) than tothe LMC. Indeed, the present shortcomings of the LMC TRGB calibration are that it hasbeen measured only with ground-based systems (Jang & Lee 2017), which have low angularresolution that results in blending of ∼ A I ≥ . A I ≈ . ± .
02 mag (Jang & Lee 2017; Freedman, Madore& Hatt 2019; Yuan et al. 2019).Replacing the calibration of the TRGB of F W = − . ± .
04 mag derived from theimproved distance to NGC 4258 on the HST (i.e., native) photometric system with the valueused by Freedman, Madore & Hatt (2019) of F W = − . ± .
04 mag and using their SNIa TRGB sample yields H = 71 . ± . − Mpc − . This value is in excellent agreementwith that derived using Cepheids calibrated by the distance to NGC 4258 of H = 72 . ± . − Mpc − (see Table 3). We also provide the individual values of H using the twopreviously described TRGB measurements in NGC 4258 in Table 4.An additional consideration for comparing these two distance ladders is that each useda different sample of SN Ia to measure the Hubble flow. Riess et al. (2016) used a homoge-neously calibrated “Supercal” compilation of surveys (Scolnic et al. 2015), and Freedman,Madore & Hatt (2019) used a sample from the Carnegie Supernova Program (CSP; Burnset al. 2018). Because most of the data for the SNe in TRGB or Cepheid hosts is also derivedfrom other non-CSP surveys, there is a preference for the use of a homogeneously calibratedcompilation at both ends of the ladder to reduce systematic errors between samples. TheCSP sample used with the TRGB produces an intercept which is ∼
1% lower (in H ) thanthe intercept from the compilation set (Burns et al. 2018; Kenworthy et al. 2019) usedwith Cepheids, and this 1% difference is the same as the remaining difference in H from theTRGB and Cepheid route. Thus, we find using the geometric calibration from NGC 4258and the same Hubble diagram intercept for both the TRGB and Cepheid distance laddersbrings them into agreement. Facilities:
VLBA, HST 12 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
14 –Table 3. Estimates of H Including Systematics Using Cepheids
Anchor(s) H value Difference from(km s − Mpc − ) (Planck+ΛCDM) ∗ NGC 4258 72 . ± . σ Two anchorsLMC + NGC 4258 72 . ± . σ LMC + MW 74 . ± . σ NGC 4258 + MW 73 . ± . σ Three anchors (best)NGC 4258 + MW + LMC 73 . ± . σ Note. — ∗ : H = 67 . ± . − Mpc − (Planck 2018)
15 –Table 4. Estimates of H Including Systematics Using TRGB
Anchor(s) H value a Difference from(km s − Mpc − ) (Planck+ΛCDM) ∗ NGC 4258 b . ± . σ NGC 4258 c . ± . σ NGC 4258 d . ± . σ Note. — ∗ : H = 67 . ± . − Mpc − (Planck 2018).Note. — a : TRGB and Cepheids use differentSN Ia intercepts as discussed in the text.Note. — b : Based on a foreground extinctioncorrected TRGB peak of F W = 25 . ± . c : Based on a foreground extinctioncorrected TRGB peak of F W = 25 . ± . d : Using Jang and Lee (2017) andMacri et al. (2006) variance-weighted average of F W = 25 . ± ..