COSMICFLOWS-2: THE DATA

Details about how we derived a measure of maximum rotation velocities are provided by Courtois et al. (2009, 2011b). Here we focus entirely on the interpretation of 21 cm H i profiles. The data are provided by our observations with the Green Bank Telescope (GBT) or the Arecibo and Parkes telescopes or are acquired from archival sources related to the Arecibo Telescope, the National Radio Astronomy Observatory 140 foot and 300 foot telescopes, and GBT, or the Nançay, Parkes, and Effelsberg telescopes. Each of these alternative sources gives us digital records of fluxes in wavelength bins.

With Cosmicflows-1 and earlier we used an analog measure of an H i profile line width, W₂₀, the width of a profile at 20% of peak intensity. We now use a measure called W_m50, the width at 50% of the mean flux per channel over the range of channels containing 90% of the total flux. The exclusion of 5% of the flux at each of the high and low velocity extremes reduces ambiguity in the channel count due to profile wings. The new line width definition is a more robust measure than W₂₀ with asymmetric or single peaked profiles. Springob et al. (2005) and Catinella et al. (2007) have discussed alternative line width characterizations. However, our choice is justified in Courtois et al. (2009), in which we describe the adjustments that are made to account for instrumental and redshift broadening

where z is redshift, Δv is the smoothed spectral resolution, and λ = 0.25 is an empirically determined constant. Next we translate the observed line width parameter to a statistical measure of twice the maximum rotation speed

with W_{c, m50} = 100 km s⁻¹ describing the transition from boxcar (horned) profiles to roughly Gaussian profiles and W_{t, m50} = 9 km s⁻¹ describing the effects of thermal broadening. Finally a de-projection translates the observed inclination, i, to edge-on orientation in order to arrive at the desired parameter $W^i_{mx}$

The uncertainty in a line width measurement W_m50 is determined in the first instance by the ratio of the signal S defined by the mean flux per channel in the spectral line to the noise N in channels outside the spectral line

In order to be considered useful for a TFR distance determination, we require that e_W ⩽ 20 km s⁻¹. After an initial automatic fit to line profiles from a computer algorithm, profiles and fits are inspected by eye. Attention is given to the possibility of confusion from neighbors or to anomalies such as might arise from interactions. The result of the inspection might be that a profile is rejected as a candidate for a distance determination, in which case e_W is set greater than 20 km s⁻¹, or on rare occasions, a profile with low signal-to-noise ratio (S/N) is considered adequate, whence e_W is set by hand to 20 km s⁻¹.

The line profile information, from whatever telescope and whatever source, is accumulated in EDD in the catalog called All Digital H i. The H i profiles and our fits for the parameter W_m50 are available for inspection at that site. There can be up to three independent profiles for a given target. The preferred measure of W_mx is derived from an average of all contributions with e_W ⩽ 20 km s⁻¹, weighted by the inverse square of the error estimates. As we closed out acquisitions for the present sample, the All Digital H i catalog contained entries for 14,219 galaxies and contained entries with e_W ⩽ 20 km s⁻¹ for 11,343 galaxies. Figure 5 is an example of the graphic material made available for every galaxy in EDD.

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The All Digital H i catalog is incomplete in one important respect that requires us to retain a legacy of the analog W₂₀ profile measurements. Very large galaxies or special cases such as galaxies with profiles near zero velocity confused by local emission are not adequately observed with large single dish radio telescopes. In a small number of cases it is necessary to interpret profiles observed with interferometers or to fall back to old observations with a small telescope like Dwingeloo. In these cases, we make use of a tight correlation between the old analog parameter W_R that approximates twice the maximum rotation (before de-projection to edge-on) and the new W_mx parameter to recover a W_mx proxy

If S/N is sufficient, comparable and consistent profile information is obtained with all seven radio telescopes providing data. However some facilities are more sensitive than others. The Arecibo Telescope is the most sensitive. As a consequence our sample extends dramatically in the declination range 0° < δ < 38° accessed by this telescope. GBT is the second most sensitive facility and allows observations over the entire northern sky down to δ = −45°. The Parkes Telescope is used for targets below this limit, however its lesser sensitivity results in reduced redshift reach toward the southern celestial pole. Since the Parkes Telescope is the smallest of current facilities it has the largest beam, which means that observations with this telescope are favored for large nearby galaxies.

The rotation curve information that has been discussed characterizes the total mass within the observed domain of galaxies. Photometry provides a characterization of the mass in stars. It also provides a handle on galaxy inclinations, information needed for the de-projection of disk motions and for estimates of reddening. After accounting for the effects of tilt, the tightness of the TFR is an evident manifestation of a correlation between stellar and total mass over the observed radii in galaxies.

Photometry may be usefully acquired over a range of wavelengths. Most of the stellar mass is in cool stars so it can be anticipated that the best TFR correlations are in bands that are least contaminated by flux from young populations, emission lines, and dust. With ground-based observations it has been demonstrated that the TFR scatter is minimal at R and I bands (Pierce & Tully 1988; Tully & Pierce 2000). Scatter increases at B, presumably because of the stochastic incidence of star formation and increased obscuration. Scatter also increases in the near-infrared, at H and K. A primary cause is high and variable sky foreground, which limits observations to the high surface brightness components of galaxies. A secondary, inevitable cause of degradation results from the steepening of the TFR toward longer wavelengths (Tully et al. 1982).

For the current analysis we concentrate on photometry at Cousins I band, although our final product will be informed by photometry at 3.6 μm obtained with Spitzer Space Telescope. As with the TRGB and H i data sets that have already been described, we combine photometry from our own observations with published material. Our photometry procedures were described by Courtois et al. (2011a) and our I band data products were made available in EDD within the catalog Hawaii Photometry. The analysis uses the Archangel photometry software (Schombert 2007; Schombert & Smith 2012). Our products, in addition to total I band magnitudes, include disk scale lengths and central surface brightnesses, radii enclosing 20%, 50%, and 80% of light, a concentration index, and a ratio of dimensions on the minor and major axes. An example of the graphic material in EDD is seen in Figure 6.

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Considerable attention was given to ensure consistency with literature contributions to the I band photometry compilation. Only sources that offered substantial contributions were considered so that there would be significant overlap between samples. We began with the assembly of magnitudes already used in Cosmicflows-1 (Tully & Pierce 2000; Tully et al. 2008). The most significant contributions in that earlier compilation were from Mathewson et al. (1992), Giovanelli et al. (1997), and our own program (Pierce & Tully 1988; Tully et al. 1996). Now, in addition to the new photometry described by Courtois et al. (2011a), we include the accumulated and reanalyzed photometry from the Cornell group compiled by Springob et al. (2007) and the photometry carried out on Sloan Digital Sky Survey (SDSS) material by Hall et al. (2012). This latter contribution involves observations at Gunn i band so requires a translation from Sloan g, r, i to Cousins I band as prescribed by Smith et al. (2002)

where cases with r − i ⩾ 0.95 are excluded. After making these translations, a slight tilt was found in comparisons between $I_c^{{\rm sdss}}$ and I_c from the three alternative sources mentioned above (Cosmicflows-1, Cornell, or recent Hawaii). From the 725 cases in common with Cosmicflows-1 and the 857 cases in common with the Cornell compilation the following empirical correction was determined

The uncertainty on the slope is ±0.003 so the deviation from slope unity is significant at 5.5σ.

After these adjustments, all contributions were determined to be on the same scale. In cases of targets with multiple measurements, all photometry sources were regarded as equal in taking an average. On instances there were strongly deviant measures. With three or more independent determinations the bad measures could be unambiguously culled and with two determinations it was often obvious which contribution was bad. Otherwise, if the difference was not extreme the two determinations were averaged. The relative quality of the sources could be evaluated by pairwise comparisons. The rms scatter in the differences between the Cosmicflows-1 and Cornell sources with 2106 cases (rejecting 20 differences >0.45) was 0.077 mag. The scatter between Cosmicflows-1 and SDSS with 713 cases (after 12 rejections) was 0.117 mag. The scatter between Cosmicflows-1 and the Hawaii photometry with 223 cases (after 4 rejections) was 0.120 mag.

A raw observed magnitude requires adjustments to account for obscuration in our Galaxy, $A^I_b$, and in the target galaxy, $A^I_i$, as well as a small k-correction that accounts for spectral displacement with redshift, $A^I_k$. To summarize, the corrected magnitude is

where I_T is the total observed magnitude and the three correction factors are $A_b^I = R_I E(B-V)$ with differential reddening E(B − V) from Schlegel et al. (1998) cirrus maps and R_I = 1.77, $A_i^I = \gamma _I \log (a/b)$ with a/b the major to minor axis ratio and $\gamma _I = 0.92 + 1.63(\log W^i_{mx} - 2.5)$ or γ_I = 0 if $W^i_{mx} \le 86$ km s⁻¹ (Tully et al. 1998), $A_k^I = 0.302z + 8.768z^2 - 68.680z^3 + 181.904z^4$ (Chilingarian et al. 2010).

Our procedures for using Spitzer Space Telescope 3.6 μm photometry are described by Sorce et al. (2012a). Again we resort to a combination of observations made within our collaboration and observations by others made available through the Spitzer archive.¹¹ In this case all observations are made with the same facility and with very similar observation strategies. Again, reductions were carried out using the Archangel software and products are made available in EDD, this time in the catalog Spitzer [3.6] Band Photometry. See Figure 7 for an example of a surface brightness profile.

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The Spitzer mid-infrared (mid-IR) photometry has a couple of clear advantages. It is a concern with I band photometry that the assembly is a composite of material acquired at several observatories, with subtly different filters and detectors, over different parts of the sky, many observing seasons, and with the vagaries of sky conditions. The Spitzer photometry, by contrast, is consistently obtained all-sky.

In addition to that most important advantage, the Spitzer photometry profits from being negligibly affected by obscuration, by being sensitive to low surface brightness features because of minimal background and because the target fluxes are dominated by light from old stars. As a result, the Spitzer magnitudes after corrections have uncertainties of only 0.05 mag, which are a minor contributor to the TFR error budget.

The Spitzer photometry will lead to an important improvement of the TFR in the future. At present the number of galaxies with reduced Spitzer photometry is limited. The TFR calibration with the material currently available will be discussed in a later subsection and this calibration is important for the establishment of the overall distance scale. However, today, by far, the larger compilation of photometry is at the I band. The TFR component of Cosmicflows-2 is a construct at I band with only a minor zero point adjustment arising from the Spitzer calibration.

The final important ingredient needed for a TFR distance determination is a decent inclination. Uncertainties in inclination can be a dominant source of observational error. There are two regimes sensitive to inclination effects: those of increased obscuration toward edge-on and increased projection uncertainty in rotation velocities toward face-on.

The issue of obscuration toward edge-on is the lesser problem. Corrections at optical bands can be quite large but they are remarkably predictable. It is well demonstrated that obscuration is greatest in the most intrinsically luminous spirals, whereas at the extreme of the smallest spirals, obscuration effects are marginally detected (Giovanelli et al. 1995; Tully et al. 1998). The coupling of reddening with luminosity is a problem since the measurement of distances is inextricably linked with the measurement of luminosities. Tully et al. (1998) sidestep the connection between reddening and luminosity by the replacement connection between reddening and line width, the latter a distance-independent monitor of intrinsic luminosity. The prescription for reddening at I band, $A_i^I$, was already given in the qualifications for Equation (7). At the Spitzer 3.6 μm band the reddening, $A_i^{[3.6]}$, is small. The coefficient that describes the amplitude as a function of axial ratio is $\gamma _{[3.6]} = 0.10 + 0.19(\log W^i_{mx} - 2.5$) or γ_[3.6] = 0 if $W^i_{mx} \le 94$ km s⁻¹ (Sorce et al. 2013). Reddening in magnitudes at 3.6 μm is reduced by a factor nine from I band.

There can be greater problems with corrections to line widths for galaxies seen toward face-on. Some galaxies are sufficiently regular that their orientations can be determined with precisions of 1°–2°. However others present difficulties. The tilt of barred systems can be ambiguous, as can that of some prominent spirals, depending on the orientation of these features with respect to the major and minor axes. There are also galaxies with asymmetries and warps that still seem reasonable candidates for a TFR application. Overall, comparisons between axial measurements by different authors, along with tests that involve sorting images by inclination, suggest that a typical uncertainty in inclination is 5°. Axial ratios can be a poor descriptor of inclinations with large-bulge systems, as witnessed in the Sombrero galaxy.

This level of uncertainty prescribes the limit we set of 45° for our TFR samples. An uncertainty of 5° at this limit amounts to an uncertainty in line width of 9%. For all the concern that we have about inclinations, it is comforting that with tests involving cluster samples (galaxies assumed to be at the same distance) discussed by Tully & Courtois (2012) there are no TFR systematics or increases in scatter over the full range of inclinations from 45° to 90°. Nonetheless, the issue of inclination corrections is sufficiently disconcerting that we are attracted to a particular sample that eliminates the problem: galaxies from the various incarnations of the Flat Galaxy Catalog (Karachentsev et al. 1993, 1999; Mitronova et al. 2004; Kudrya et al. 2003, 2009). These galaxies are all seen extremely edge-on. They constitute a particularly interesting sample when combined with Spitzer photometry where reddening is minor. We consider such a sample, as will be discussed.

We define inclinations, i, from measurements of axial ratios, b/a, the minor to major axis dimensions, with the formula

where q₀ = 0.2 is the assumed flattening of an edge-on system. Other authors use involved variants of q₀, noting that some edge-on systems are distinctly thinner than b/a = 0.2 (Bottinelli et al. 1983; Giovanelli et al. 1997). However we prefer this simpler formulation, recalling that our primary interest is not the inclination per se but rather the correction needed to measure a distance. This simple formulation avoids discontinuities. Also, as discussed at length by Tully & Pierce (2000), a variation in q₀ translates to an almost fixed displacement of line width at all inclinations: an uncertainty in the numerator on the right side of Equation (8) because of the choice of q₀ has a large effect on i toward edge-on, but a small effect on the line width correction 1/sin i, whereas the uncertainty in this numerator because of the choice of q₀ is small toward face-on where the potential effect on 1/sin i would be great.

Our sources of axial ratios track our sources of I band photometry with one important extension. We find that the axial ratios given in the Lyon Extragalactic Database (LEDA) are of good quality (Paturel et al. 1996). Tiny adjustments were made to arrive at consistency with the b/a values used earlier in our program:

Inter-comparisons with the large Cornell sample (Springob et al. 2007) result in the following rms scatter in differences: ±0.04 (582 cases) with Cosmicflows-1, ±0.05 (810 cases) with SDSS, ±0.07 (167 cases) with the Hawaii photometry, and ±0.08 (2104 cases) with LEDA. The LEDA contribution ensures that there are at least two independent b/a measurements for all galaxies with I band photometry. The availability of multiple measures is helpful for culling bad data. When the occasion demanded it, we introduced new measurements of b/a from our evaluation of the images.

The TFR calibration used with Cosmicflows-1 dates from Tully & Pierce (2000). An update is required, especially because of the new definition of the H i line width parameter. The required re-calibration for the TFR at I band was published by Tully & Courtois (2012). A parallel calibration for the TFR using Spitzer [3.6] band photometry is given by Sorce et al. (2013). The procedures followed for the construction of the TFR are described in detail in these three references so only a brief outline of important points will be given here.

In order to minimize the Malmquist "selection" bias (Willick 1994) we derive distances with the "inverse" TFR, the linear fit to the correlation between magnitudes and log line width with errors taken in line width. Only a tiny correction is required for a residual bias, as will be discussed later. A feature of the inverse relation (and a check that it is minimally biased) is invariance of the slope with the limiting magnitude of the sample. Hence a template can be constructed through the superposition of subsamples drawn from clusters, with appropriate shifts on the magnitude axis to account for relative differences in distance moduli.

With the current investigation we draw on observations within 13 clusters. A rapidly converging iteration between choice of slope and modulus shifts between the 13 clusters leads to an optimal matching of the clusters and a definition of the inverse TFR slope. The final step is to define the zero point using galaxies with distances independently established by Cepheid PLR or TRGB observations. The correlation slope determined with the 13 cluster template is assumed in deriving the least squares best fit with the zero point calibrators. The results of these procedures are demonstrated for the I band calibration by Tully & Courtois (2012) in the top panel of Figure 8.

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There is a complication with the calibration at 3.6 μm. There is a color term. The rms magnitude dispersion of the I and [3.6] correlations are comparable at ∼0.4 mag but only after correction for a color term in the mid-IR case. The color dependency arises because of the well known correlation between galaxy morphology and color that causes the TFR to steepen in bands toward longer wavelengths (Tully et al. 1982). Given two galaxies of different colors but the same line width, the two galaxies must displace in magnitude in different passbands. Empirically, displacements are minimum around 1 μm. At shorter wavelengths the bluer of our hypothetical pair will tend to be brighter while at longer wavelengths the redder system becomes brighter. I band measurements are near the inflection wavelength so there is no clear advantage for the introduction of a third parameter adjustment, but by the mid-IR there is a clear need for a color adjustment. The issue of a third parameter has been debated in the literature. It is generally couched as a surface brightness or morphological dependence (Rubin et al. 1985; Theureau et al. 1998; Masters et al. 2006) but surface brightness and morphology are closely correlated with color. The use of morphology rather than color as an additional parameter creates unfortunate discontinuities in the distance tool. Also, morphology assignments are subjective and may vary with distance. An alternative, more quantitative stand-in for morphology makes use of an H i flux to near-infrared luminosity ratio (Kashibadze 2008); later types have relatively higher H i flux ratios.

The [3.6] band TFR with the color correction that was derived by Sorce et al. (2013) is shown in the lower panel of Figure 8. The slope template is based on the same 13 clusters, using the same galaxies in all cases where Spitzer photometry is available (213 of the 267 galaxies in the I calibration). The zero point is set by the subset of galaxies with Cepheid PLR or TRGB distances with Spitzer photometry (26 of the 36 galaxies in the I calibration). The calibrations in the two bands use the same line width and inclination values and the same reddening procedures save for the transparency differences with wavelength.

The results of the TFR calibrations are summarized with the following relations:

with 1σ uncertainties of ±0.07 in the zero point and ±0.16 in the slope and magnitudes in the Vega system.

with 1σ uncertainties of ±0.08 in the zero point and ±0.22 in the slope. The pseudo-magnitude $M_{C_{[3.6]}}$ is an absolute magnitude in the [3.6] band where the apparent magnitude after color correction is

The [3.6] magnitudes are in the AB photometric system.

There is a small Malmquist selection bias to distances, which arises for two reasons and requires correction in either band. First, the magnitude cutoffs in the cluster samples were made at constant values in the blue but the cutoffs then slant with line width at redward wavelengths since larger line width galaxies are redder. Second, as a consequence of the exponential cutoff in the galaxy luminosity function, more galaxies are available to scatter brightward than scatter faintward, especially at large distances where most candidates are near the bright limit. A correction b to distance modulus μ is required

where β_I = −0.005 and β_[3.6] = −0.0065.

With either version of the TFR calibration offered here, the magnitude rms scatter is 0.4 so a distance can be measured to a target with an uncertainty of 20%. The scatter is reasonably approximated by a Gaussian in the core but there are non-Gaussian wings, with ∼2% of candidates manifesting scatter greater than 3σ. These are cases that escaped screening during the processing of sample selection, profiles, photometry, and inclinations, so the reasons for deviations are often not clear.

The considerable contribution of archival material is in itself an inchoate sample. With our own observations as a supplement we have focused on five interconnected programs. One of these has already been discussed: the assembly of the data required for a construction of the TFR cluster template and zero point calibrators. A second gave attention to spiral galaxies that have hosted SNIa in order to reinforce the bridge between TFR and SNIa distances. Current results with this program have been published by Courtois & Tully (2012) and Sorce et al. (2012b) and will be given further attention in the next section.

The other three programs involve large all-sky samples designed to improve our knowledge of velocity fields in successive outward steps. The three programs were discussed in some detail by Courtois et al. (2011b) so we only need to summarize here.

The first of the three aims was to achieve close to completion coverage of applicable large spirals within 3000 km s⁻¹. The word "applicable" is in recognition that there are systems too face-on, confused, or disrupted to be considered for TFR analysis. The limit of 3000 km s⁻¹ is somewhat a legacy of very early work (Fisher & Tully 1981) conditioned by the sensitivity of radio telescopes of the day. The issue of sensitivity is still germane. A spiral galaxy within 3000 km s⁻¹ is now easily detected at high S/N wherever it lies in the sky. The 3000 km s⁻¹ limit nicely includes the entire historic Local Supercluster (de Vaucouleurs 1953) and its limit just reaches the next important structures including the Centaurus and Hydra clusters (straddling the 3000 km s⁻¹ boundary but fully included as components of the cluster template). Specifically, we seek to achieve completion with all galaxies within 3300 km s⁻¹ with $M_{K_s} < -21$, i > 45°, type later than Sa, and not confused or distorted. The absolute magnitude cut comes from the observed Two Micron All Sky Survey (2MASS) magnitude (Jarrett et al. 2003) and redshift interpreted through a preliminary model of local flows. This program builds on Cosmicflows-1 that already gave high density coverage of the Local Supercluster. Cosmicflows-2 now contains TFR distance estimates for 1360 galaxies with V_h < 3300 km s⁻¹.

The second extensive sample was built with two principal considerations. One is a recognition of the major structures in the Centaurus–Hydra–Norma (Dressler et al. 1987a) and Perseus–Pisces (Haynes & Giovanelli 1988) regions that lurk just beyond the 3000 km s⁻¹ limit of the nearby sample. We wanted these important structures to be included in the extended sample. The other consideration is the sensitivity of radio telescopes today. If a target enters our preliminary sample we want to expect with high probability that it will be detected with acceptable S/N in H i. For these reasons, our second sample is limited to V_cmb < 6000 km s⁻¹. The candidates are selected from the PSCz (Point Source Catalog redshift) survey of galaxies brighter than 0.6 Jy at 60 μm detected with the Infrared Astronomical Satellite (Saunders et al. 2000). There is an inclination limit, i > 45°, and a color limit, 60–100 μm flux less than unity, the latter to discriminate against starburst systems in favor of normal spirals with emission predominantly from cirrus. With the selection based on far-infrared flux the sample is minimally affected by obscuration (except in the ability to identify sources at extreme low latitudes) so we get good coverage toward the Galactic plane. The Centaurus–Hydra–Norma and Perseus–Pisces regions both flirt with the zone of avoidance. The galaxies that have been observed randomly sample within the selection criteria that have been discussed. Presently we have distance estimates for 1363 galaxies with 3300 < V_h < 6000 km s⁻¹. We have less than satisfactory coverage at δ < −45°, the exclusive domain of Parkes Telescope. A continuing effort is being made to improve this situation.

A third sample extends to greater redshifts in order to investigate the nature of galaxy flows on very large scales. It draws from the Revised Flat Galaxy Catalog (Karachentsev et al. 1999). The thinnest galactic systems are spirals of type Sc with small bulges and well constrained disks. These properties give targets selected from the Flat Galaxies Catalog a homogeneity and amenability to H i detection that is favorable for a sparse all-sky sampling (Kudrya et al. 2009). At present the implementation is instrumentally restricted. Most of the currently observed galaxies in the flat galaxies program with large velocities lie in the declination range of the Arecibo Telescope and have been observed with that facility. GBT has contributed over a wide range of declinations but generally with targets at lower redshifts. Cosmicflows-2 includes 1446 galaxies at V_h > 6000 km s⁻¹ largely drawn from the Flat Galaxy Catalog. Figure 4 gave a histogram that summarizes the redshift distribution of our entire TFR contribution to Cosmicflows-2. The peaks within 3000 km s⁻¹ and 6000 km s⁻¹ are reflections of prominent overdensities in the Local Supercluster, Centaurus–Hydra–Norma, and Perseus–Pisces convolved with the emphasis of our samples. Courtois et al. (2011b) provide a more detailed breakdown of the sample coverage.

The construction of our version of the TFR and the foundation of 36 Cepheid PLR and TRGB calibrators was reviewed in Section 3 and discussed earlier in more detail (Tully & Courtois 2012; Sorce et al. 2013). This material is given the shorthand name CF2. Before moving on to other methods we acknowledge and integrate a second important compendium of TFR distances, the SFI++ compilation (Springob et al. 2007). It is to be appreciated that there is a substantial overlap between SFI++ and the current work in the frequent use of the same raw photometric and H i data. However there are sufficient differences in analysis procedures to make for interesting comparisons. We initially accept SFI++ distances as given by Springob et al. and now check for agreement in zero point and look for any systematics. As an aside, we note that an absolute zero point for the TFR distances by the Cornell group was given by Masters et al. (2006) based on the HST key project scale but with a reduced sample of Cepheid PLR calibrators. Here we accept the SFI++ distances as relative equivalent velocities and ultimately rescale to achieve agreement with other measures.

The SFI++ collection offers distances that are directly measured or distances adjusted for bias. There are several reasons for bias. There is the "selection" Malmquist bias that arises if, among galaxies with the same line width at the same distance, the brighter are given attention but not the fainter. Then there are the "homogeneous" and "inhomogeneous" Malmquist biases that depend on the distribution of galaxies. In the homogeneous case, for galaxies with a given measured distance, there are more cases that arrived there from scatter inward from larger true distances than arrived from scatter outward from smaller true distances. In the inhomogeneous case, the scatter is away from high densities toward low densities. Each of these biases produce kinematic artifacts. Springob et al. (2007) make an adjustment that globally compensates for all of these factors.

We follow a different strategy. In the case of the selection Malmquist effect, the danger is the systematic mismeasurement of distances. In the case of the distribution Malmquist effect, whether homogeneous or inhomogeneous, distance measures can be individually unbiased but care must be taken to avoid bias in the velocity field. Our strategy is to deal with the two distinct Malmquist biases separately. We feel that the issues of distance measurements and velocity field inferences should be kept apart. Our goal is to measure individual distances that, while suffering errors, are unbiased. Therefore we do not choose to give distances that are "wrong" in a way that compensates for suspected kinematic effects. Specifically in dealing with the distribution Malmquist problem, we believe that a proper method is to evaluate distances at observed redshifts, rather than to infer peculiar velocities at measured distances. Consequently we make no adjustments for the distribution Malmquist effects in our reported distances. Regarding the selection Malmquist problem we note that it is possible to null the bias, and if so desired even reverse the sign of it, through the choice of the slope of the TFR. Our procedures to define optimal slopes at our I and [3.6] passbands were discussed in Section 3.4, including a description of small corrections for a residual bias that we find necessary.

Returning to SFI++, we remark that what interests us are their directly measured distances, free of their complex corrections. However, since a bi-variate fit that considers errors in both magnitudes and line widths was used in the determination of their TFR slope, it is to be anticipated that the distances suffer from the selection bias. This expectation is acknowledged by the SFI++ authors and evidence for it is found in the comparison of distance moduli for galaxies in common to our samples shown in Figure 9. After rejection of only five discordant differences (one CF2 value judged to be bad and four SFI++) a least squares linear fit to the differences in 2071 moduli, $\Delta \mu = \mu _{{\rm cf2}} - \mu _{{\rm sfi}}^{100}$, has the form

where V_LS is the velocity of a galaxy in the Local Sheet frame (Tully et al. 2008), μ_cf2 is the CF2 distance modulus with the zero point established by Tully & Courtois (2012), and μ_sfi is the Springob et al. (2007) unadjusted modulus with a nominal zero point consistent with H₀ = 100. After a correction of the form prescribed by Equation (13) is applied to the SFI++ sample the rms agreement between CF2 and SFI++ distance moduli is ±0.22 mag. This scatter is roughly half the scatter found for cluster samples. The agreement affirms that our distinct analysis methodologies produce comparable results and TFR scatter is dominated by factors unrelated to our procedures.

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For each galaxy it is possible to construct its "Hubble parameter," V_LS/d, from its observed velocity and distance. Values for this parameter are shown as a function of velocity for the galaxies with both CF2 and SFI++ distances in the top panel of Figure 10. The distances are averages of CF2 and SFI++. Our own CF2 distances are given double weight since we have a clearer understanding of how they were obtained, including the issue of bias treatment. The horizontal line at H₀ = 74.6 km s⁻¹ Mpc⁻¹ results from a fit to Hubble parameters with V_LS > 4000 km s⁻¹, beyond the domain of obvious velocity perturbations. The zero point scaling comes from Tully & Courtois (2012). This value of H₀ is not our final value; this issue will be reviewed in a later section. Here, attention is drawn to the constancy with a specific value of the Hubble parameter with velocity. The normal indication of a selection Malmquist bias is an increase of the Hubble parameter with redshift (Teerikorpi 1993; Sandage 1994). There is no hint of a residual selection bias in this data.

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SFI++ includes almost 2000 galaxies not found in CF2. It could have been supposed that the selection Malmquist bias correction implied for SFI++ from the roughly 2000 galaxies in common would serve for the SFI++ only sample, but this expectation is not met. If that correction is made then there is a highly significant decrease in the Hubble parameter with increasing velocity with the SFI++ only sample. In the middle panel of Figure 10 we show the run of Hubble parameter with velocity for the SFI++ only sample raw distances with no bias correction. The zero point has been shifted to match the best fit H₀ = 74.6 km s⁻¹ Mpc⁻¹ for V_LS > 4000 km s⁻¹ found in the top panel. It is seen that with no bias correction the run of Hubble parameter with velocity is nicely constant in the mean.

The apparent explanation for this situation derives from the reason why there are a large number of systems in SFI++ that are not in CF2. While the CF2 sample falls off abruptly at V_LS ∼ 12, 000 km s⁻¹, the SFI++ sample extends with significant coverage to almost 30,000 km s⁻¹. Mostly, the difference is caused by the inclusion in SFI++ of the cluster samples of Dale et al. (1999a, 1999b). For these samples, rotation information was derived from optical spectroscopy rather than H i line widths. The merits of the optical material were discussed by Catinella et al. (2005, 2007). Since we have not made any attempt to integrate optical and radio spectroscopic observations, the optical kinematic-based component of the global TFR sample rests solely with SFI++. It is not clear to us why this component of SFI++ does not manifest the selection Malmquist bias but it is probably related to the fact that these galaxies were analyzed in the context of cluster memberships. The identifications with cluster membership will be revisited in a later section.

In the bottom panel of Figure 10 we see the entire CF2 plus SFI++ TFR sample. In addition to the 2071 cases in common to the two sources in red and the 1970 cases added by SFI++ alone in green, there are 2097 sources provided by CF2 alone in blue for a total of 6138 galaxies. All subsamples have been brought to the same zero point scale resulting in a mean Hubble parameter value of 74.6 km s⁻¹ Mpc⁻¹ with an rms scatter of 24.2 km s⁻¹ Mpc⁻¹. After culling, as discussed in Section 5, the combined TFR sample retains 5998 galaxies.

The important SBF sample of Tonry et al. (2001) was already incorporated within Cosmicflows-1. This material is again included with only the small modifications advocated by Blakeslee et al. (2010). What is new with the SBF technique is the contribution made by HST observations which carry the promise of extending the utility of the method from ∼40 Mpc from the ground to ∼100 Mpc from space (Blakeslee 2012). Currently, the contributions from HST are either of a calibration nature or restricted to the nearest two large clusters, Virgo and Fornax (Mei et al. 2007; Blakeslee et al. 2009). These studies provide a valuable constraint on the relative distances of these two key clusters to our dynamics. The Virgo study clarifies the status of the Virgo W' Group which is in the cluster line-of-sight but is 50% more distant. The detailed information will be of great value in subsequent dynamical modeling.

SBF studies are limited to systems with predominantly old populations and this constraint is shared with the FP technique. The elliptical and S0 galaxies that are useful for FP studies are prominently represented in clusters. The accuracy of the FP procedure is greatly enhanced by averaging over cluster members. Consequently for the present compilation we consider contributions from three literature sources that emphasize observations within clusters. These studies go by the acronyms SMAC, EFAR, and ENEARc (Hudson et al. 2001; Colless et al. 2001; Bernardi et al. 2002). There have already been discussions of these samples, briefly in the TFR calibration paper by Tully & Courtois (2012) and more extensively in the SNIa calibration paper by Courtois & Tully (2012). Here we need only review.

Our goal is to ensure a consistency between these three FP sources and then a consistency with the other methodologies at our disposal. The SMAC sample serves as a good intermediary between the FP samples, with good overlap with both ENEARc and EFAR, and as an intermediary with an absolute calibration because of a literature linkage with SBF (Blakeslee et al. 2001). This latter connection to an absolute scale is used to a first approximation, but small modifications are required to match the CF2 system. Regarding the sample overlaps, an inter-comparison of the three FP contributions is summarized in Figure 3 of Courtois & Tully (2012). SMAC and ENEARc consider 19 clusters in common and SMAC and EFAR have 11 in common, with a total of 28 clusters observed by at least two teams. Pairwise modulus correlations in each case are consistent with slope unity with distance. Standard deviations of the fits are 0.05 mag with ENEAR−SMAC and 0.09 mag with EFAR−SMAC. An additional 105 clusters are observed by one team only. FP distances are available from these sources for 1508 separate galaxies.

There are some details to be noted in the FP distances carried in Cosmicflows-2. With both the SMAC and ENEARc samples we gave attention to the selection bias issue. The authors of those studies addressed the problem in their own way. We re-analyzed their data our way, analogous to the "inverse" TFR analysis. Fits to cluster templates were made assuming errors on the velocity dispersion axis only. Overall our results agree with SMAC and ENEAR literature results. Figure 11 shows the agreement in the case of the ENEARc sample. In a comparison between SMAC and ENEAR distances there is a slight reduction in scatter using values from our "inverse" fit analysis rather than the previously published values: the rms scatter between SMAC and ENEAR literature moduli for clusters in common is ±0.222 mag, while with our re-analysis the scatter is ±0.189 mag. The distances that we report for SMAC and ENEAR galaxies are those determined by our new analysis.

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In the case of EFAR we have not attempted a re-analysis. Furthermore, the distances we use from this literature source are group averaged. The clusters contributing to this study are at comparatively large distances and errors for individual galaxies are large. In the Cosmicflows-2 data catalog EFAR entries report distances that do not pertain just to the individual target but rather to their assigned group.

The connection between FP distances and TFR distances is shown in Figure 4 of Courtois & Tully (2012) with 32 clusters in common between the unified FP collection and SFI++ TFR and 11 clusters in common with CF2 TFR. Again, correlations between the moduli are consistent with slope unity with distance. Standard deviations for both SFI++ and CF2 comparisons are 0.04 mag.

While we have argued that TRGB might be the gold standard for distances on small scales, it is evident that SNIa distances have become the gold standard on large scales. The uncertainty in a single SNIa distance estimate is half the uncertainty of a TFR value. A single SNIa measure is worth four TFR measures. Even with relatively small samples there have already been interesting studies of large scale flow fields (Colin et al. 2011; Dai et al. 2011; Turnbull et al. 2012).

There has been tremendous activity in SNIa studies. Observations are being made to relativistic redshifts that test cosmological models. For our purposes we assembled SNIa distance measures from the literature (Courtois & Tully 2012; Sorce et al. 2012b). Our compilation uses UNION2 (Amanullah et al. 2010) as a backbone supplemented by four other studies of SNIa within z = 0.1 (Prieto et al. 2006; Jha et al. 2007; Hicken et al. 2009; Folatelli et al. 2010). The merging of these samples is illustrated in Figure 1 of Courtois & Tully. Our dual purposes in those earlier papers were first to establish a bridge between the TFR and SNIa scales (the I band TFR in Courtois & Tully and the Spitzer [3.6] band in Sorce et al.) and second to use the ensuing SNIa calibration to arrive at a determination of the Hubble constant in a redshift regime where peculiar velocities should be insignificant.

The full compilation of SNIa distances used in those earlier papers continues to help define H₀. However, the CF2 compendium of distances truncates at z = 0.1. Our other methodologies are limited to this regime. Also, it is roughly at this redshift that there is a transition in the nature of SNIa surveys, from an approximation to all-sky for nearby SNIa to narrow angle (usually equatorial) deep imaging for distant SNIa. We presently have a distillation of 306 SNIa distances within 30,000 km s⁻¹ from the five sources identified above. These have all been brought to the common CF2 zero point through comparisons demonstrated in Figure 5 of Courtois & Tully (2012).

Relative distances are adequate for studies of peculiar velocities and the underlying density field but an accurate absolute determination of the Hubble constant constrains the value of the equation of state for dark energy and the number of neutrino species (Riess et al. 2011; Freedman et al. 2012). Our initial construction of the CF2 ladder of distances began with the HST key project scale based primarily on the Cepheid PLR and a distance modulus for the Large Magellanic Cloud of 18.50. The details of this construction are found in the paper describing our calibration of the TFR at I band (Tully & Courtois 2012) and the subsequent paper on the extension of this calibration to SNIa (Courtois & Tully 2012). The current discussion to this point has been based on the absolute scale described in those papers.

The re-calibration of the TFR at the Spitzer [3.6] band (Sorce et al. 2013) and its impact on the SNIa scale (Sorce et al. 2012b) provided an opportunity to re-evaluate issues concerning the absolute scale. Thanks to trigonometric parallax observations with HST of Galactic Cepheids (Benedict et al. 2007), studies of detached eclipsing binaries in the Large Magellanic Cloud (Guinan et al. 1998; Fitzpatrick et al. 2002; Ribas et al. 2002; Pietrzyński et al. 2009b, 2013) and mid-IR observations of Cepheids in the LMC (Freedman et al. 2012), there is increased precision in the distance to the LMC. Here we accept the modulus 18.48 ± 0.03 found by Freedman et al. (2012), which is slightly smaller than the HST key project fiducial distance although consistent within the assigned error. This change would decrease distances by 1%.

The TFR distances that are such an important part of this compilation are all based on photometry at I band. Photometry for large samples based on Spitzer (and WISE) satellite observations in the mid-IR are products for a future release of Cosmicflows. In advance of that, there are several advantages with the mid-IR photometry that can already be incorporated. The most important advantage is the 1% consistency of satellite photometry across the sky. The I band photometry has been acquired by many observers at a multitude of telescopes with different detectors and subtly different filters, north and south, over many observing seasons, and diverse not always well documented observing conditions. The mid-IR photometry has other advantages. Reddening essentially disappears as a concern. The background (zodiacal light and distant galaxies) is at such a low level that almost all the light from a target is recorded in a short exposure. The flux at 3.6 μm is dominated by old stars, presumed proxies for the mass. Because of these advantages we trust the absolute calibration of the TFR at [3.6] more than we trust the I band calibration. It was determined that there is a small systematic difference between the [3.6] and I band scales (Sorce et al. 2012b, 2013) though less than the error estimates. With the HST key project LMC reference, distances are increased by 2%. In combination with the downward revision in the LMC distance, we end up with an increase in the CF2 scale by a factor 1.009, or 1%.

Accordingly, we make small adjustments as follows. Distances that are directly based on Cepheid PLR observations are decreased in the modulus by 0.02 mag for consistency with the revised LMC distance; these include the Cepheid PLR measures themselves and the SBF measures. Distances based on a Population II calibration (TRGB, RR Lyrae, horizontal branch) or geometric considerations (eclipsing binary, maser) are unmodified. Distances on large scales tied to our TFR calibration (TFR, FP, SNIa) are increased by 0.02 in the modulus to reflect the preferred mid-IR scale re-calibration.

The entire sample is a mix of distance measures with uneven quality. Where possible, group assignments and subsequent averaging considerably reduce both distance and velocity errors. Uncertainty estimates are summarized in Figure 15 for the 5224 distinct entities after grouping. The top panel gives a histogram of the uncertainty estimates. The distribution is strongly bimodal. One peak at distance errors 8%–10% results from the input of precision estimators such as Cepheid PLR, TRGB, and SNIa and from groups with many contributions (the entities with the lowest error estimates are either in the proximity of the Local Group or, such as the Virgo Cluster, with many contributions—in such cases systematic errors dominate these statistical error estimates). The other peak in the bimodal error distribution is at 20%, the error assigned to individual TFR distances. In the figure, this contribution goes far off scale. The lower panel illustrates the distribution of the error estimates with systemic velocity.

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As is well known, if percentage errors are roughly a constant as a function of distance with a given methodology then errors in assigned peculiar velocity grow linearly with distance, rapidly becoming much larger than intrinsic peculiar velocities. The problem is aggravated by an asymmetry in the error-induced peculiar velocities. The errors are symmetric in measurements of distance modulus, a logarithmic quantity. However there is a resultant skewness in distance errors. As a thought experiment, consider two galaxies that are intrinsically at 100 Mpc and have no peculiar velocity with H₀ = 75 when they are observed at 7500 km s⁻¹. Suppose that rather than observed to be at their correct distance modulus of 35.0, one has a distance modulus error of 2σ with a TFR measurement that brings it forward to 34.2, while the other has a comparable error that takes it back to 35.8. The corresponding distances are 69 and 145 Mpc and the implied peculiar velocities (V_pec = V_mod − H₀d) are +2311 and −3338 km s⁻¹, respectively. It follows that there is a skewness in peculiar velocities, with the tail to negative velocities more extensive than the tail to positive velocities.

This phenomenon is observed in the peculiar velocities inferred from CF2 distance measurements. The effect can already be seen in the lower panel of Figure 13 but is more clearly seen with the log N scale used in the histogram of Figure 16. Here, the black outer histogram is built with the entire grouped sample assuming H₀ = 74.4 km s⁻¹ Mpc⁻¹ while the inner green histogram reflects only the subsample with distance error estimates ⩽14%.

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That the skewness toward negative velocities evident in both the full sample and subsample are due to the error bias is demonstrated by the good matches provided by the dashed histograms. These secondary histograms are generated from a mock file that duplicates the systemic velocities and error estimates of the real data. Distances are assigned to the mock galaxies by giving them random deviations from the Hubble value, drawing from a Gaussian distribution in the modulus, with the Gaussian dispersion given by the error estimate in the particular case.

The mock experiment guides an adjustment that effectively removes the error bias to peculiar velocities. Positive peculiar velocities are untouched but negative peculiar velocities are shrunk by a small factor controlled by the fractional distance uncertainty e_d as described by the following equation:

The adjusted velocity V_adj approaches 0.77V_pec if the product e_dV_mod is large, and approaches V_pec if that product is small. The adjustment is justified by the reality that errors strongly dominate the peculiar velocity signal in the regime of significant modifications. In the tables to be discussed, both raw and adjusted peculiar velocities are made available.

The increase in the amplitude of measured peculiar velocities with redshift because of errors is evident in the two panels of Figure 17, with the lower panel simply displaying an enlargement of the crowded region in the top panel. The black points in these displays show the individual peculiar velocities as a function of systemic velocity for the full grouped sample while the green points emphasize the subsample with fractional errors ⩽0.14. The blue and red squares with error bars (some errors too small to be seen) are averages in systemic velocity bins. The effect of distance error bias is easily seen in the peculiar velocity asymmetries of the guide lines in this figure. The solid red lines show the peculiar velocity loci with distance modulus errors ±0.8 mag, a 2σ error with a TFR measurement. In the figure, the black and green points have been plotted after applying the distance error adjustment to negative peculiar velocities, with the consequence that the bias toward larger amplitude negative velocities is removed. It can be seen in Figure 17 that extreme peculiar velocities are clipped. In all, 138 galaxy distance measurements are rejected either because they have greater than 3σ excursions from a group mean or because of extreme Hubble parameter excursions. In two-thirds of these cases the reason for a bad distance measurement was evident on close inspection, attributable to a bad inclination, strange morphology, confusion, or interaction with a neighbor. The number of rejections is 1.7% of the total sample.

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The Cosmicflows-2 data are made available in two tables: one providing an entry for every galaxy with a distance and the other condensed to an entry for each separate group, including groups of one galaxy. Table 1 is the complete catalog, made available in its entirety online, and also available at EDD, with versions that might be updated. The following is a description of the 8315 current entries in Table 1. All the columns are available in the online table but, for clarity, the following columns are suppressed in the abbreviated presentation: Columns 13–21, 24–26, 36–41, and 44–46.

Column 1: Principal Galaxies Catalog (PGC, LEDA) number.

Columns 2–4: distance, distance modulus, and fractional distance error for the galaxy. Weighted average values are given if there are multiple sources.

Columns 5–12: codes indicating source of distance: C = Cepheid PLR; T = TRGB from this program; L = TRGB from literature; M = miscellaneous (RR Lyr, horizontal branch, eclipsing binary, maser); S = SBF; N = SNIa; H = TFR; F = FP.

Columns 13–18: coordinates, successively celestial (J2000), Galactic, and supergalactic.

Column 19: morphological type in the RC3 numeric code.

Column 20: B band reddening from Schlegel et al. (1998).

Columns 21 and 22: magnitudes at B and K_s bands from RC3 and 2MASS respectively.

Columns 23–27: velocities in the successive reference frames: helio, Galactic, Local Sheet, CMB, and CMB adjusted for cosmological effects with Ω_m = 0.27 and Ω_Λ = 0.73.

Column 28: common name.

The following columns pertain to the group associated with the individual galaxy.

Column 29: bookkeeping group number; preferred group identification.

Column 30: 2M++ group ID (Lavaux & Hudson 2011); alternate group identification.

Column 31: number of galaxies with measured distances in group.

Columns 32–34: weighted average distance, distance modulus, and fractional distance error of group.

Column 35: number of galaxies with known positions and velocities in group.

Columns 36–39: Galactic and supergalactic coordinates of group.

Column 40: mean morphological type of group members.

Columns 41 and 42: summed B and K_s magnitudes for group.

Columns 43–47: mean group velocity; respectively helio, Galactic, Local Sheet, CMB, and CMB adjusted for cosmological effects with Ω_m = 0.27 and Ω_Λ = 0.73.

Column 48: rms group velocity dispersion.

Columns 49 and 50: PGC ID of brightest member and alternate names for group or cluster.

Table 2 contains 5224 group entries, including 4690 of these as singles. The column entries track entries 28–47 in Table 1 and in addition include three more velocity related parameters. The following information is included. All the columns are available in the online table but, for clarity, the following columns are suppressed in the abbreviated presentation: Columns 8–11, 14–15, and 22.

Column 1: number of galaxies with measured distances in group.

Columns 2–4: weighted average distance, distance modulus, and fractional distance error of group.

Column 5: number of galaxies with known positions and velocities in group.

Columns 6–9: Galactic and supergalactic coordinates of group.

Column 10: mean morphological type of group members.

Columns 11 and 12: summed B and K_s magnitudes for group.

Columns 13–16: mean group velocity; respectively helio, Galactic, Local Sheet, CMB.

Column 17: V_mod, group velocity with adjustment for cosmological model (Ω_m = 0.27, flat topology) as given by Equation (14).

Column 18: rms group velocity dispersion.

Columns 19 and 20: value in Column 19 is peculiar velocity =V_mod − H₀d assuming H₀ = 74.4 km s⁻¹ Mpc⁻¹. Value in Column 20 is peculiar velocity adjusted for the distance measurement error bias (only differs from value in previous column if negative).

Column 21: bookkeeping group number; preferred group identification.

Column 22: 2M++ group ID (Lavaux & Hudson 2011); alternate group identification.

Columns 23 and 24: PGC number identifies brightest galaxy in group. Alternate names for group or cluster.