Cosmicflows-4

Conventionally in the past, TFR samples have been acquired by individual targeting of selected candidates for both the photometric (imaging) and kinematic (linewidth) required components. Nowadays, wide-field optical, infrared, and H i radio surveys provide access to much larger samples. Specifically, here, we make use of serendipitous SDSS DR12 u,g,r,i,z optical imaging (Alam et al. 2015), WISE W1 and W2 infrared imaging (Wright et al. 2010; Mainzer et al. 2011), and Arecibo Legacy Fast ALFA Survey (ALFALFA) neutral hydrogen spectral detections (Haynes et al. 2018), supplemented in the radio with pointed observations with the Green Bank Telescope and Parkes Telescope (Dupuy et al. 2021).

In a series of three papers, we explored the properties of the optical and infrared photometric material particularly pertaining to issues of extinction (Kourkchi et al. 2019), then provided optical and infrared calibrations of the TFR based on ∼600 galaxies in 20 clusters (Kourkchi et al. 2020a) and used the calibrations to derived TFR distances for ∼10,000 galaxies (Kourkchi et al. 2020b). It was subsequently revealed that the distances in the latter publication are affected by a bias: a trend in Hubble parameter values, H_i = f_i cz_i /d_i , as functions of apparent magnitude. ¹⁵ The bias strongly affects distance estimates to intrinsically fainter galaxies within 4000 km s⁻¹ and arises from faint end curvature in the TFR. Distances to galaxies with velocities greater than 4000 km s⁻¹ are mildly affected.

Much, and even a majority, of baryonic mass in faint galaxies is in the form of interstellar gas. It has been noted that adding this constituent to the stellar component represented by optical or infrared light, formulating the baryonic Tully–Fisher relation (BTFR), effectively linearizes the relation between the logarithms of baryonic mass and H i profile linewidth (McGaugh et al. 2000; McGaugh 2005; Lelli et al. 2016, 2019). Consequently, and in response to our concern regarding the bias with faint galaxies in the TFR study by Kourkchi et al. (2020b), the same sample has been reanalyzed with the BTFR methodology by Kourkchi et al. (2022).

The BTFR requires the additional component of H i fluxes, an observable acquired simultaneously with linewidths. The need for a robust H i detection gives focus to the condition that the sample is H i flux limited: photometry for any target with sufficient H i flux is easily obtained. However, a H i flux limit translates to a cut in gas mass that increases with distance. This trend results in a bias that must be addressed in order to obtain distance measures of value. Kourkchi et al. (2022) developed a procedure that was demonstrated with mock data to provide unbiased distance estimates.

Application of the BTFR also requires the translation of luminosities into approximations of stellar mass, involving color terms. The conversions are linear, but uncertainties are further compounded by the summation of stellar and gas mass components. On the one hand, there is a greater complexity with the BTFR, but on the other hand, the linkage with the dark matter-dominated total mass is expected to be tighter. Scatter as evaluated from H_i values at V > 4000 km s⁻¹ is 22% in distances, comparable to that with the TFR. Kourkchi et al. (2022) provide BTFR distance estimates for 9967 galaxies.

Cosmicflows-4 assembles TFR distances to 12,412 galaxies, the largest, most coherent compilation to date by this methodology. The most important contribution (9967 galaxies) is the new BTFR sample discussed by Kourkchi et al. (2022, hereafter cf4). This new sample is compared and merged with five TFR samples: the assembly of 5980 cases in Cosmicflows-2 that itself is broken into a part (4069) derived within our collaboration (hereafter cf2) and a part (3957) emanating from the SFI++ study (Springob et al. 2007, hereafter sfi), 2251 galaxies discussed in Cosmicflows-3 incorporating photometry from Spitzer Space Telescope images (spitzer) (Sorce et al. 2014), 1715 galaxies utilizing 2MASS photometry (2mtf) (Hong et al. 2019), and 551 extreme edge-on galaxies (Makarov et al. 2018) (flat).

Figure 1 is a histogram of the run of velocities for the TF subsamples and the full TF sample. The distribution of the combined TF sample is displayed in supergalactic coordinates in Figure 2. The bands of high object density crossing the two supergalactic hemispheres lie in the 0 < δ < 38 decl. zone accessed by the Arecibo Telescope.

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Our analysis began with each sample alone. At systemic velocities above 4000 km s⁻¹ cosmic expansion velocities are expected to overwhelmingly dominate deviant velocities. Hence, a necessary (not sufficient) criterion a sample should satisfy is approximate constancy in the Hubble parameter for individual galaxies, H_i = f_icz_i /d_i , averaged in velocity bins. The results of this test for all but the most recent flat galaxy sample were presented in Kourkchi et al. (2020b). A significant drift toward smaller 〈H_i 〉 (larger derived distances) was evident in the 2mtf sample. An adjustment to negate this trend was introduced by Kourkchi et al. (2020b), see Section 5, and is incorporated in the current work. The flat sample passes the 〈H_i 〉 constancy test. Note that absolute 〈H_i 〉 values are not an issue at this stage; they can be (and are) different for each sample.

This test of the constancy of H_i with redshift provides as a side product an evaluation of the rms dispersion in measurements within each sample. The measured values include dispersion in velocities and intrinsic dispersion but these components are unimportant if, as we do, we restrict our attention to velocities greater than 4000 km s⁻¹. We find the following characteristic rms dispersions for seven samples (treating separately the optical and infrared components of cf4 (whence, cf4-op and cf4-ir), the two components of Cosmicflows-2, cf2 and sfi, and the components of spitzer with and without color corrections (spitzer-cc and spitzer-nc). The rms dispersions for the BTFR cf4-ir and cf4-op are 0.45 and 0.47 mag, respectively, while for the two-parameter TF studies dispersions are 0.40 for all three, cf2, sfi, and spitzer-cc, 0.50 for both spitzer-nc and 2mtf, and 0.55 for flat. Distance values for 2mtf at Local Group frame velocities less than 2000 km s⁻¹, as evaluated by the Hubble parameter test and the test to be discussed next, are systematically too low and we reject all those 2mtf measurements.

Another test of the samples applied by Kourkchi et al. (2020b) was to focus on differences in distance moduli between cf4 and an alternate sample: 〈μ_cf4 − μ_alt〉 where alt is any of the other samples (now extended to include flat). This test is particularly useful for the isolation of egregiously bad distance values in one of the samples. Much less than 1% of cases in cf2, sfi, spitzer, and cf4 are rejected by this test. With 2mtf ∼2%, and with flat ∼5%, are rejected.

We now turn our attention to the integration of these samples into a global maximally consistent compilation of TFR distances.

Our goal is to combine the distinct TFR subsamples into a single TFR sample. Before this integration, each subsample has its own zero-point scaling. Here, we revise the zero-points of subsamples to achieve statistical equality between them. We stress that we do not make relative changes in moduli within a subsample in this process. Doing so would subvert the utility provided by multiple subsamples in reducing systematics.

It is evident that TFR samples have non-Gaussian outliers. Steps have been described to remove strongly deviant cases but our initial integration of subsamples reveals additional instances. Applying a 3.5σ rejection criterion caught 275 cases among 22,233 measures (1.2%) where there would be nine with a normal distribution. These outliers are removed.

Next, we want to profit from the advantages of averaging over groups discussed in Section 2. We begin by weighted averaging of the distance moduli of all galaxies within a 1PGC group within a single subsample. Individual weights are formed from the inverse square of rms uncertainties. This average and associated weight is one object in the ensuing analysis. Accordingly, each subsample is reduced to a quantity of objects (halos, groups, clusters) composed of from one to many individual galaxies, each identified by a 1PGC number.

Ultimately, we want to merge all samples by all methodologies into a coherent set with a zero-point established by geometric distance measurements. At this stage, it is sufficient to bring all TFR subsamples onto a common scale. The baseline TFR scale will be set by our new cf4 subsample that should lie close to our final scale, given its linkage to Cepheid and TRGB measures as discussed by Kourkchi et al. (2020a, 2020b, 2022).

Here, we pursue our goal to find the global modulus offset of each sample, "s", from that of cf4, where "s" stands for any of the samples we introduced earlier in this chapter (cf2, sfi, spitzer, 2mtf, flat). We will adjust the reported distance moduli within each sample, ${\mathrm{DM}}_{\mathrm{in}}^{(s)}$, following ${\mathrm{DM}}^{(s)}={\mathrm{DM}}_{\mathrm{in}}^{(s)}+{\rm{\Delta }}{\mu }_{s}$ in order to set all cataloged distances on the same scale. By our convention, Δμ_cf4 = 0. We treat these adjusting values as a set of free parameters that are optimized together in a Bayesian framework. The best offset parameters minimize the total deviation of adjusted object distance moduli (groups and individual galaxies) from the weighted distance modulus averages offered by all samples together.

Our objective is to find the posterior probability distribution ${ \mathcal P }({\rm{\Theta }}| { \mathcal D })$, with Θ being the vector of all moduli offsets, (Δμ_s1, Δμ_s2, ...). ${ \mathcal D }$ holds the original cataloged distance moduli, ${\mathrm{DM}}_{\mathrm{in}}^{(s)}$. According to conditional probability theory, ${ \mathcal P }({\rm{\Theta }}| { \mathcal D })\propto { \mathcal P }({ \mathcal D }| {\rm{\Theta }}){ \mathcal P }({\rm{\Theta }})$. Having no prior knowledge about the distribution of the moduli offsets implies ${ \mathcal P }({\rm{\Theta }})=1$ and subsequently ${ \mathcal P }({\rm{\Theta }}| { \mathcal D })\propto { \mathcal P }({ \mathcal D }| {\rm{\Theta }})$, where the right-hand side is the likelihood function, ${ \mathcal L }$. We assume that all measured object distances are independent with Gaussian uncertainties. Therefore, for each object, n, the likelihood function is the multiplication of a set of independent probabilities given as

$$\begin{eqnarray}&&{{ \mathcal L }}_{n}=\displaystyle \prod _{\mathrm{All}{\rm{ \mbox{\unicode{x00060}\unicode{x00060}} }}s{\rm{\mbox{''}}}}\displaystyle \frac{1}{\sqrt{2\pi {\sigma }_{n,s}^{2}}}\,\exp \displaystyle \frac{-1}{2}{\left(\displaystyle \frac{{\mathrm{DM}}_{n}^{(s)}-\langle \mathrm{DM}{\rangle }_{n}}{{\sigma }_{n,s}}\right)}^{2},\end{eqnarray} \tag{ 1 }$$

iterating over all distance catalogs. 〈DM〉_n is the weighted average distance modulus of the nth object that is derived from the adjusted distance moduli, ${\mathrm{DM}}_{n}^{(s)}$ is the distance modulus of the object in the sample "s", and ${\sigma }_{n,s}^{2}$ is the variance of ${\mathrm{DM}}_{n}^{(s)}-\langle \mathrm{DM}{\rangle }_{n}$, which is determined by adding the uncertainties of the associated parameters in quadrature. Likewise, the total likelihood function for all objects is ${{ \mathcal L }}_{\mathrm{tot}}={\prod }_{n=1}^{N}{{ \mathcal L }}_{n}$, where N is the total number of objects (groups and individuals). It is simpler to work with the logarithm of the likelihood function, which is expressed as $\mathrm{log}{{ \mathcal L }}_{\mathrm{tot}}=-{\sum }_{n=1}^{N}{\chi }_{n}^{2}/2$, where

$$\begin{eqnarray}&&{\chi }_{n}^{2}={\left(\displaystyle \frac{{\mathrm{DM}}_{n}^{(s)}-\langle \mathrm{DM}{\rangle }_{n}}{{\sigma }_{n,s}}\right)}^{2}\,.\end{eqnarray} \tag{ 2 }$$

Adopting a flat prior distribution for the moduli offsets leaves us with a χ² minimization problem. We are interested in a set of moduli offsets that minimizes ${\chi }_{\mathrm{tot}}^{2}={\sum }_{n=1}^{N}{\chi }_{n}^{2}$.

To sample the posterior distribution, ${ \mathcal P }({\rm{\Theta }}| { \mathcal D })$, we use the Python package emcee (Foreman-Mackey et al. 2013), which implements Markov chain Monte Carlo (MCMC) simulations to explore the parameter space. Starting from our likelihood function, we generate 128 chains each with a length of 10,000. We remove the first 1000 steps, which are conservatively chosen to ensure that the remaining steps adhere to Markov chain statistics. Figure 3 illustrates corner plots for the resulting posterior distribution of Δμ_s . The topmost panel of each column shows the one-dimensional distribution of the corresponding sampled parameter, overlaid with the median values (red solid line) and the lower/upper bounds corresponding to 16/84 percentiles (black dashed line). Horizontal and vertical red lines in the two-dimensional distributions exhibit the location of the median values that are adopted as the optimum moduli offsets of the corresponding catalogs with respect to cf4.

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The variance for a given subsample that is recorded in Figure 3 depends on both the uncertainties in individual measurements and the number of intersections with other subsamples. The individual uncertainties between the alternate TF subsamples are only modestly different, so it is the numbers of intersections that dominate.

While the 6dFGSv sample was originally included in Cosmicflows-3, in this latest work we provide a new recalibration of this sample based on the findings of Qin et al. (2018) designed to explore and remove spurious flows. In total, the 6dFGSv sample subtends the entire δ < 0° sky, except for regions with galactic latitude ∣b∣ < 10°. Its 8885 objects incorporate many of the brightest early-type galaxies in 6dFGS (Jones et al. 2009), nominally selected to have a spectral signal-to-noise ratio > 5, total J-band magnitude <13.65, redshift cz < 16,500 km s⁻¹, and velocity dispersion greater than 112 km s⁻¹ (Campbell et al. 2014). Further refinements to this selection include visual classification and removal of galaxies based on their morphological type (although as demonstrated in Tully et al. 2016 not all remaining galaxies are classified as ellipticals) and removal of objects with undesirable spectral features or poor spectral template fits (Campbell et al. 2014). Photometry for the FP sample was obtained by cross-matching with the 2MASS survey (Jarrett et al. 2000; Skrutskie et al.2006).

Springob et al. (2014) produce peculiar velocity measurements with this sample by modeling the logarithmic difference in observed and cosmological distances to each galaxy as a function of the logarithmic difference between their observed effective radii and the effective radii predicted from the best-fit FP. For each galaxy, one can compute the probability of it having a particular distance modulus by assuming a Gaussian probability distribution function (PDF) about the FP. However, this procedure is complicated by Malmquist bias and the selection function of the 6dFGSv data, particularly the magnitude limit. The presence of a magnitude limit cuts a slice through the FP, such that the PDF is no longer normalized. Because the magnitude limit is in apparent magnitudes, the portion of the FP that cannot be observed varies with distance, and so the normalization of the PDF for each galaxy also depends on distance. To counteract this effect, Springob et al. (2014) produced a calculation of this normalization as a function of distance using simulations drawn from the best-fit 6dFGSv FP with a J < 13.65 limit.

In subsequent work by Qin et al. (2018), similar simulations reproducing the FP, selection function, and methodology applied to the 6dFGSv data were used to measure the bulk flow. A significant offset between the measured and true bulk flows in the simulations was identified in the direction directly toward the southern celestial pole. It was found to be possible to remove this effect in the simulations (and subsequently the data) by recalculating the normalization of the probability distribution for each galaxy using an ad hoc, brighter, magnitude limit of J < 13.217. In this work we repeat this calculation, paying special attention to not only the bulk flow, but also the Hubble parameter in radial shells.

We start by generating 128 mock 6dFGSv surveys matching the methods in Magoulas et al. (2012) and Qin et al. (2019), which are then run through a reconstruction of the 6dFGSv pipeline using a magnitude limit of J < 13.65. Unlike previous works, the normalization of the PDF for each galaxy is computed using numerical Monte Carlo integration of the truncated 3D Gaussian PDF, rather than summing over simulations. This procedure was found to result in less noise and was far more reliable for computing an accurate value at large comoving distances where by design the number of galaxies available to compute the probability, even over 128 simulations, quickly falls to zero.

The bulk flow in each simulation was then computed using the η-maximum likelihood estimator (MLE; Kaiser 1988; Qin et al. 2018), as was the weighted mean value of

$$\begin{eqnarray}&&{\mathrm{log}}_{10}({H}_{i})={\mathrm{log}}_{10}\left(\displaystyle \frac{{f}_{i}{{cz}}_{i}}{{d}_{i}}\right).\end{eqnarray} \tag{ 3 }$$

in redshift bins, where ${f}_{i}=1+1/2[1-{q}_{0}]{z}_{i}-1/6[1\,-{q}_{0}-3{q}_{0}^{2}+{j}_{0}]{z}_{i}^{2}$, z_i is the redshift of the galaxy, q₀ and j₀ are the acceleration and jerk parameters, and c is the speed of light. Here, d_i is the luminosity distance to each galaxy, computed assuming H₀ = 75 km s⁻¹ Mpc⁻¹. For comparison, the same quantities are computed for each simulation using the true luminosity distance and peculiar velocity of each simulated galaxy.

The results of this procedure are shown in Figures 4 and 5. Also plotted are the results for the original 6dFGSv data. From the binned Hubble parameters, it is clear that the mocks with distance moduli computed using the 6dFGSv pipeline exhibit an outflow and do not lie on the expected H₀ = 75 km s⁻¹ Mpc⁻¹ line. The distribution of these same mocks matches the data, which leads us to conclude the same is likely true for the data too. This trend is not obvious without the presence of simulations (and so not highlighted previously), particularly because cosmic variance at cz_cmb < 5000 km s⁻¹ seems to be scattering the observed Hubble parameters high in the data.

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The effect on the bulk flow is particularly pronounced. It was found by Qin et al. (2018) that the measured bulk flow in the simulations is biased quite negatively in the direction of the southern celestial pole compared to the bulk flow known to exist in the simulations. Although the true bulk flow in the 6dFGSv is not known and can only be estimated, the measured value is consistent with the biased mock results, leading us to conclude that the data is similarly biased.

As with the previous work by Qin et al. (2018), we correct for this problem by modifying the magnitude limit used in the Malmquist bias correction/normalization of each galaxy's PDF. By iterating, a magnitude limit of J < 13.38 was found to produce binned Hubble parameters that are flat with redshift while also substantially reducing the difference between the adjusted and measured bulk flows. The results of applying this limit to the mocks and data are shown in Figures 4 and 5. The small differences between the optimal value found here and that used in Qin et al. (2018) are likely the result of the more rigorous calculation of the normalization using numerical integration adopted in this work.

We believe this recalibration to be robust and so use the updated 6dFGSv data in Cosmicflows-4. The source of the discrepancy between the magnitude limit used to construct the simulations and the optimal value found for the Malmquist bias correction is unclear, but indicates a discrepancy between the best-fit 6dFGSv FP (from which the simulation apparent magnitudes are derived) and the assumed magnitude limit of the data. It is not inconceivable that the true magnitude limit of the 6dFGSv data is in reality brighter than the nominal selection function, particularly in light of the other aspects of the sample selection required to go from the 2MASS photometry and 6dFGS spectra to the FP sample, and then again to the peculiar velocity sample. However, there may be more to the picture. A preferable solution would be to perform a joint fit for the FP parameters and peculiar velocities simultaneously, again validated against simulations; however, such an analysis is beyond the scope of the current work.

The systematic problem as a function of morphological type identified by Tully et al. (2016) remains in the revised bias-adjusted 6dFGSv distances. Candidates in the 6dFGSv compilation are given a morphology M description, with M = 0 for ellipticals, M = 2 for lenticulars, and M = 4 for spirals. As can be seen in Figure 6, there is a clear drift in 〈f_iV_i /d_i 〉 with M, where V_i and d_i are individual galaxy velocities and distances and f_i is defined in association with Equation (3). The drift is in the sense that d_i values are increasingly measured too low with increasing M.

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This situation is not too surprising given that the FP pertains to galaxy bulges. The admixture of disk contributions evidently creates a systematic. The revisions discussed above to 6dFGSv distances has not addressed this morphology-related problem. Adjustments are made to distances that statistically counter the trend shown in Figure 6. Only the fitting parameters are changed from the adjustments made in Tully et al. (2016). The distance moduli accepted into Cosmicflows-4 incorporate both the revised bias corrections and the morphological corrections described in this section.

A new addition to the Cosmicflows program is the SDSS peculiar velocity catalog (Howlett et al. 2022). Containing a total of 34,059 objects, it is currently the largest single source of extragalactic distance measurements available. It also contributes, along with a modest number of supernova observations, the most distant objects in the Cosmicflows-4 catalog, extending up to a maximum cosmic microwave background (CMB)-frame redshift of z_CMB = 0.1. These measurements were made by combining public photometric data from SDSS Data Release 14 (Abolfathi et al. 2018) over a contiguous ∼7000 deg² area with existing spectroscopic Hα and velocity dispersion measurements (Thomas et al. 2013).

The full catalog is magnitude limited to 10.0 ≤ m_r ≤ 17.0 and velocity dispersion limited to σ > 70 km s⁻¹. The selection of the sample also included a number of additional cuts on the surface brightness profiles, concentration index, color, axial ratio, Hα equivalent width, and visual morphology to ensure a clean sample of elliptical galaxies is retained. As part of this process, cuts were applied using the morphological M type (Tempel et al. 2011) before fitting of the FP, so no correction of the form seen in Figure 6 for the 6dFGSv sample is required for the new SDSS data.

After data cuts, the sample was first fit using the 3D Gaussian FP model. Then, for each galaxy, the PDF of the logarithmic distance ratio between cosmological and observed comoving distances was obtained. The normalization of this PDF is set by integrating over the portion of the FP within which each galaxy would be observable at a given cosmological distance, and hence encodes the selection bias. Larger proposed distances for each galaxy end up being up-weighted to account for the increased cosmological volume in which a galaxy could be found, and the lower sample completeness at those distances. Previous work (i.e., Springob et al. 2014) computed this normalization using Monte Carlo simulations, which requires assuming the same normalization for each galaxy. However, Howlett et al. (2022) demonstrated it can also be computed numerically, which makes the calculation much faster and allows one to use a different normalization for each object. The mean, standard deviation, and skew of each PDF were used to describe the full PDF of each galaxy.

Using cosmological simulations of the SDSS data and selection function, Howlett et al. (2022) demonstrated the measurements were unbiased as a function of redshift and absolute magnitude. However, they did identify a bias between the distance and group richness. This bias arises from a correlation between the mean surface brightness of galaxies in the sample and their corresponding group richness, possibly hinting at intrinsic correlations in the FP due to a fourth unmodelled parameter such as stellar age. This group properties bias was corrected in the catalog by fitting separate FPs to subsamples of the data as a function of group richness. These corrected distance measurements are the ones used in this work.

Calibration of the zero-point in the SDSS PV catalog is made difficult due to the small sky area. In Howlett et al. (2022) this calibration was performed by cross-matching to groups containing galaxies with Cosmicflows-3 distances. The use of groups was found to be important considering the group richness bias discussed above, although consistency with individual measurements in the two catalogs was also demonstrated as long as the uncorrected SDSS distances were used.

Just as multiple acquisitions of a distance to a spiral galaxy with photometry and rotation curve information are highly correlated, so it is the case with multiple applications of the FP to early-type galaxies. Hence, as an initial step we bind all the FP subsamples into a joint sample. The same methodology as discussed in Section 3.4 is adopted to combine the distances of the FP catalogs of this study: SDSS, 6dFGSv, smac, enear, and efar.

As noted in the introduction of Section 4, there is only the tiniest overlap of 41 cases between 6dFGSv, which is entirely restricted to negative declinations, and SDSS, which is limited to the north galactic and celestial pole caps, dipping slightly into the celestial south to δ = −37. Consequently, the smac, enear and efar samples play the important roles of providing scaling links between the two large samples. Specifically, smac provides 118 and 90 links with SDSS and 6dFGSv, respectively, enear provides 75 and 41, respectively, and efar provides 28 and 41, respectively. Collectively, these three smaller samples provide 367 overlaps with SDSS and 200 overlaps with 6dFGSv.

The 3.5σ rejection criterion is used to identify two outliers in enear, one in smac, and 17 in the 6dFGSv subsamples. These cases are removed prior to the subsequent calculations.

With the interlacing of FP samples, the SDSS distances are taken as the reference of comparisons, whence Δμ_SDSS = 0 by our convention. The posterior distribution of the offset parameters (Δμ_s ) are explored with our MCMC procedure in a similar fashion to that explained in Section 3.4. Figure 7 illustrates the posterior distributions of the moduli offsets of 6dFGSv, smac, enear, and efar subsamples from SDSS. It is to be appreciated that while the zero-point of the SDSS sample was set to be in approximate agreement with Cosmicflows-4 distances (Howlett et al. 2022), it remains to receive further adjustment in the coupling with the full complement of methodologies described further along. In particular, the union between methodologies afforded by group memberships will be exploited.

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A histogram of the distribution of all FP measurements with velocity is shown in Figure 8. The 6dFGSv and SDSS contributions are comparable at V_cmb < 16,000 km s⁻¹. Only SDSS information is available between 16,000 km s⁻¹ and the sample cutoff at z = 0.1. Figure 9 shows the distribution of the galaxies with FP measurements on the sky, split into two panels for those above and below 16,000 km s⁻¹.

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We have gathered SBF distances for 508 galaxies based on five sources of SBF measurements. Two of these sources already found a presence in the earlier Cosmicflows compilations. Tonry et al. (2001) pioneered the methodology with observations from the ground at the optical I band. That program targeted 300 E/S0 galaxies including essentially all E galaxies within 2000 km s⁻¹ and a sampling out to 4000 km s⁻¹. The other earlier source of SBF distances was the HST study of E/S0 galaxies in the Virgo and Fornax clusters (Mei et al. 2007; Blakeslee et al. 2009, 2010). The exceptional spatial resolution of HST resulted in distance measurements with sufficient accuracy to resolve the three-dimensional structure of the Virgo Cluster, distinguishing the M, W, and ${W}^{{\prime} }$ background galaxy groups, and to provide an accurate differential distance between the Virgo and Fornax clusters.

Now with Cosmicflows-4 we add SBF distance information from two new sources. The first of these (Cantiello et al. 2018a) is an extension of the HST Virgo Cluster study derived from the u^⋆,g,i,z imaging of the Next Generation Virgo Cluster Survey carried out with the Canada–France–Hawaii Telescope (Ferrarese et al. 2012). Our knowledge of the Virgo Cluster environs, and particularly, of the separation of the ${W}^{{\prime} }$ group 50% farther away in projection and the W and M structures twice as faraway, are given improved clarity with this enhanced distance compilation.

The second new SBF source, infrared SBF with the HST WFC3/IR camera, is a harbinger of a particularly promising improvement to the SBF methodology (Jensen et al. 2021). The wide field of view and improved sensitivity of WFC3/IR over earlier ground-based and NICMOS IR observations makes it an optimal instrument for SBF measurements. Red giant stars are particularly bright in the near-IR, making IR SBF much brighter and easier to measure than at optical wavelengths, especially from space where the sky background is greatly reduced. Jensen et al. (2015) established the calibration for SBF in the F110W and F160W filters using observations of 16 Virgo and Fornax galaxies. As discussed in Blakeslee et al. (2021), these IR SBF distances are tied to the I-band SBF observations (Blakeslee et al. 2009), the Cepheid distances to Virgo and Fornax, and the LMC distance modulus of 18.477 (Pietrzyński et al. 2019).

The HST F110W SBF observations reach a distance of 80 Mpc in a single orbit, with an observational error of 4%–5% (Blakeslee et al. 2021; Jensen et al. 2021). A number of different WFC3/IR F110W programs have now been mined for usable observations. Two of these programs targeted galaxies specifically for SBF distance measurements. The program MASSIVE seeks to understand the stellar dynamical properties and central black hole masses of massive galaxies within ∼100 Mpc (Ma et al. 2014). This study includes the determination of 41 high-quality IR SBF distances. A second targeted SBF study measured distances to 19 early-type type Ia supernova (SN Ia) host galaxies (P. Milne et al. 2022, in preparation; Garnavich et al. 2022). These data sets, along with independent observations of NGC 4874 in Coma (Cho et al. 2016; Bartier et al. 2017) and NGC 4993, the 2017 gravitational wave event host (Cantiello et al. 2018b), sum to a total of 62 galaxies out to 100 Mpc for which high-precision distances are now known.

There is significant overlap between only three of the four SBF subsamples. The new IR SBF contribution has an overlap of only eight targets with Tonry. The corner plot for the three SBF subsamples that can be compared can be seen in Figure 10. The Virgo/Fornax collection based on HST optical band observations is taken as the reference. Whereas with the TF and FP studies galaxy-group comparisons help reduce errors, in the case of SBF the galaxies are relatively nearby and the individual accuracies are greater so comparisons are strictly galaxy−galaxy. The integration of the full four SBF contributions awaits the global MCMC compilation of all methodologies.

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Figure 11 is a histogram of the cumulative SBF sample dependence on systemic velocity, with a breakdown by source.

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SNe Ia arise in binary systems where at least one of the stars is a white dwarf. The path, or possibly paths, resulting in explosions is not resolved, with the two most discussed alternatives being, alternatively, single-degenerate models where matter from a companion accretes onto a white dwarf, or double degenerate models where two white dwarfs merge (Maoz et al. 2014). In spite of this uncertainty in the physical mechanism, the minimal dispersion in the absolute luminosities of SNe Ia after a calibration based on post-maximum decline rate (Phillips 1993) can be exploited to measure galaxy distances with high accuracy. The events are sufficiently bright that SN Ia at z < 0.1 can be discovered and monitored with moderate-sized telescopes. This methodology can provide distances with 2–3 times the accuracy of TF and FP with considerably greater reach. SNe are serendipitous events, though, and the number of well-studied occurrences remains small.

Modest samples of SN Ia were included in earlier releases: 308 cases in Cosmicflows-2, increased to 389 in Cosmicflows-3. Here, those samples are augmented by contributions from four new sources, more than doubling the available sample to 1008 SN Ia events within z = 0.1. Core-collapse SNe II are proving to be useful distance tools also, although not with the same accuracy. A limited sample of SNe II will be discussed in Section 6.2.

The new samples beyond those of Cosmicflows-3 include 235 hosts from the Lick Observatory Supernova Search (Ganeshalingam et al. 2013), 137 hosts from the first release of the Carnegie Supernova Project (Burns et al. 2018), 669 hosts from the collection by Stahl et al. (2021), 597 hosts in the PantheonPlus compilation (Brout et al. 2022; Scolnic et al. 2022), the 560 late-type hosts in the largely overlapping and augmented Pantheon+SH0ES sample (Riess et al. 2022), 134 hosts of SN Ia with spectral twin properties by the Nearby Supernova Factory (Boone et al. 2021), and 89 hosts of events studied at infrared bands (Avelino et al. 2019).

There are considerable overlaps between these and the earlier contributions, as is evident if the contributions are summed. Diverse light curve fitters are involved. Selection cuts differ between programs and analyses could involve different photometry in different passbands but it is reassuring that alternate distance measurements to the same events are in agreement. For 200 pairwise comparisons of multiple distance entries to the same SN Ia the weighted average difference of 0.063 mag in the modulus was found, about 3% in distance. The agreement is considerably less good in the distance estimates to multiple SN Ia in the same host. Twenty-one hosts have measurements for at least two SN Ia events, with three such events in NGC 3417 and four in NGC 1316 in the Fornax Cluster. To avoid issues related to differences in analysis procedures and zero-point scaling we consider just the 13 hosts with multiple events in the SH0ES study (Riess et al. 2022). The rms scatter of 0.278 mag corresponds to an uncertainty of 13.6% in distance. This large scatter is puzzling, especially since peculiar velocities are removed as a factor with events in a common host. By comparison, Avelino et al. (2019) and Boone et al. (2021) contend that the scatter they find with conventional light-curve fitters at optical bands correspond to ∼7% in distance, with improvements to 5% with the inclusion of infrared photometry or spectral matching (twins embedding), respectively.

A histogram of the distributions in velocities of the SN Ia subsamples is shown in Figure 12. The 15 SN Ia samples are merged in a manner analogous to that discussed in Section 3.4. The corner plot of moduli offsets following from the MCMC analysis is seen in Figure 13. The zero-point scaling is set to that provided by the SH0ES sample although, as with all the individual methodologies, the absolute scaling is subject to revision with the merging of all available material. Figure 14 illustrates the distribution of SN Ia on the sky along with the SN II and SBF contributions. Undersampling in the celestial south accounts for the underrepresentation of events at supergalactic longitudes 180°–270° to the left of the zone of obscuration in the plot. Concentrations of SBF contributions can be seen at the positions of the Virgo and Fornax clusters, at the right central and lower left, respectively.

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Although SNe II vary intrinsically by more than two magnitudes, the underlying events are better understood than is the case with SNe Ia and their luminosities correlate with observable properties. More luminous SNe II have higher photospheric expansion velocities (Hamuy & Pinto 2002) and are bluer. Colors are a monitor of extinction. In Cosmicflows-4 we include a sample of 96 SNe II in 94 hosts studied by de Jaeger et al. (2020a, 2020b). The 1σ scatter about the Hubble diagram is ∼15% in distance, about a factor 2 worse than realized with SNe Ia.

Our procedure to achieve the union of the five methodologies, TF, FP, SBF, SN Ia, and SN II, is conceptually the same as that described in Section 3.4 with application to each of the individual methodologies. The fundamental difference is that, whereas with the individual methodologies the bonding involves matching between galaxies, now the linkage units are galaxy groups. So, for example, the 42,254 galaxies with FP distances are bundled into 27,701 groups. The distance to a group is the weighted average of the moduli of all the FP measures in the group, with a statistical error given by the inverse square root of the sum of the weights. ¹⁶ The velocity of the group is found by averaging over all known group members (not just those with distance estimates). In the case of TF, the 12,395 individual galaxies with distance estimates are bundled into 10,189 groups. With SBF, 425 individual galaxies lie in 227 groups. The 1008 individual SN Ia hosts are found in 945 groups while the 94 SN II hosts are all in 94 separate groups.

As with the previous MCMC analyses of individual methodologies, one of the contributions is selected as a zero-point reference. Here, we take the grouped SN Ia sample as that reference. This choice is impelled by three factors. For one, the SN Ia sample provides the best cross-reference overlaps with the numerically dominant FP and TF components. Also, there is utility in the greater accuracy of SN Ia distances. Finally, it is most useful that the SN Ia are widely dispersed and have great reach in distance. At the onset of the joint analysis, each methodology had been set on its own arbitrary zero-point scale. For example, the SN Ia scale is consistent internally with H₀ ∼ 71 km s⁻¹ Mpc⁻¹, the TF scale is consistent with H₀ ∼ 75 km s⁻¹ Mpc⁻¹, and FP, SBF, and SN II are intermediate. Now, all the methods will be brought to the same, but still arbitrary, zero-point scaling within statistical uncertainties.

The corner plot resulting from the MCMC merging of methodologies through joint group affiliations is shown in Figure 16. The constraints on the unions with FP and TF are robust because the intersections are numerous and can involve groups with many distance measurements. The constraints with SBF are reasonably good although intersections are limited because individual SBF measures have high accuracy. The constraints are weakest in the connection with the currently limited SN II methodology.

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The five methodologies that reach substantial distances have been merged to give coherent distances but on an arbitrary scale. It is through a matching of sources with TRGB or CPLR measurements that the ensemble is put on an absolute scale, which manifests in an estimate of the Hubble constant. Each of TRGB and CPLR are themselves anchored by other geometric techniques. Both take grounding in the nuclear maser distance of 7.576 Mpc to NGC 4258, which has a quoted uncertainty of 1.5% (Reid et al. 2019). A separate foundation for CPLR is the quoted 1% eclipsing binary distance of 49.59 kpc to the LMC (Pietrzyński et al. 2019), the host galaxy for the largest coherent sample of Cepheid variables. Both CPLR and TRGB take recourse in parallax measurements of appropriate stellar populations within the Milky Way. In the case of CPLR, eight measurements come from HST WFC3 spatial scanning and 75 parallaxes are available from Gaia EDR3 (Riess et al. 2018, 2021). In the case of TRGB, the parallax targets are abundant Population II stars that provide specification of the HB (Carretta et al. 2000). Rizzi et al. (2007) identified the levels of the HB in five Local Group galaxies observed with HST (Fornax, Sculptor, and NGC 185 dwarfs, IC 1613 and M33) to determine color-dependent values for the absolute magnitude of the TRGB.

The Rizzi et al. (2007) TRGB scale can be compared with TRGB on the scale in Freedman et al. (2019). For 12 galaxies, there is the difference 〈μ_rizzi−μ_freedman〉 = −0.028 ± 0.035 with rms scatter ±0.052 (Anand et al. 2022). This 1.5% difference in distances is not statistically significant but the Rizzi scale distances closer are consistent with the Anand et al. (2022) determination of a Hubble constant value 0.9% higher than that by Freedman et al. (2019).

There are indications that the Population II zero-point scaling will be altered somewhat as Gaia satellite Data Release 3 (DR3) geometric parallaxes replace older Hipparcos parallaxes. The following presents a best-effort estimate of the Gaia informed rescaling in advance of access to Gaia DR3. We begin by recalling that the basis for the Rizzi et al. (2007) calibration was the following formula for the luminosities of stars on the HB derived from Hipparcos satellite trigonometric parallaxes (Carretta et al. 2000)

$$\begin{eqnarray}&&{M}_{V}(\mathrm{HB})=0.13([\mathrm{Fe}/{\rm{H}}]+1.5)+0.54\end{eqnarray} \tag{ 4 }$$

The same reference gave a related formula for the luminosities of the RR Lyrae stars (RR) that lie embedded in the HB.

$$\begin{eqnarray}&&{M}_{V}(\mathrm{RR})=0.18([\mathrm{Fe}/{\rm{H}}]+1.5)+0.5\end{eqnarray} \tag{ 5 }$$

Applied to the LMC, this reference finds μ_LMC = 18.54 ± 0.03 ± 0.06 (statistical and systematic errors, respectively). An updated version of Equation (5) based on parallaxes from the early release of Gaia DR3 (eDR3) is (Garofalo et al. 2022)

$$\begin{eqnarray}&&{M}_{V}(\mathrm{RR})=0.33([\mathrm{Fe}/{\rm{H}}]+1.5)+0.63\end{eqnarray} \tag{ 6 }$$

and these authors find a distance to the LMC of μ_LMC =18.501 ± 0.018. The change in the magnitude coefficient at fiducial [Fe/H] = −1.5 implies RR Lyrae stars fainter by 0.06 mag (although note the greater metallicity dependence). The application to the LMC would have the LMC closer by 0.04 mag.

Consider another study. Nagarajan et al. (2022) derive new luminosity–metallicity and period–luminosity–metallicity relations with luminosities in Wesenheit magnitude units to negate reddening effects for 36 RR Lyrae in the Milky Way with eDR3 parallaxes and for RR Lyrae in 39 nearby dwarf galaxies, making a simultaneous Bayesian fit to all Milky Way anchors and all RR Lyrae stars in each dwarf. They find an offset for the Milky Way calibration stars from values obtained by Carretta et al. (2000) ${\mu }_{\mathrm{gaia}}-{\mu }_{\mathrm{hipparcos}}$ = −0.048 (without an error). The subsequent distances obtained for the nearby dwarfs (Nagarajan et al. 2022) can be compared with our TRGB distance estimates with the Rizzi scaling (Anand et al. 2021). In two of the cases, NGC 147 and NGC 185 with the RR Lyrae information from Monelli et al. (2017), the modulus discordances are almost 0.4 mag and the reported RR Lyrae distances do not make sense given the quality of the TRGB values. Discounting these two cases, eight available comparisons give 〈μ_rizzi − μ_rrl〉 = 0.085 ± 0.034 with rms scatter ±0.095.

Rather than bootstraps to TRGB luminosities through RR Lyrae or the HB, there is the prospect of direct measurements of the luminosities of the brightest stars on the red giant branch, including dependencies on metallicity, from Gaia parallax measurements of large numbers of stars within the Milky Way. A preliminary example is provided by the study by Soltis et al. (2021) of the determination of the TRGB magnitude for the ω Centaurus globular cluster with Gaia eDR3 parallaxes. Their determination of M_I,trgb = −3.97 ± 0.06 is lower by ∼0.08 than the Rizzi et al. (2007) expectation.

A common thread can be seen through the discussion in this section. Distances following from the Hipparcos-based Rizzi et al. (2007) calibration are too great, although the amplitude of the problem is uncertain. From Garofalo et al. (2022), the revised scaling of RR Lyrae magnitudes and corresponding distance to the LMC suggests a modification between 0.04 and 0.06 mag. From Nagarajan et al. (2022), changes in distances to Milky Way RR Lyrae and nearby dwarfs with observed RR Lyrae indicate a modification between 0.048 and 0.085. From Soltis et al. (2021), there is the single direct TRGB measurement giving the modification 0.08. In no individual case is the justification for a modification compelling, yet the hints are strong. Gaia DR3 observations are expected to provide clarity. Under the circumstances and as a provisional solution, we will force agreement of our TRGB measurement for the galaxy NGC 4258 (Anand et al. 2022) to match the maser distance to that galaxy (Reid et al. 2019) rounded to the nearest 100th of a magnitude, a zero-point shift to smaller distances with respect to Rizzi et al. (2007) of 0.05 mag. The weakly color-dependent TRGB calibration becomes

$$\begin{eqnarray}&&{M}_{I,\mathrm{trgb}}=-4.00+0.22[(V-I)-1.6].\end{eqnarray} \tag{ 7 }$$

The uncertainty in the lead coefficient is dominated by systematics between Hipparcos and Gaia parallaxes that remain to be resolved, but an estimate ±0.03 brackets the range of shifts discussed above. The color term is also subject to review. As opposed to the linear ramp with color incorporated in Equation (7), published alternatives are a constancy with color on the premise that the measurement is made of the most metal-poor of TRGB stars (Freedman et al. 2019) or a curved refinement of our linear relation (Jang & Lee 2017).

Direct comparisons can be made between CPLR distance moduli and TRGB moduli with the revised zero-point calibration using Gaia information. In the case of eight galaxies (including NGC 4258) with TRGB estimates (Anand et al. 2021) and CPLR estimates from the SH0ES Collaboration (Riess et al. 2022) there is an average difference in distance modulus 〈μ_trgb−μ_cplr〉 = 0.028 ± 0.038 with rms scatter ±0.107. There are 10 galaxies in common between the TRGB distances of Anand et al. (2021) and CPLR distances of Freedman et al. (2001) (assuming LMC is at μ_LMC = 18.477). In two cases (M66 and NGC 5253), the CPLR moduli are greater than TRGB by ∼0.2 mag. It can be seen by eye with the TRGB fits displayed in the Extragalactic Distance Database (Anand et al. 2021) that a deviation this large is not supported by the TRGB measurements. Considering the other eight cases, there is an average difference in modulus of 〈μ_trgb − μ_cplr〉 = 0.019 ± 0.025 with rms scatter ±0.069. Combining the Riess et al. (2022) and Freedman et al. (2001) CPLR results, providing 16 comparisons, we find 〈μ_trgb−μ_cplr〉 =0.023 ± 0.022 with rms scatter ±0.087. There is agreement between the TRGB and CPLR observations in common to within 1% with an rms scatter including errors in both inputs of 4%.

Going forward, then, our cumulative sample of zero-point calibrators will consist of 489 galaxies with TRGB distance estimates (Anand et al. 2021) shifted closer by 0.05 mag in the modulus, 76 galaxies with CPLR distances on the scale of Riess et al. (2022), and six maser distances (Pesce et al. 2020) including the important case of NGC 4258 (Reid et al. 2019). Accounting for overlaps, 542 zero-point calibrator galaxies are involved. In the union of all methodologies, the distance moduli of these calibrators in the ensemble will not shift.

The absolute distance scale calibration is established by evaluating linkages between individual calibrators and targets within the five methods ensemble discussed in Section 10.1. We find 128 individual galaxies among the TRGB, CPLR, and maser calibrators with matches in the combined FP, TF, SBF, SN Ia, and SN II samples. After rejection of three outliers (PGCs 5896, 44982, and 68535) and four Local Group galaxies, the remaining 121 matches are linked through an MCMC chain as summarized in the corner plot of Figure 17. It follows that the arbitrary five source scale that was set with SN Ia requires the revision

$$\begin{eqnarray}&&{\mu }_{\mathrm{zp}}={\mu }_{5\mathrm{so}}-0.138\pm 0.022\,\mathrm{mag}.\end{eqnarray} \tag{ 8 }$$

There is significant variation in the coefficient of the revision if the five sources of inputs are considered separately. Individually (with respect to the scale of Figure 13) μ_zp−μ_xxx equals −0.182 ± 0.057 (TF: 81 cases), −0.053 ± 0.022 (SN Ia: 41 cases), −0.223 ± 0.069 (SBF: 29 cases), and −0.142 ± 0.128 (SN II: eight cases). There are only three overlaps with FP. The result with SN Ia alone is anomalous. The anomaly is reduced somewhat in a comparison between groups containing calibrators and SN Ia: offset −0.064 ± 0.022 mag with 44 SN Ia. The difference from the five source value given in Equation (8) would have a 3.5% effect on the Hubble constant (ΔH₀ = −2.5 km s⁻¹ Mpc⁻¹). This difference in the establishment of an absolute scale calibration between referencing SN Ia alone versus the coupling of SN Ia to other methodologies is essentially identical to that found in the construction of Cosmicflows-3 (Tully et al. 2016), see the discussion involving Figure 12 in that paper. Going forward, we retain the more robust zero-point scaling given by Equation (8) but we recognize the plausibility of systematic errors at least as large as the ambiguity raised by the SN Ia anomaly.

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Our results support the contention that the apparent differences in the cosmic expansion velocity today between direct measurements of distances and velocities and expectations of the standard Lambda cold dark matter model from conditions in the early universe are real, but the case is not yet compelling. In our analysis the numbers of measurements are large so statistical uncertainties are very small. However, the possibilities of systematic errors are a concern.

Issues can be separated between the establishment of the absolute distance scale from local observations and possible systematics in measurements well beyond the calibrators. Consider first the local problem. There is the expectation that, within a year or two, precision parallaxes provided by the Gaia experiment will provide precise intrinsic luminosities for the kinds of stars important for distance work: Cepheids, RR Lyrae, and horizontal branch stars, and stars at the TRGB. The samples should be numerous enough to unravel metallicity and age effects. Present uncertainties in the absolute scaling are at the level of several percent. Surely it is not at the 10% level required to reconcile the Hubble constant controversy. In any event, we can expect that this problem will soon be resolved at the 1% level.

Presently, the FP and TF methodologies provide the large samples required for depth and wide field coverage of the sky. However, with only 20%–25% accuracy per target, uncertainties at redshifts 0.05c–0.1c are 3000–7500 km s⁻¹. Systematics that can arise from selection biases at the level of only a few percent can have several hundred kilometers per second effects. If a sample is free of systematics it should pass our tests of Hubble parameter constancy with redshift or other parameters, but such tests are not a guarantee against problems.

Going forward, prospects for building a large all-sky sample at substantial redshifts look best with SN Ia. Surveillance at high cadence with moderate aperture telescopes suffice to detect the ∼10³ SNe Ia within z = 0.1 that erupt each year. With samples of many thousand, with photometry at multiple passbands and accompanying spectra, it should be possible to empirically calibrate variations in the properties of the explosion events, providing relative distances accurate to 5%. Vulnerability to systematics should be reduced roughly linearly with the improved statistical accuracy.

What remains is the coupling between the near-field calibration and the far-field mapping. The approach of coupling between Cepheids and SNe Ia may be reaching its limits. Proximate SNe Ia are few and the requisite monitoring required to characterize distant Cepheids will challenge resources. Prospects are better with TRGB, bright in the infrared and accessible to 40 Mpc or more with a single exposure sequence per target with JWST. TRGB targets will couple equally to SN Ia and SBF hosts. JWST will facilitate the SBF methodology with 5% distance accuracy out to z ∼ 0.07. An all Population II TRGB−SBF path to galaxy distances will complement the Population I CPLR−SN Ia path and the availability of two independent methodologies covering the same distance range will provide a good test of systematics.

The distribution of all the entities in the group catalog is shown in the two orthogonal projections of Figure 21. The coverage is dense and relatively uniform around the sky within ∼8000 km s⁻¹ except for the Milky Way avoidance zone at SGY ∼ 0. The two FP samples provide coverage in two distinct fans: 6dFGSv out to 16,000 km s⁻¹ in the celestial south and SDSS out to 30,000 km s⁻¹ in the north galactic and celestial polar cap. There is only spotty coverage in the sector in the northern sky south of the Galaxy at velocities greater than 8000 km s⁻¹ where the range of TF coverage falls off (see the histogram in Figure 1).

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The most dramatic feature seen in these projections contains as a component the Sloan Great Wall (Vogeley et al. 2004; Gott et al. 2005) at SGY ∼ 20,000 km s⁻¹. The Center for Astrophysics Great Wall (de Lapparent et al. 1986) is seen as SGY ∼ 6000 km s⁻¹.

While there is vast terrain to explore opened up by the full SDSS FP sample, uncertainties in individual distance measurements are extreme in the outer regions, and there will continue to be particular interest in the volume of reasonably uniform coverage. As can be seen in Figure 22, the ensemble sample is reasonably dispersed within 16,000 km s⁻¹ except for the falloff beyond ∼8000 km s⁻¹ in the celestial north outside the boundaries of the SDSS study, at positive SGX and positive SGZ. There is the usual caveat regarding the zone of obscuration.

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Figures 23 and 24 expand views to the inner 8000 km s⁻¹ volume where the TF coverage is of principal importance and individual peculiar velocities are tractable. Figure 24 shows 5000 km s⁻¹ slices in SGY on the north and south sides of the galactic plane in the direction orthogonal to the views of Figures 21–23. The complex networks of filaments await detailed morphological and dynamic studies.

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