THREE-DIMENSIONAL VELOCITY AND DENSITY RECONSTRUCTIONS OF THE LOCAL UNIVERSE WITH COSMICFLOWS-1

In Tully et al. (2008), we published a catalog of distances that will be hereafter identified as "Cosmicflows-1."⁵ The catalog contains distances for 1797 galaxies within 3300 km s⁻¹, drawing from four different methods: the Cepheid period–luminosity relation, tip of the red giant branch and surface brightness fluctuation luminosity indicators, and the galaxy luminosity–rotation rate or Tully–Fisher correlation. The first three methods give distances with accuracies of 10%, but in the case of the first two only to very local galaxies, and in the case of the third only to early types that are preferentially located in high-density regions. The Tully–Fisher method, invoking the correlation between the neutral H i gas rotation rate and the integrated magnitude of a spiral galaxy, has an rms accuracy of 20% with measurement in a single band but the lesser accuracy per measurement is offset by the availability of large samples widely distributed over the sky and in distance.

Peculiar velocities (V_pec in km s⁻¹) and corresponding errors are computed from distances (d in Mpc) and their errors assuming a value for the Hubble constant H₀ = 74 km s⁻¹ Mpc⁻¹ (Tully et al. 2008):

$$\begin{equation} V_{{\rm pec}}=V_{{\rm CMB}}-d* H_{0} \hfill \,\, (\rm km\ s^{-1}) \end{equation} \tag{ 1 }$$

$$\begin{equation} \Delta V_{{\rm pec}}=-\Delta\, d* H_{0} \hfill \,\,(\rm km\ s^{-1}). \end{equation} \tag{ 2 }$$

The distribution of rms percent errors in distance is shown in Figure 1. The peak at 20% results from single measures with the Tully–Fisher method. Distance measures with uncertainties larger than 20% are not accepted as sufficiently accurate and thus are not used in this catalog. As seen in Figure 1, the median rms error in both distance and peculiar velocity is 13%.

Zoom In Zoom Out Reset image size

It is to be appreciated that the Cosmicflows-1 catalog is not a complete sample. Incompleteness does not affect the reconstruction of the three-dimensional velocity field as long as the underlying individual distances are not subject to biases in the distance measures.⁶ The observed peculiar velocity field can be sparse in an inhomogeneous way without leading to deviant results in the reconstruction as it will be demonstrated later in this paper. Figure 2 shows the distribution of galaxies with distance measures in the Cosmicflows-1 catalog in supergalactic coordinates SGL, SGB. The velocity distribution of the sample is shown in Figure 1.

Zoom In Zoom Out Reset image size

The general features of the LSS within 8000 km s⁻¹ are illustrated in Figure 2. This figure is a representation of the 30,124 galaxies in the redshift catalog drawn from the literature that we call "V8K."⁷ The 1797 data points with measured peculiar velocities within 3000 km s⁻¹ are included in this larger catalog. The WF is sensitive to structure on scales larger than the domain of the distance measurements. The V8K catalog is used in this paper to compare the observed and derived densities from small to large scales.⁸

The catalog of distances is a work in progress. Data relevant to the measurement of distances are being accumulated in the Extragalactic Distance Database, and an extended version of this catalog "Cosmicflows-2" is in preparation. Observations are being extended to ∼15, 000 km s⁻¹ in a program of H i observations and photometry that we are calling "Cosmic Flows" (Courtois et al. 2011b). H i observations are being made in the context of the Cosmic Flows Large Program with the 100 m Green Bank Telescope at the National Radio Astronomy Observatory and complementary southern observations with the Parkes Telescope in Australia. H i profiles for more than 13,000 galaxies have been measured consistently by the same pipeline (Courtois et al. 2009). The photometry for the program is being acquired from the University of Hawaii 2.2 m Telescope (Courtois et al. 2011a), the multiband survey Pan-STARRS in the north and, it is anticipated, SKYMAPPER in the south.

Peculiar velocity surveys constitute a very noisy and sparse data set diluted across the whole sky. Nonetheless, the peculiar velocities directly sample the gravitational field induced by the LSS of the universe and therefore can be used to uncover the underlying density and velocity field. The WF algorithm provides the optimal tool for the reconstruction of the LSS from peculiar velocity surveys. The general WF framework has been thoroughly reviewed in Zaroubi et al. (1995) and Hoffman (2009). The application of the WF to peculiar velocity observations was first presented in Zaroubi et al. (1999).

In the standard model of cosmology the LSS emerges out of a primordial random Gaussian perturbation field via gravitational instabilities. In the linear regime of deviations from the homogeneous and isotropic Friedmann expansion the density and velocity fields retain their Gaussian nature. The WF constitutes the optimal linear minimal variance estimator given a general data set and an assumed prior cosmological model. It can be shown that in the case of an underlying Gaussian random field the WF coincides with the optimal Bayesian estimator of the field given the data (Zaroubi et al. 1995).

The linearity assumption implies that the density and velocity fields are reconstructed as if the linear regime is valid today on all scales. However, the present epoch LSS is nonlinear, with a smooth transition from the very large scales where the linear theory prevails down to small scales that are governed by nonlinear dynamics. The scale of transition from the linear to the nonlinear regime depends on the nature of the field under consideration. The scale of nonlinearity of the velocity field is roughly a few Mpc and for the density field it is about a factor of 10 larger. It follows that the WF algorithm provides an excellent tool for the reconstruction of the present epoch velocity field.

A final comment on a property of the WF reconstruction is due here. The WF constitutes a conservative estimator which is designed to find the optimal balance between the observational data and the prior model one is assuming in constructing the WF (Zaroubi et al. 1995). In regimes where the data are dense and accurate, the WF is dominated by the data while in regimes where the data are very noisy or very sparse the WF recovers the prediction of the prior model. In the standard model of cosmology the expected mean density departures and velocity fields are zero. In observational surveys of peculiar velocities the statistical error of individual measurements increases linearly with distance and the data sparseness increases with the distance. The degradation in the quality of the data with the distance implies that the WF reconstruction is expected to attenuate the reconstructed fields toward the null fields.

The WF presents the conditional mean field given the data and the assumed prior model. The variation of the actual field around the WF mean field can be sampled by performing CRs of the underlying Gaussian density field (Bertschinger 1987; Hoffman & Ribak 1991; Zaroubi et al. 1999). The scatter of the CRs provides a measure of the statistical uncertainties of the WF reconstruction.

An interesting question given attention by modelers of the velocity field concerns the identification of the sources causing the motion. What cosmological objects such as clusters, superclusters, and voids are affecting the velocity field within a given region of space?

For a given reconstruction of the three-dimensional velocity field in a given region of radius R, the velocity field can be decomposed into two components, one that is induced by the mass distribution within the volume and one induced by the outside mass. This decomposition can easily be done in the linear regime (Hoffman et al. 2001) where the velocity field is proportional to the density field and therefore the velocity and density fields are related by a Poisson-like equation:

$$\begin{equation} \nabla \cdot {\bf v}= - H_0 f(\Omega _m, \Omega _\Lambda) \delta. \end{equation} \tag{ 3 }$$

Given a density field within a given volume and assuming boundary conditions on that volume, the Poisson-like equation can be solved and the velocity field can be calculated. This amounts to the particular solution, to which one can always add the homogenous solution of the source-free equation, namely the Laplace equation. The consequence is the separation of the velocity field inside a given volume into its local component, namely the one induced by the internal mass distribution, and the tidal component (Hoffman et al. 2001). It is to be noted that we used the term "tidal" to represent the full velocity field that is induced by the external mass distribution, including the dipole term (i.e., the bulk motion of the volume due to external causes at any depth).

The WF reconstructs the density and velocity fields on a grid that spans some computational box. Alternatively, one can use the WF velocity field and calculate the linear density field by taking the divergence of the velocity field. Suppose that the velocity field is to be separated into the component induced by the density within a sphere of radius R, enclosed within the box, and the residual of that component. This separation is done by solving Equation (3) for a density field set to the WF field within R and taken to be zero outside R. In practice, the operation is carried out using a fast Fourier transform solver and imposing zero-padding outside the desired volume. The resulting field is the divergent or local component, and the residual of the full WF velocity field after vector subtraction of the local component constitutes the tidal field induced by the mass distribution beyond the sphere of radius R. Visual inspection of the local field often suffices to infer the existence of "great attractors" that lie outside the sphere. A multipole decomposition of the tidal field provides a more quantitative characterization of the tidal field.

Peculiar velocity data sets consist of a finite number of data points taken within a finite volume, the data zone. Peculiar velocities have a much larger correlation length than the underlying density field. It follows from the latter fact that the WF reconstruction can be applied on a volume larger than the data zone. The extent and quality of the reconstruction depends on the assumed prior model, i.e., the power spectrum, and the quality of the data: the magnitude of the errors and the sparseness and the space coverage of the data. In the following, the WF reconstruction is to be conducted within two boxes of size L = 64 and 160 h⁻¹ Mpc, centered on the origin of the supergalactic coordinate system.

A detailed analysis of the tests conducted with mock catalogs is available in the accompanying paper (Hoffman et al. 2012). We give an overview here. A mock velocity catalog has been drawn from a constrained simulation of the local universe. The simulation was performed within the CLUES project,⁹ specifically the case discussed by Cuesta et al. (2011). The simulation is contained within a box of L = 160 h⁻¹ Mpc, hereafter BOX160, assuming the WMAP3 cosmological parameters. This model is a revision of the Klypin et al. (2003) constrained simulation, with a different realization of the random field and more modern estimation of the cosmological parameters. A visualization of the simulation and some discussion of the structural properties is presented in Cuesta et al. (2011). The simulation reproduces the main players of the local cosmography. In particular, it recovers a representation of the Local Supercluster, the Virgo and Coma clusters, the Great Attractor, and the Perseus-Pisces Supercluster. It also reproduces an environment like the neighborhood of the Local Group (LG) and its cold Hubble flow. The simulation provides a very good dynamical replication of the local universe and thereby constitutes a good "laboratory" for tests of our tools of analysis.

This is a dark matter only simulation. A hierarchical friends-of-friends halo finder was applied to the simulation and a halo catalog has been constructed. A mock galaxy catalog has been constructed using the algorithm of the conditional luminosity function (van den Bosch et al. 2007). An "observer" has been placed at the position of the LG and the peculiar velocity of 1797 galaxies has been measured, assuming a Poisson distribution of errors centered around 13%. The selection of the galaxies is done randomly in a sphere of R = 30 h⁻¹ Mpc centered on the LG in order to mimic the observed redshift distribution.

The upper right panel of Figure 3 shows the supergalactic plane of the constrained BOX160 simulation. The density and velocity fields are Gaussian smoothed with R_g = 1 h⁻¹ Mpc. At such high resolution the density field lies well within the nonlinear regime. The WF reconstruction is applied to the simulation with alternative smoothing lengths of R_g = 5 and 10 h⁻¹ Mpc. A visual inspection of the maps reveals that the reconstruction recovers the overall LSS of the box. The comparison in Table 1 shows that the reconstruction is best with the larger Gaussian smoothing and is excellent within the 30 h⁻¹ Mpc annulus of the data zone. The extrapolation at larger distances recovers the gross features of the LSS but at a degraded quality. It follows that the WF can be used to provide a rough expectation of the LSS in unobserved regions.

Zoom In Zoom Out Reset image size

A linear regression of the N-body and WF reconstructed fields is given in Table 1. The table also provides the mean, standard deviation, skewness, and kurtosis of the residual of the WF from the N-body fields.

The series of panels within Figure 4 illustrate the broad features of the WF reconstruction within the radius containing distance measurement constraints; the data zone. The contours of colors illustrate the WF reconstructions based on the observed velocity field. The left panels display linear density and the right panels display radial peculiar velocities. In the left panels, the individual points that are plotted are galaxies drawn from the V8K catalog that lie within the specific distance shells. The pronounced correlation of galaxies with the high-density WF contours provides a visual demonstration that the WF methodology can recover the three-dimensional density distribution of the local universe from a sole knowledge of peculiar radial velocities. It is to be noted here that the V8K galaxies are a rough proxy for the actual spatial distribution of nearby galaxies. Galaxies are plotted based on their redshifts, assumed to be rough estimators of their distances. The WF reconstructed radial velocity field is shown in the right panels.

Zoom In Zoom Out Reset image size

Figure 5 provides orthogonal views of slices of the WF reconstruction. The panels on the left illustrate densities as contours and velocities as vectors. Model velocities are black and on a grid and observed radial velocities are in color (red: outward; blue: inward). The panels on the right show densities again but now with color and galaxies within the slices from the V8K redshift catalog are superimposed. With the top two rows the slices in the normal direction are on the cardinal axes (SGZ = 0 and SGX = 0, respectively), while with the bottom plots the slice is displaced to SGY = 16.9 Mpc to coincide with the main structure of the Local Supercluster which runs through the Virgo Cluster on a line roughly parallel to the SGX axis. This structure is seen top-down in the upper panels and edge-on in the lower panels.

Zoom In Zoom Out Reset image size

The WF reconstruction only provides a description of the linear regime and hence it has a limited power to reconstruct the present epoch structure on small scales. However, the WF is shown to be very good for a recovery of the important large-scale features. Perhaps the most interesting revelation with the current study is the nature of the under dense region that wraps around the Local Supercluster centered on the Virgo Cluster (SGY = 17 Mpc near SGX = SGZ = 0). The larger part at SGZ positive and −20 < SGY < +10 Mpc is the Local Void (Tully & Fisher 1987; Tully et al. 2008). There is also a pronounce void directly behind the Virgo Cluster at SGY ∼  + 30 Mpc and SGX ∼ 0 that we will call the "Virgo Void." It becomes evident here (best seen in the middle panels of Figure 5) that the Local and Virgo voids are linked by a generally under dense region that raps above the Local Supercluster at SGZ ∼ 15 Mpc. Sparse filaments do course through this volume (Tully & Fisher 1987). However, the overriding aspects of our local region are, first of all, the general under density across a region of ∼60 Mpc but, second, the bridge of over density of the Local Supercluster at SGY ∼18 Mpc, SGZ ∼0 and extending to negative SGZ, and a width in SGX that establishes links to major structures including the Centaurus and Hydra clusters (seen best in the top panels of Figure 5).

These features are seen in other reconstructions of the local neighborhood (di Nella et al. 1996; Branchini et al. 1999; Erdoğdu et al. 2006). Romano-Díaz & van de Weygaert (2007) emphasize the important dynamic role of voids. Kitaura et al. (2009) provide hints of voids on scales of 150 Mpc. The present study is distinguished by a higher density of distance information on a smaller scale and, in particular, benefits from the observation of the kinematic signature of void expansion; the discontinuity in peculiar velocities between galaxies in the local wall of the void and galaxies beyond the wall (Tully et al. 2008). The same distance constraints were used in the study by Lavaux et al. (2010) but only to confirm the properties of the bulk flow of the local region within 3000 km s⁻¹.

The series of panels in Figure 6 present the evolution of the local field on Aitoff all-sky projections at increasing distances. The plotted velocity field is the radial part only of the local field associated with mass within 40 Mpc. The top panel at 6.7 Mpc shows a motion from supergalactic north to south and toward the direction of the Virgo Cluster, which is beyond this slice. The panels at 13.5 and 20.3 Mpc display the front fall away from us toward the Virgo Cluster and backside infall toward us. The same effect is seen around the Fornax Cluster: a front- and a back-side infall around sgl = 270, sgb = −45. The color bar gives the velocities in km s⁻¹. Within this local volume the largest motion is toward a location at sgl = 150, sgb = 0. The farthest shells show the fading of the WF reconstruction toward the null fields as it should from construction.

Zoom In Zoom Out Reset image size

The WF reconstruction has two modes of operation. On the one hand, it interpolates the structure in between the data points, providing the optimal interpolation given the data, its errors, and the underlying prior model. On the other hand, the WF extrapolates the structure to regions not covered by the data (Zaroubi 1995; Zaroubi et al. 1995; Hoffman et al. 2001). The long-range coherent nature of the peculiar velocities makes the WF an effective tool to probe LSS beyond the spatial confines of the data. In the current situation, the catalog of observed peculiar velocities extends to 3000 km s⁻¹. In order to probe to larger scales the WF has been applied to a 160 h⁻¹ Mpc box, centered on the LG, on a 128³ grid.

Results on this larger scale are illustrated in Figure 7. The top left and right panels are expanded scale versions of the top and middle panels of Figure 5. As was described in Section 3, the WF density reconstruction degrades to the mean density on scales beyond the data zone. The local component of the reconstructed velocity field is dominated by the repulsion of the voids and flows toward the densest structure. The dominant convergence at SGX ∼  − 40 Mpc, SGY ∼  + 20 Mpc is coincident with prominent structure in Norma, Hydra, Centaurus, and Pavo-Indus, the principal components of what came to be called the "Great Attractor" (Shaya 1984; Dressler et al. 1987). Consideration of the tidal part gives the visual impression that the large-scale flow on a scale of 80 Mpc is dominated by attractions outside the computation box.

Zoom In Zoom Out Reset image size

Alternative views are given in Figure 8. The top panels are the same as with Figure 7 minus the superposition of the V8K galaxies. The lower panels show the gridded local and tidal decompositions, respectively, with a decomposition radius of 80 Mpc. The choice of a decomposition radius is very large. It comfortably includes the canonical Great Attractor region, located in the region of velocity convergences in the middle panels. Even with this large decomposition radius the tidal flow shows a convergence of the flow just beyond this radius, thereby indicating the existence of structure beyond the Great Attractor. It also suggests that the reconstructed Great Attractor backside infall arises from the lack of data on this scale and is not physical.

Zoom In Zoom Out Reset image size

The hypothesis that the tidal velocity field is predominantly induced by a Great Attractor convergence can be tested. The tidal velocity field has been fitted by the bulk and shear flow model of Hoffman et al. (2001). Consider Equation (6) from that paper:

$$\begin{equation} v_\alpha (\vec{r}) = B_\alpha + (\tilde{H}\, \delta _{\alpha \beta } + \Sigma _{\alpha \beta }) r_\beta, \end{equation} \tag{ 4 }$$

where B_α is the bulk velocity, Σ_αβ is the traceless shear tensor, $\tilde{H} \delta _{\alpha \beta }$ is a possible local isotropic perturbation, and the indices α and β range over the three Cartesian coordinates. The bulk and shear correspond to the dipole and quadruple terms of potential theory. The isotropic term $\tilde{H} \delta _{\alpha \beta }$ vanishes in an analysis of tides.

Table 2 presents the bulk velocity, V_bulk, the three eigenvalues of the shear tensor, λ₁, λ₂, and λ₃, and the direction of their corresponding eigenvectors. The eigenvalues are multiplied by R = 30 h⁻¹ Mpc, so they acquire the dimensionality of velocity. A tidal flow induced by a single attractor is expected to have

(1) the eigenvector corresponding to λ₁, hereafter the first eigenvector, and the bulk velocity to be aligned;

(2) the shear eigenvalues have the following structure:

λ₂ = λ₃ = −λ₁/2.

The results of Table 2 imply that the local tidal flow cannot be modeled by a single attractor. The 61° misalignment between the first eigenvector and the bulk velocity is one indicator. Yet, the strongest evidence comes from the structure of the eigenvalues, which are closer to a (λ, 0, −λ) structure which is expected for a random field point (Hoffman et al. 2001).

The bulk velocity of any velocity field is defined here as the volume-weighted average velocity of the full three-dimensional velocity field in a given volume. For this discussion, a sphere of radius R = 30 h⁻¹ Mpc is assumed and the velocity field is assumed to be evaluated on a regular Cartesian grid. Table 3 presents the bulk velocity, V_bulk, of the WF reconstructed velocity field. To study the robustness of WF estimator of V_bulk a suite of 16 CRs have been constructed, using the same observational data and prior model as in the WF reconstruction. The construction of a CR involves the complementary use of an unconstrained realization of the underlying field. These random realizations (dubbed RANs) are used as a tool for calculating the mean and scatter of the amplitude of the bulk velocity of a random sphere in the assumed ΛCDM (WMAP5) cosmology. Both the CRs, and therefore also the RANs, are evaluated on a box of L = 640 h⁻¹ Mpc on a 128³ grid. This large computational box is used to avoid suppression of the residual power contributed by very large scales. The definition of V_bulk used here differs from the usual definition adopted in the literature, whereas V_bulk is often taken to be the mean velocity of a given sample of galaxies, the bulk velocity is defined here by the mean velocity of all galaxies within a given volume.

Table 4 provides information on the sources of the motion of the LG with respect to the CMB frame of reference. The full reconstructed velocity field yields a velocity V_LG that is very close to the CMB dipole motion, V_CMB, with only 5° misalignment. The tidal and local components have similar amplitudes and deflections but are mutually misaligned by 60°. The last two rows of Table 4 present the values of the velocity field at the position of the LG induced by the mass distribution in spheres of radius 7 and 10.6 Mpc h⁻¹ centered on the Virgo Cluster. The mass in the smaller sphere induces a motion closely aligned with the cluster but the amplitude of 29 km s⁻¹ is small. The mass in the larger sphere induces a velocity with an increased amplitude of 60 km s⁻¹ but the direction is 65° away from the cluster. It follows that the Virgo Cluster plays a minor role in shaping the motion of the LG with respect to the CMB. The growth of the amplitude and the shift in direction with the increase in radius of the shell around the cluster are consistent with the proposition of a dominant role of the Local Void, as discussed in the next section.