A NEW DYNAMICAL MASS MEASUREMENT FOR THE VIRGO CLUSTER USING THE RADIAL VELOCITY PROFILE OF THE FILAMENT GALAXIES

The simplest, and probably most reasonable, explanation for the current acceleration of the universe is the anti-gravitational effect of the cosmological constant (Λ) that is supposed to be dominant in the present epoch. The fundamental assumption that underlies this explanation is that general relativity (GR) is the universal law of gravity, the validity of which has been confirmed by recent cosmological observations (Reyes et al. 2010; Wojtak et al. 2011; Rapetti et al. 2013). In addition, the latest news from the Planck Cosmic Microwave Background mission has drawn the following bottom line (Planck collaboration XIV 2015 results): the risk that one has to take by accepting the ΛCDM (Λ+cold dark matter) cosmology founded upon GR is lower than that by accepting any other alternative.

Nevertheless, the conceptual difficulty of acquiescing to the extremely fine-tuned initial conditions of the universe associated with Λ still dares cosmologists to take the high risk of suggesting modified gravity (MG) models and developing independent tests of GR (Clifton et al. 2012). One of the most plausible tests of GR on the cosmological scale is to compare the lensing mass of a cluster with its dynamic mass (e.g., see Schmidt et al. 2009; Schmidt 2010; Zhao et al. 2011). The former is estimated from the weak gravitational lensing effect generated by the mass of a cluster on the shapes of the foreground galaxies, while the latter is conventionally derived from the velocity dispersions of the central satellites of a cluster. A substantial discrepancy between the two mass measurements for a cluster, if found and accurately measured, would challenge the key tenet of GR.

However, it is a formidable task to measure the dynamical mass of galaxy clusters with high accuracy. The success of the conventional method of using the satellite velocity dispersions is contingent upon the completion of relaxation in the dynamical state and spherical symmetry in the shape. There is another method for the dynamical mass measurement of clusters that uses the peculiar velocities of  neighboring galaxies located beyond the virial radius of a cluster. This method can be applied even to those clusters which have yet to be completely relaxed without spherical symmetry (Corbett Moran et al. 2014). However, large uncertainties inherently involved in the measurements of the peculiar velocity field remain as a technical bottleneck that stymies the accurate measurement of the dynamic mass of a cluster from the peculiar velocities of neighboring galaxies (see also Corbett Moran et al. 2014; Gronke et al. 2014; Hellwing et al. 2014; Zu et al. 2014)

Falco et al. (2014) devised a novel method to measure the dynamic mass of a cluster by using the universal radial velocity profile of the filament galaxies around the cluster. The sole condition for the practical success of their method is that the neighboring galaxies should be distributed along a narrow filament. Since it has been theoretically and observationally shown that the formation of galaxy clusters preferentially occurs at the junction of cosmic filaments (e.g., see Bond et al. 1996; Dietrich et al. 2012), the majority of galaxy clusters should meet this condition and thus be eligible for the application of this method. The method of Falco et al. (2014) for measuring the virial mass of a cluster from the radial velocity profile of the filament galaxies is applicable even to those clusters which have yet to complete their relaxation process. In this respect, their method fits the Virgo cluster especially well, as it is still undergoing merging events. Furthermore, the measurements of the redshifts and equatorial coordinates of the galaxies in the vicinity of the Virgo cluster were performed with high accuracy thanks to their proximity, which will help to reduce uncertainties in the dynamical mass measurement using the method of Falco et al. (2014) for the Virgo cluster. The goal of this paper is to test this new method against the Virgo cluster, around which narrow filamentary structures were recently identified by S. Kim et al. (2015, in preparation).

The upcoming sections will present the following. A review of the method of Falco et al. (2014) in Section 2. A reconstruction of the radial velocity profile of the filament galaxies around the Virgo cluster and a new dynamical mass estimate for the Virgo cluster by comparing the reconstructed profile with the numerical formula from Falco et al. (2014) in Section 3. Discussions on the final results and on the future works for the improvements in Section 4. Throughout this paper, we assume a GR+ΛCDM cosmology with ${{\rm{\Omega }}}_{m}=0.26,{{\rm{\Omega }}}_{{\rm{\Lambda }}}\;=0.74,h=0.73$ and zero curvature, to be consistent with Falco et al. (2014).

The galaxies located in the vicinity of a cluster must feel the gravity of the cluster, even if they are not satellites bound to the cluster, and thus their motions relative to the cluster should deviate from pure Hubble flows. To quantify the degree of deviation of the galaxy motions from the Hubble flows and its dependence on the cluster mass, Falco et al. (2014) determined the mean radial velocity profile of the galaxies in the vicinity of the clusters from a high-resolution N-body simulation using GR+ΛCDM cosmology. Their numerical experiment discovered that the mean radial velocity profile of the neighboring galaxies located in the zone where the effect of Hubble expansion exceeds that of the gravitational attraction of a cluster is well approximated by the following universal formula:

$$\begin{eqnarray}&&{v}_{{\rm{r}}}(d;z,{M}_{{\rm{v}}})=H(z)d-0.8\;{V}_{{\rm{v}}}{\left(\displaystyle \frac{{r}_{{\rm{v}}}}{d}\right)}^{{n}_{{\rm{v}}}},\end{eqnarray} \tag{ 1 }$$

where v_r is the mean radial velocity at (comoving) a separation distance d from the cluster center, ${M}_{{\rm{v}}}$, ${r}_{{\rm{v}}}$, and ${V}_{{\rm{v}}}$ are the virial mass, radius, and velocity of the cluster, respectively, and H(z) is the Hubble parameter at redshift z. The first term on the right-hand side represents a pure Hubble flow from the cluster center while the second term corresponds to the mean peculiar velocity induced by the gravitational attraction of the cluster.

According to Falco et al. (2014), the radial velocity profile is universal in the respect that the power-law index, ${n}_{{\rm{v}}}$, in Equation (1) has a constant value of 0.42, independent of ${M}_{{\rm{v}}}$ and z. They also showed that the zone where the radial velocities of the galaxies are well approximated by Equation (1) with ${n}_{{\rm{v}}}=0.42$ corresponds to the range of $3\;{r}_{{\rm{v}}}\leqslant d\leqslant 8\;{r}_{{\rm{v}}}$. Adopting the conventional definition that ${r}_{{\rm{v}}}$ is the radius of a sphere within which the mass density reaches ${{\rm{\Delta }}}_{{\rm{c}}}$ times the critical density of the universe ${\rho }_{{\rm{c}}}\equiv 3\;{H}^{2}/(8\pi \;G)$, Falco et al. (2014) used the following simple relations to calculate the virial mass ${M}_{{\rm{v}}}$ and the virial velocity ${V}_{{\rm{v}}}$ for Equation (1):

$$\begin{eqnarray}&&{M}_{{\rm{v}}}=\displaystyle \frac{4\pi }{3}{\rm{\Delta }}\;{\rho }_{{\rm{c}}}\;{r}_{{\rm{v}}}^{3},\quad {V}_{{\rm{v}}}^{2}=\displaystyle \frac{G\;{M}_{{\rm{v}}}}{{r}_{{\rm{v}}}}.\end{eqnarray} \tag{ 2 }$$

The value of ${{\rm{\Delta }}}_{{\rm{c}}}$ was set at 93.7 in agreement with the formula of ${{\rm{\Delta }}}_{{\rm{c}}}=18{\pi }^{2}+82[{{\rm{\Omega }}}_{m}(z)-1]-39{[{{\rm{\Omega }}}_{m}(z)-1]}^{2}$ given by Bryan & Norman (1998).

Falco et al. (2014) suggested that the virial mass of a massive cluster M_v could be estimated by comparing the observed radial velocity profile of its neighboring galaxies to Equation (1). However, since the real values of v_r and d are not directly measurable, it is difficult to determine the radial velocity profiles of the galaxies based on observational data. They claimed that this difficulty could be overcome by using the anisotropic spatial distribution of the galaxies belonging to a one-dimensional filament in the vicinity of a cluster. The three-dimensional separation distance d and radial velocity v_r of the filament galaxies can be estimated from two direct observables, the projected distance R in the plane of the sky and the line-of-sight velocity ${v}_{\mathrm{los}}$, as $R=\mathrm{sin}\beta \;r$ and ${v}_{{\rm{r}}}={v}_{\mathrm{los}}/\mathrm{cos}\beta$, respectively, where β is the angle at which the filament is inclined to the line-of-sight direction to a cluster (see Figure 1). Finally, Equation (1) was rewritten in terms of the observables as

$$\begin{eqnarray}&&{v}_{\mathrm{los}}(R,\beta ,{M}_{{\rm{v}}})=\mathrm{cos}\beta \left[H(z)\displaystyle \frac{R}{\mathrm{sin}\beta }-0.8{V}_{{\rm{v}}}{\left(\displaystyle \frac{R}{\mathrm{sin}\beta \;{r}_{{\rm{v}}}}\right)}^{{n}_{{\rm{v}}}}\right].\end{eqnarray} \tag{ 3 }$$

Zoom In Zoom Out Reset image size

Since galaxy clusters usually form at the junction of cosmic filaments, this new scheme of Falco et al. (2014) seems quite plausible. However, the validity of Equation (3) depends strongly on the geometrical shape of the filament. The more straight and narrow a filament is, the better Equation (1) is approximated by Equation (3). Figure 1 illustrates a three-dimensional configuration of a filamentary structure around a cluster inclined at an angle β to the line-of-sight direction to the cluster, defining the projected distance R between a filament galaxy and the cluster center in the plane of sky. Falco et al. (2014) performed a numerical test of Equation (3) by applying it to three cluster-sized halos around which filamentary structures were found. The estimated values of ${M}_{{\rm{v}}}$ turned out to agree very well with the true virial masses for all three cases, which numerically verified the validity of Equation (3).

Falco et al. (2014) also performed an observational test of their algorithm by applying it to the sheet structures identified around the Coma cluster. Despite their claim that the dynamic mass of the Coma cluster determined by their method turned out to be consistent with the previous estimates, their method, strictly speaking, is valid only for those galaxies distributed along a one-dimensional filamentary structure that interacts gravitationally with the cluster. In fact, it is obvious from Equation (3) that large uncertainties should contaminate the measurement of the cluster mass if the radial velocity profile is reconstructed from the sheet galaxies.

3.1. The Zeroth Order Approximation

S. Kim et al. (2015, in preparation) developed an efficient technique to identify a filamentary structure whose galaxies are under the gravitational influence of a neighboring cluster. The merit of their filament-finding technique is that it only requires information on the observable redshifts and two-dimensional positions of the galaxies projected onto the plane of sky. Applying this filament-finding technique to the NASA-Sloan-Atlas catalog of local galaxies³ , S. Kim et al. (2015, in preparation) have identified two narrow filamentary structures to the east and west of the Virgo cluster (that is, Filaments A and B, respectively). Filaments A and B were both found to consist primarily of dwarf galaxies with absolute r-band magnitudes of ${M}_{{\rm{r}}}\gt -20$. For a detailed description about how Filaments A and B were identified from the NASA-Sloan-Atlas catalog, see S. Kim et al. (2015, in preparation).

Figure 2 plots the right asensions (R.A.) and declinations (decl.) of the galaxies belonging to Filaments A and B as filled blue and red dots, respectively. The certain and possible member galaxies of the Virgo cluster from the Extended Virgo Cluster Catalog (EVCC; Kim et al. 2014) are shown as black and gray dots, respectively, inside an open rectangular box. The gray dots outside the box represent the field galaxies that do not belong to the EVCC or to the two filaments but lie in the redshift range of $500\leqslant {cz}/[\mathrm{km}\;{{\rm{s}}}^{-1}]\leqslant 2500$. Note that Filament A appears to be more straight than Filament B in the equatorial plane.

Zoom In Zoom Out Reset image size

Showing that the mean radial velocities of the galaxies belonging to each filament are similar to the system velocity of the Virgo cluster, S. Kim et al. (2015, in preparation) have confirmed that the filament galaxies are in gravitational interaction with the Virgo cluster. Figure 3 plots the redshifts of the galaxies belonging to Filaments A and B as a function of their projected distances in units of degrees from  M87 (i.e., the Virgo center) as filled blue and red dots, respectively. As can be seen, the redshift variation of the filament galaxies ${\rm{\Delta }}\;{cz}$ at constant R are less than 1000 km s⁻¹.

Zoom In Zoom Out Reset image size

Denoting the equatorial positions (i.e., R.A. and decl.) of a filament galaxy and the Virgo center as $({\alpha }_{{\rm{g}}},{\delta }_{{\rm{g}}})$ and $({\alpha }_{\mathrm{cl}},{\delta }_{\mathrm{cl}})$, respectively, we determine the projected distance R between each filament galaxy from the Virgo center in the plane of sky (see Figure 1) as $R=c\;{z}_{\mathrm{cl}}\left(\mathrm{cos}{\theta }_{{\rm{g}}}\mathrm{cos}{\theta }_{\mathrm{cl}}+\mathrm{sin}{\theta }_{{\rm{g}}}\mathrm{sin}{\theta }_{\mathrm{cl}}\mathrm{cos}{\rm{\Delta }}{\phi }_{{\rm{g}}}\right)$, where ${\theta }_{{\rm{g}}}=\pi /2-{\delta }_{{\rm{g}}}$, ${\theta }_{\mathrm{cl}}=\pi /2-{\delta }_{\mathrm{cl}}$, ${\rm{\Delta }}{\phi }_{{\rm{g}}}={\alpha }_{{\rm{g}}}-{\alpha }_{\mathrm{cl}}$, and ${z}_{{\rm{g}}}$ and ${z}_{\mathrm{cl}}$ are the redshifts of the filament galaxy and the Virgo center, respectively. The line-of-sight velocity magnitude ${v}_{\mathrm{los}}^{\mathrm{ob}}$ of each filament galaxy at a projected distance, R, from the Virgo center is approximated as $c| \;{z}_{{\rm{g}}}(R)-{z}_{\mathrm{cl}}\;|$.

Equation (3) is valid only for those filament galaxies whose separation distances from the cluster center are larger than three times the virial radius of the Virgo cluster according to Falco et al. (2014). Taking the typical cluster size $2\;{h}^{-1}\;\mathrm{Mpc}$ as a conservative upper limit on the virial radius of the Virgo cluster, we select only those filament galaxies which satisfy the condition of $R\geqslant 6\;{h}^{-1}$ Mpc. A total of 130 and 96 galaxies are selected in Filaments A and B, respectively, and the method of Falco et al. (2014) is applied to each as follows.

Setting the power-law index ${n}_{{\rm{v}}}$ at the universal value of 0.42, we compare the observed values of ${v}_{\mathrm{los}}^{\mathrm{ob}}$ of the filament galaxies with Equation (3). Assuming a flat prior, we search for the best-fit values of ${M}_{{\rm{v}}}$ and β which maximize the following posterior density function $p({M}_{{\rm{v}}},\beta | \mathrm{data}):$

$$\begin{eqnarray}&&p({M}_{{\rm{v}}},\beta | {v}_{\mathrm{los}}^{{\rm{o}}},{R}_{i})\\ &&\quad \propto \mathrm{exp}\left\{-\displaystyle \sum _{i=1}^{{N}_{{\rm{g}}}}\displaystyle \frac{{[{v}_{\mathrm{los}}^{{\rm{o}}}({R}_{i})-{v}_{\mathrm{los}}({R}_{i};{M}_{{\rm{v}}},\beta )]}^{2}}{2{\sigma }_{{\rm{v}}}^{2}({N}_{{\rm{g}}}-2)}\right\},\end{eqnarray} \tag{ 4 }$$

where ${N}_{{\rm{g}}}$ is the number of selected filament galaxies, R_i is the projected distance of the ith filament galaxy, and ${\sigma }_{{\rm{v}}}$ represents the errors involved in the measurement of ${v}_{\mathrm{los}}^{\mathrm{ob}}({R}_{i})$ from the redshift space. Given that ${\sigma }_{{\rm{v}}}$ includes all of the systematic errors that must exist but are unknown, we set ${\sigma }_{{\rm{v}}}$ at unity for all of the selected filament galaxies. Here, the posterior probability density function $p({M}_{{\rm{v}}},\beta | {v}_{\mathrm{los}}^{{\rm{o}}},{R}_{i})$ is normalized to satisfy the condition of $\int {{dM}}_{{\rm{v}}}\int d\beta \;p({M}_{{\rm{v}}},\beta | {v}_{\mathrm{los}}^{\mathrm{ob}},{R}_{i})=1$.

The 68%, 95%, and 99% confidence level contours of $\mathrm{log}{M}_{{\rm{v}}}$ and β obtained from the above fitting procedure for the case of Filament A (B) are shown as solid, dashed, and dot–dashed lines, respectively, in the left (right) panel of Figure 4, which reveals the existence of a strong degeneracy between M_v and β. To break this degeneracy, we would like to place some prior constraint on the range of β by making the zeroth-order approximation to the three-dimensional positions of the filament galaxies relative to the Virgo cluster.

Zoom In Zoom Out Reset image size

In the zeroth-order approximation of $d\approx {d}^{0}=c\;{z}_{{\rm{g}}}$, under the assumption of no peculiar velocity, the three-dimensional separation vector of each selected filament galaxy, ${{\boldsymbol{d}}}^{0}=({d}_{1}^{0},{d}_{2}^{0},{d}_{3}^{0})$, can be expressed as ${d}_{1}^{0}=c\;{z}_{{\rm{g}}}\mathrm{sin}{\delta }_{{\rm{g}}}\mathrm{cos}{\alpha }_{{\rm{g}}}$ $-\;c\;{z}_{\mathrm{cl}}\mathrm{sin}{\delta }_{\mathrm{cl}}\mathrm{cos}{\alpha }_{\mathrm{cl}},{d}_{2}^{0}$ = $c\;{z}_{{\rm{g}}}\mathrm{sin}{\delta }_{{\rm{g}}}\mathrm{sin}{\alpha }_{{\rm{g}}}$ $-\;c\;{z}_{\mathrm{cl}}\mathrm{sin}{\delta }_{\mathrm{cl}}\mathrm{sin}{\alpha }_{\mathrm{cl}},{d}_{3}^{0}$ = $c\;{z}_{{\rm{g}}}\mathrm{cos}{\delta }_{{\rm{g}}}$ $-\;c\;{z}_{\mathrm{cl}}\mathrm{cos}{\delta }_{\mathrm{cl}}$. The zeroth-order approximation to the angle, β, between ${\boldsymbol{d}}$ and the line-of-sight direction to the Virgo cluster, i.e., $\hat{{\boldsymbol{h}}}$, is now calculated as, $\beta \approx {\beta }^{0}\equiv {\mathrm{cos}}^{-1}\langle {\hat{{\boldsymbol{d}}}}^{0}\cdot \hat{{\boldsymbol{h}}}\rangle$ with ${\hat{{\boldsymbol{d}}}}^{0}\equiv {{\boldsymbol{d}}}^{0}/| {{\boldsymbol{d}}}^{0}|$, where the ensemble average is taken over the selected filament galaxies.

Filaments A and B yield ${\beta }^{0}=36\buildrel{\circ}\over{.} 8\pm 3\buildrel{\circ}\over{.} 9\mathrm{and}$ ${\beta }^{0}=36\buildrel{\circ}\over{.} 9\pm 5\buildrel{\circ}\over{.} 6$, respectively. It is interesting to note that this result is in line with previous observational evidence for an alignment tendency between the major axis of the Virgo cluster and the line-of-sight direction (Mei et al. 2007). It is also consistent with the recent discovery that the Virgo satellites tend to fall into Virgo along line-of-sight directions aligned with the minor principal axis of the local velocity shear field (Lee et al. 2014). Taking these zeroth-order values as a prior, we break the degeneracy between ${M}_{{\rm{v}}}$ and β shown in Figure 4 and finally determine the dynamical mass of the Virgo cluster: ${M}_{{\rm{v}}}\approx ({0.84}_{-0.51}^{+2.75})\times {10}^{15}\;{h}^{-1}\;{M}_{\odot }$ for the case of Filament A and ${M}_{{\rm{v}}}\approx ({3.2}_{-1.3}^{+5.0})\times {10}^{15}\;{h}^{-1}\;{M}_{\odot }$ for the case of Filament B.

3.2. Radial Motions of the Filament Galaxies Around the Virgo Cluster

It is worth comparing our result with previous measurements of the Virgo mass based on various methods. For instance, Urban et al. (2011) converted the X-ray temperature of the hot gas in the Virgo cluster to its mass via the mass-temperature scaling relation obtained by Arnaud et al. (2005) and found ${M}_{200}/{M}_{\odot }\approx 1.4\times {10}^{14}$. Here, M₂₀₀ represents the mass enclosed by a spherical radius r₂₀₀ at which the mass density becomes $\rho ({r}_{200})=200\;{\rho }_{\mathrm{crit}}$, where ${\rho }_{\mathrm{crit}}\equiv 3\;{H}^{2}/8\pi G$. Karachentsev et al. (2014) studied the infall motions of seven nearby galaxies located along the line-of-sight direction to the Virgo cluster whose distances were measured by means of the Tip of the Red Giant Branch (TRGB) method (Baade 1944; Salaris & Cassisi 1997) and found ${M}_{200}/{M}_{\odot }=(7.0\pm 0.4)\times {10}^{14}$. It was also claimed that the lensing mass of the Virgo cluster should be in the range of $\mathrm{log}\left({M}_{200}/{M}_{\odot }\right)={13.87}_{-1.95}^{+0.42}$ by employing the technique developed by Kubo et al. (2009) to estimate the lensing mass of the galaxy clusters in the local universe (D. Nelson 2014, private communication⁴ ).

Noting that the definition of M₂₀₀ is different from that of the virial mass ${M}_{{\rm{v}}}$ considered here, we convert the values of ${M}_{200}/{M}_{\odot }$ reported in the previous works into ${M}_{{\rm{v}}}/({h}^{-1}\;{M}_{\odot })$ for a fair comparison under the assumption that the mass density profile of the Virgo cluster is well approximated by the NFW formula (Navarro et al. 1997) and list them in Table 1. As can be seen, our best-fit value ${M}_{{\rm{v}}}$ obtained from Filament A is consistent with the previous dynamic mass estimated by the TRGB-distance method but substantially higher than the other two estimates. Meanwhile, for the case of Filament B, the best-fit ${M}_{{\rm{v}}}$ is significantly off not only from the previous estimates but from the best-fit value obtained for the case of Filament A. Given that the method of Falco et al. (2014) is strictly valid only for the galaxies distributed along a thin straight filament, we suspect that the bent shape of Filament B is responsible for the significant discrepancy in ${M}_{{\rm{v}}}$ between the two filaments and suggest that Filament B should not be eligible for the application of the method of Falco et al. (2014).

Table 1.  Observable, Virial Mass Estimate of the Virgo Cluster in Units of ${10}^{14}\;{h}^{-1}\;{M}_{\odot }$ With h = 0.73 and Literature that Reports the Estimate

Indicator	${M}_{{\rm{v}}}$	Literature
	(${10}^{14}\;{h}^{-1}\;{M}_{\odot }$)
X-ray temperature	1.50	Urban et al. (2011)
Weak lensing effect	${0.81}_{-0.80}^{+1.29}$	Nelson (2014)a
TRGB distance	7.73 ± 0.43	Karachentsev et al. (2014)

Note.

^aD. Nelson 2014, private communication; https://www.cfa.harvard.edu/~dnelson/doc/dnelson.fermilab.talk.pdf

Download table as:  ASCIITypeset image

Before excluding the Filament B from our analysis, it may be worth investigating how different the reconstructed velocity profile of the Virgo cluster is from the expected profile, i.e., the universal formula given by Falco et al. (2014) when the mass of the Virgo cluster is fixed at some constant value ${M}_{{\rm{v}}}$. Treating ${n}_{{\rm{v}}}$ in Equation (3) as a free parameter, we search for the best-fit values of ${n}_{{\rm{v}}}$ and β that maximize the following posterior distribution:

$$\begin{eqnarray}&&p({n}_{{\rm{v}}},\beta | {v}_{\mathrm{los}}^{\mathrm{ob}},{R}_{i})\propto p(\mathrm{data}| {n}_{{\rm{v}}},\beta )p(\beta )\\ &&\quad \propto \mathrm{exp}\left\{-\displaystyle \frac{{[{v}_{\mathrm{los}}^{{\rm{o}}}({R}_{i})-{v}_{\mathrm{los}}({R}_{i};{n}_{{\rm{v}}},\beta )]}^{2}}{2({N}_{{\rm{g}}}-2)}\right\}\\ &&\quad \displaystyle \frac{1}{{\sigma }_{{\beta }^{0}}}\mathrm{exp}\left[-\displaystyle \frac{{\left(\beta -\bar{{\beta }^{0}}\right)}^{2}}{2{\sigma }_{{\beta }^{0}}^{2}}\right].\end{eqnarray} \tag{ 5 }$$

Here, instead of setting β at the fixed value $\bar{{\beta }^{0}}$, we assume a Gaussian prior on β, setting its mean and standard deviation at $\bar{{\beta }^{0}}$ and ${\sigma }_{\beta 0}$, respectively, both of which are determined in Section 3.1.

Figure 5 shows the 68%, 95%, and 99% confidence level contours of β and ${n}_{{\rm{v}}}$ obtained from Filament A as solid, dashed, and dot–dashed lines, respectively, for the four different cases of the dynamical mass of the Virgo cluster: ${M}_{{\rm{v}}}=5\times {10}^{13},{10}^{14},5\times {10}^{14},{10}^{15}\;{h}^{-1}\;{M}_{\odot }$. As can be seen, for the cases of ${M}_{{\rm{v}}}\lt {10}^{14}\;{h}^{-1}\;{M}_{\odot }$ consistent with the previous estimate of the lensing mass (see Table 1), the original value of ${n}_{{\rm{v}}}=0.42$ is located beyond the 95% confidence level contours. For the case of ${M}_{{\rm{v}}}\geqslant 5\times {10}^{14}\;{h}^{-1}\;{M}_{\odot }$ consistent with the dynamical mass estimate obtained by the TRGB method, ${n}_{{\rm{v}}}=0.42$ is included within the 68% confidence level contour. Hence, one can conclude that the numerical formula of the radial velocity profile suggested by Falco et al. (2014) matches fairly well with the reconstructed profile from observation for the case of Filament A around the Virgo cluster, yielding a best-fit value of ${M}_{{\rm{v}}}$ consistent with the previous dynamical mass measurement for Virgo. Figure 6 is the same as Figure 5 but for Filament B. As can be seen, for all of the four cases of ${M}_{{\rm{v}}}$, the original value of 0.42 stays outside the 95% contours, indicating that Equation (3) fails to describe the radial motions of the galaxies in Filament B.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We also show the radial velocity profile, v_r(d), reconstructed from Filament A as a black solid line in Figure 7. To plot v_r, the Virgo mass ${M}_{{\rm{v}}}$ is set at ${10}^{14}\;{h}^{-1}\;{M}_{\odot }$ (close to the lensing mass), and the slope ${n}_{{\rm{v}}}$ and the angle β are set at the best-fit values determined by the above fitting procedure. The associated $1\sigma$ and $2\sigma$ uncertainties around the reconstructed profile are also shown as dark and light gray regions, respectively, and the red dashed line represents the expected profile with the fixed value of ${n}_{{\rm{v}}}=0.42$ claimed by Falco et al. (2014). As can be seen, the expected profile is only marginally consistent with the reconstructed one if ${M}_{{\rm{v}}}={10}^{14}\;{h}^{-1}\;{M}_{\odot }$. Figure 8 is similar but for the case of Filament B. It is clear that the difference between the numerical profile and the reconstructed one becomes much larger for the case of Filament B.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We measured the dynamic mass of the Virgo cluster, ${M}_{{\rm{v}}}$, by applying the method of Falco et al. (2014) to two narrow filaments (Filaments A and B) consisting mainly of dwarf galaxies located to the east and west of the Virgo cluster in the equatorial reference frame, which were recently identified by S. Kim et al. (2015, in preparation) from the NASA-Sloan-Atlas catalog. The method of Falco et al. (2014), which can even be applied to unrelaxed clusters like Virgo, is based on numerical findings that the radial velocity profile of galaxies at distances larger than three times the virial radius of a neighboring cluster has a universal shape and that it can be readily reconstructed from direct observables as long as the galaxies are distributed along a filamentary structure.

To break the strong degeneracy found between the value of ${M}_{{\rm{v}}}$ and the angle β, we have constrained the range of β by making a zeroth-order approximation in which the redshift-space positions of the filament galaxies are assumed to be equal to their real-space ones. Our estimates based on the method of Falco et al. (2014) have yielded ${M}_{{\rm{v}}}=({0.84}_{-0.51}^{+2.75})\times {10}^{15}{h}^{-1}\;{M}_{\odot }$ and ${M}_{{\rm{v}}}=({3.2}_{-1.3}^{+5.0})\times {10}^{15}\;{h}^{-1}\;{M}_{\odot }$ from Filaments A and B, respectively.

Given that Filament B appears to be not as straight as Filament A in the equatorial plane, the substantial difference between the two results for the value of ${M}_{{\rm{v}}}$ from Filaments A and B is suspected to be an indication that Filament B is not eligible for the algorithm of Falco et al. (2014), the success of which depends sensitively on the degree of filament straightness. The method of Falco et al. (2014) itself also suffers from a couple of unjustified assumptions. First, it assumes that the filaments are extremely narrow and straight, which is an obvious over-simplication of the true shapes of filaments. Second, in their original study, the universal radial velocity profile was obtained by taking an average over all of the particles and halos around the clusters but not over the filament halos only. If only a subsample composed of the filament galaxies is used to reconstruct the radial velocity, then it would not necessarily agree with the average universal profile. Besides, the zeroth-order approximation that we have made for β in our estimation of the Virgo mass with the method of Falco et al. (2014) is an additional simplification of reality. If a broader range of β were considered, then the mismatches between the two masses would likely be reduced.

Our estimate of ${M}_{{\rm{v}}}$ from Filament A has turned out to be consistent with previous dynamical mass estimates based on the TRGB-distance method of Karachentsev et al. (2014), but three to four times higher than other mass estimates based on the weak gravitational lensing effect (D. Nelson 2014, private communication) and the X-ray temperature of hot gas (Urban et al. 2011). The usual suspect for this disagreement between the mass estimates for the Virgo cluster is the contamination of the measurements due to the presence of systematics associated with such over-simplified assumptions as the spherical NFW density profile (Navarro et al. 1997) in the lensing mass estimates, the hydrostatic equilibrium condition for the X-ray temperature measurement, etc.

Another, more speculative, explanation for the difference between the lensing and dynamic mass measurements for the Virgo cluster is the failure of GR. We have shown that if the dynamic mass of the Virgo cluster is assumed to be identical to the lensing mass, then the reconstructed radial velocity profile from observational data appears to decrease less rapidly with the distance than the universal radial profile from the simulation for the GR+ΛCDM cosmology. The less rapid decrease of the radial velocity profile with distance indicates higher peculiar velocities of the filament galaxies at large distances than predicted for the case of GR. In fact, several numerical works already explored how much the peculiar velocities of galaxies around massive clusters would be enhanced in the presence of MG (Lam et al. 2012; Corbett Moran et al. 2014; Gronke et al. 2014; Zu et al. 2014). Before exploring this speculative idea, however, it will be necessary to study comprehensively how the systematics produced by the over-simplified assumptions mentioned above would affect the reconstruction of the radial velocity profile of the filament galaxies in the vicinity of the clusters. Our future work is in this direction.

J.L. thanks D. Nelson for very helpful discussions. This work was supported by a research grant from the National Research Foundation of Korea to the Center for Galaxy Evolution Research (No. 2010-0027910). J.L. also acknowledges financial support by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013004372). S.C.R. acknowledges support by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2012R1A1B4003097). S.K. acknowledges support from the National Junior Research Fellowship of NRF (No. 2011-0012618).