Measuring the Gravitomagnetic Distortion from Rotating Halos. I. Methods

We work within the parameterized post-Newtonian (PPN) approximation and consider dark matter halo with an SIS density profile. For static halos, the lensing convergence κ( ξ ) reads (Bartelmann & Schneider 2001)

$$\begin{eqnarray}&&\kappa ({\boldsymbol{\xi }})=\displaystyle \frac{{\rm{\Sigma }}({\boldsymbol{\xi }})}{{{\rm{\Sigma }}}_{\mathrm{crit}}({z}_{l},{z}_{s})}=\displaystyle \frac{{\sigma }_{v}^{2}}{2G\,{{\rm{\Sigma }}}_{\mathrm{crit}}({z}_{l},{z}_{s}){\boldsymbol{\xi }}},\end{eqnarray} \tag{ 1 }$$

where ${{\rm{\Sigma }}}_{\mathrm{crit}}({z}_{l},{z}_{s})=\tfrac{{c}^{2}}{4\pi G}\tfrac{{D}_{s}}{{D}_{l}{D}_{{ls}}}$ is the geometry factor given a lens at redshift z_l and a source at z_s , D_l , D_s , and D_ls are the angular diameter distances between the observer and the lens, the observer and the source, and the lens and the source, respectively; ${\sigma }_{v}^{2}$ is the velocity dispersion of matter in the halo; and the vector ξ denotes the two-dimensional position with respect to the gravitational potential center of the dark matter halo.

Let us now turn to rotating halos. The halo angular momentum J can be quantified by a dimensionless parameter λ, the ratio between the actual angular velocity and the theoretical one (Padmanabhan 2002):

$$\begin{eqnarray}&&J=\lambda \displaystyle \frac{{{GM}}^{5/2}}{| E{| }^{2}},\end{eqnarray} \tag{ 2 }$$

where M and E are the total mass and the total energy of the halo. The parameter λ is almost independent of halo mass and of the large-scale structure. Its distribution is approximated by a log-normal function (Vitvitska et al. 2002):

$$\begin{eqnarray}&&p(\lambda )d\lambda =\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{\lambda }}\exp \left[-\displaystyle \frac{{\mathrm{ln}}^{2}(\lambda /\bar{\lambda })}{2{\sigma }_{\lambda }^{2}}\right]\displaystyle \frac{d\lambda }{\lambda },\end{eqnarray} \tag{ 3 }$$

where the mean value $\bar{\lambda }\approx 0.05$ and the scatter σ_λ is around 0.5. For halos with an SIS profile, the angular momentum and energy are related to the total mass ${M}_{\mathrm{SIS}}=\tfrac{2{\sigma }_{V}^{2}}{G}{R}_{\mathrm{SIS}}$ as (Bartelmann & Schneider 2001)

$$\begin{eqnarray}&&{E}_{\mathrm{SIS}}=-{M}_{\mathrm{SIS}}{\sigma }_{v}^{2},\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{J}_{\mathrm{SIS}}=\lambda \displaystyle \frac{4{\sigma }_{v}^{3}{R}_{\mathrm{SIS}}^{2}}{G},\end{eqnarray} \tag{ 5 }$$

where R_SIS ≫ ∣ξ∣ is the truncation radius. Following Sereno (2005), we take the truncated radius such that the mean density within the radius is ∼200 times larger than the critical density, yielding

$$\begin{eqnarray}&&{R}_{\mathrm{SIS}}=\displaystyle \frac{2{\sigma }_{v}}{\sqrt{n}H(z)},\end{eqnarray} \tag{ 6 }$$

where n ∼ 200 characterizes the density ratio between the halo region and the mean density of the universe, and H(z) is the Hubble parameter at redshift z. Although the halo velocity depends on multiple physical processes such as merging, we assume for simplicity that the rotation pattern is stable within the observational time. Under these assumptions, the halo rotation adds an extra term to the lensing potential via the Einstein–Thirring–Lense effect on top of the SIS potential (Sereno 2005):

$$\begin{eqnarray}&&\phi ={\phi }_{0}+{\phi }_{\mathrm{GRM}},\end{eqnarray} \tag{ 7 }$$

where the spherical SIS halo lensing potential reads

$$\begin{eqnarray}&&{\phi }_{0}({\boldsymbol{\xi }})=\displaystyle \frac{4G}{{c}^{2}}{\int }_{{ \mathcal R }}{d}^{2}{\boldsymbol{\xi }}^{\prime} {\rm{\Sigma }}({\boldsymbol{\xi }}^{\prime} )\mathrm{ln}| {\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} | =\displaystyle \frac{{D}_{{ds}}}{{D}_{s}}\displaystyle \frac{4\pi {\sigma }^{2}}{{c}^{2}}| \xi | .\end{eqnarray} \tag{ 8 }$$

The extra potential term introduced by gravitomagnetic effect ϕ_GRM by an SIS density profile with velocity v is given by (Sereno 2005)

$$\begin{eqnarray}&&{\phi }_{0}({\boldsymbol{\xi }})=\displaystyle \frac{4G}{{c}^{2}}{\int }_{{ \mathcal R }}{d}^{2}{\boldsymbol{\xi }}^{\prime} {\rm{\Sigma }}({\boldsymbol{\xi }}^{\prime} )\langle {\boldsymbol{v}}\cdot {{\boldsymbol{e}}}_{l}\rangle \mathrm{ln}| {\boldsymbol{\xi }}-{\boldsymbol{\xi }}^{\prime} | .\end{eqnarray} \tag{ 9 }$$

where 〈 v · e _l 〉 is the average velocity along the line of sight weighted by the projected density:

$$\begin{eqnarray}&&\langle {\boldsymbol{v}}\cdot {{\boldsymbol{e}}}_{l}\rangle ({\boldsymbol{\xi }})=\displaystyle \frac{\int {dl}\left[{\boldsymbol{v}}({\boldsymbol{\xi }}^{\prime} ,l)\right]\cdot {{\boldsymbol{e}}}_{l}\rho ({\boldsymbol{\xi }},l)}{{\rm{\Sigma }}({\boldsymbol{\xi }})}.\end{eqnarray} \tag{ 10 }$$

The gravitomagnetic deflection angle is then given by the derivative of ϕ_GRM, yielding

$$\begin{eqnarray}&&{\hat{{\boldsymbol{\alpha }}}}_{\mathrm{GRM}}({\boldsymbol{\xi }})=-\displaystyle \frac{8G}{{c}^{2}}{\int }_{{{ \mathcal R }}^{2}}{d}^{2}{\xi }^{{\prime} }{\rm{\Sigma }}({{\boldsymbol{\xi }}}^{{\prime} })\displaystyle \frac{\langle {\boldsymbol{v}}\cdot {{\boldsymbol{e}}}_{l}\rangle ({{\boldsymbol{\xi }}}^{{\prime} })}{c}\displaystyle \frac{{\boldsymbol{\xi }}-{{\boldsymbol{\xi }}}^{{\prime} }}{| {\boldsymbol{\xi }}-{{\boldsymbol{\xi }}}^{{\prime} }{| }^{2}}.\end{eqnarray} \tag{ 11 }$$

Let us now consider a spherically symmetric lens that rotates about an arbitrary axis $\hat{\eta }$, passing through its center (i.e., a main axis of inertia). To specify the orientation of the rotation axis, we need two Euler angles: θ, the angle between $\hat{\eta }$ and the ξ₂ axis, and β, the angle between the line of sight l and the line of nodes defined at the intersection of the ξ₁ plane and the equatorial plane (i.e., the plane orthogonal to the rotation axis and containing the lens center). The sketch map is shown in Figures 1 and 2. The former illustrates the geometry of the lensing system at large scales, while the latter, zoomed in, describes the relations among the line of sight, rotation axis, etc.

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Under the axial symmetry about the rotation axis, we get

$$\begin{eqnarray}\begin{array}{rcl}\langle {\boldsymbol{v}}\cdot {{\boldsymbol{e}}}_{l}\rangle ({\xi }_{1},{\xi }_{2},l) & = & -\omega (R)\left[{\xi }_{1}\cos (\theta )+{\xi }_{2}\sin (\theta )\cos (\beta )\right]\\ & = & -{\omega }_{1}(R){\xi }_{1}+{\omega }_{2}(R){\xi }_{2},\end{array}\end{eqnarray} \tag{ 12 }$$

where ω(R) is the modulus of the angular velocity at a distance $R=\sqrt{{R}_{1}^{2}+{R}_{2}^{2}}$ from the rotation axis. In the case of a rigid body, there is no dependence on R for ω, and Equation (12) simplifies to v · e _l (ξ₁, ξ₂, l) = − ω₁(R)ξ₁ + ω₂(R)ξ₂. Thus, for spherically symmetric rigid halos, the gravitomagnetic deflection angle, Equation (11), becomes

$$\begin{eqnarray}\begin{array}{rcl}{\hat{\alpha }}_{1}{\left({\boldsymbol{\xi }}\right)}_{\mathrm{GRM}} & = & \displaystyle \frac{2\kappa }{3c}\left[\displaystyle \frac{{\omega }_{2}\left(2{\xi }_{1}^{2}+{\xi }_{2}^{2}\right)-{\omega }_{1}{\xi }_{1}{\xi }_{2}}{| \xi | }-\displaystyle \frac{3{\omega }_{2}{R}_{\mathrm{SIS}}}{2}\right],\\ {\hat{\alpha }}_{2}{\left({\boldsymbol{\xi }}\right)}_{\mathrm{GRM}} & = & \displaystyle \frac{2\kappa }{3c}\left[\displaystyle \frac{{\omega }_{1}\left(2{\xi }_{2}^{2}+{\xi }_{1}^{2}\right)-{\omega }_{2}{\xi }_{1}{\xi }_{2}}{| \xi | }-\displaystyle \frac{3{\omega }_{1}{R}_{\mathrm{SIS}}}{2}\right],\end{array}\end{eqnarray} \tag{ 13 }$$

where $\xi =\sqrt{{\xi }_{1}^{2}+{\xi }_{2}^{2}}$.

By taking the derivative of Equation (13), we obtain the contribution from gravitomagnetic effect to the κ field:

$$\begin{eqnarray}&&\kappa {\left({\boldsymbol{\xi }}\right)}_{\mathrm{GRM}}=\displaystyle \frac{{D}_{d}}{2}({{\rm{\nabla }}}_{\xi }\cdot \hat{{\boldsymbol{\alpha }}})=\displaystyle \frac{\kappa ({\boldsymbol{\xi }})\left({\omega }_{2}{\xi }_{1}-{\omega }_{1}{\xi }_{2}\right)}{c}.\end{eqnarray} \tag{ 14 }$$

Gravitomagnetic effect induced by rotating lens yields an anisotropic contribution to the lensing convergence field κ in the form of a dipole, as given by Equation (14). This is illustrated in Figure 3. The first panel shows the κ field for a static SIS halo, with a velocity dispersion of 1000 km s⁻¹. The second panel depicts the contribution by gravitomagnetic effect with λ = 0.2. It shows an enhancement from one side of the rotation axis and reduction from the other side. It is clear from Figure 3 that the dipole induced by the rotation of the lens is much smaller than the total signal. We can however be a bit more quantitative in order to investigate if such gravitomagnetic distortion can be detected.

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One can construct an estimator to quantify the anisotropy of the signal by taking the difference of the mean κ divided by the rotation axis:

$$\begin{eqnarray}&&\delta \kappa =\langle {\kappa }_{\mathrm{enhance}}\rangle -\langle {\kappa }_{\mathrm{reduce}}\rangle .\end{eqnarray} \tag{ 15 }$$

The mean value of κ from both sides of the rotation axis is measured inside the virial radius as in Equation (6) of a halo to minimize the effect from large-scale structure. The virial radius of a 10¹⁴ solar mass is about 2 Mpc h⁻¹ according to this calculation, which is similar to that of a Navarro–Frenk–White (NFW) profile (Navarro et al. 1997), based on numerical simulations.

In the upper right corner of the right panel of Figure 3, the two regions can be clearly seen in two different colors. In Figure 4, we evaluate the dependence of δ κ on the halo velocity dispersion σ_v and the rotation parameter λ. To do so, we generate κ maps induced by gravitomagnetic effects given by Equation (14) and measure δ κ as described above. We see that δ κ gets bigger as σ_v or λ get bigger. For a halo with velocity dispersion of 1300 km s⁻¹ and rotation parameter of 0.8, δ κ can be comparable in size to κ at the edge of the halo.

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We now look for a simple functional form for δ κ. As for an SIS profile, there is no rotation if there is no velocity dispersion, δ κ depends at leading order on ${\sigma }_{v}^{2}$. We measure δ κ on a grid of 500 × 500 points of (λ, σ_v ) and fit with the following ansatz:

$$\begin{eqnarray}&&\delta \kappa (\lambda ,{\sigma }_{v})=\mu \,\lambda {\left(\displaystyle \frac{{\sigma }_{v}}{1000\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2},\end{eqnarray} \tag{ 16 }$$

where we get μ ∼ 5.69 × 10⁻⁴, an overall normalization that can be considered as the typical magnitude of gravitomagnetic effects from rotating halos. Equation (16) shows that there is a strong degeneracy between ${\sigma }_{v}^{2}$ and λ: they are thus strongly dependent on observations.

We can also express Equation (16) as a function of the halo mass:

$$\begin{eqnarray}&&\delta \kappa (\lambda ,{\sigma }_{v})=\mu \,\lambda \,\displaystyle \frac{{GM}}{2{R}_{\mathrm{SIS}}},\end{eqnarray} \tag{ 17 }$$

where R_SIS is the truncation radius defined in Equation (6). Taking into account the distribution of λ as given in Equation (3), Equation (16) becomes

$$\begin{eqnarray}&&\langle \delta \kappa ({\sigma }_{v})\rangle =\int P(\lambda )(\mu {\sigma }_{v}^{2})\lambda d\lambda .\end{eqnarray} \tag{ 18 }$$

So far we have considered that the real (halo) rotation axis is perfectly aligned with the observed (tracer) rotation axis. If there is a misalignment δ θ between those two, δ κ will be reduced by a factor of $\cos (\delta \theta )$, such that the misaligned δ κ_m is related to the aligned δ κ_o by

$$\begin{eqnarray}&&\delta {\kappa }_{m}=\cos (\delta \theta )\delta {\kappa }_{o}.\end{eqnarray} \tag{ 19 }$$

For a given distribution P(δ θ), Equation (19) becomes

$$\begin{eqnarray}&&\langle \delta \kappa \rangle =\int d\delta \theta P(\delta \theta )\cos (\delta \theta )\delta \kappa .\end{eqnarray} \tag{ 20 }$$

Combining Equation (16), Equation (19), and Equation (20), we obtain a general expression for δ κ taking into account the scatters of both the angular momentum and the misalignment of the observed-to-real rotation axis:

$$\begin{eqnarray}&&\langle \delta \kappa \rangle =\int d\delta \theta P(\delta \theta )\cos (\delta \theta )\int P(\lambda )\mu \lambda {\left(\displaystyle \frac{{\sigma }_{v}}{1000\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}d\lambda .\end{eqnarray} \tag{ 21 }$$

To summarize, Equation (19) describes gravitomagnetic distortion around a single cluster, while Equation (21) provides an estimator for stacked clusters. We will discuss next this later case, asking ourselves if stacking multiple rotating lenses can achieve sufficient S/N for measurements of gravitomagnetic effect. Meanwhile, we finish this section by discussing related issues on the measurements of gravitomagnetic distortion in the single cluster case.

We show the dependence of δ κ on the azimuthal angle, Equation (19), in Figure 5. The angular dependence (angle between the true rotation axis and an arbitrary axis) of δ κ is clear. However, this is difficult to estimate by stacking without any knowledge of the rotation axis. We will show later in Section 4 that we can infer the rotation axis by observing the same pattern of angular dependence as in Figure 5. Such specific dependence can be used to distinguish gravitomagnetic distortions from other sub-leading lensing contributions, such as from halo asymmetry, biased tracers, and so on.

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In particular, one should keep in mind that irregularities in the halo shapes can lead to a very different signal than the one we derived here, assuming perfect spherical symmetry and an SIS density profile. In Figure 6, we show for illustration the measured κ map and values of δ κ as a function of δ θ on a typical halo identified in illustrisTNG300-300 (Nelson et al. 2018). Note that we did not add shape noise here, since for a single cluster, the shape noise will be overwhelmingly dominant. The signal is comparable in size to a gravitomagnetic distortion with σ_v = 1000 km s⁻¹ and λ = 0.2, but we can see that the halo morphology leads to a very different profile for δ κ(δ θ). Furthermore, the effect of biased tracers has not been taken into account here (this will be investigated in an upcoming work). However, at this stage, we expect that biased tracers will present a similar signal as the one for halos for gravitomagnetic effect, at least for isolated clusters. Thus, to hope to detect Einstein–Thirring–Lense effect in weak lensing surveys, one should consider relatively isolated clusters, i.e., with cylindrical selection (e.g., Mandelbaum et al. 2006), or isolated galaxies (e.g., Luo et al. 2020).

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In the following section, we perform a suite of simulations to test the model above, with realistic shape noise of galaxies and galaxy number density to evaluate the detectability in LSST-like surveys.

We now investigate the detectability of gravitomagnetic distortion in weak lensing surveys by measuring δ κ on the simulations designed in previous section. The results are shown in Figure 8.

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Let us first focus on Simulations 1, 2, and 3, designed to mock LSST-like surveys: for λ > 0.1, we have 412 cluster halos of mass > 10¹⁵ h⁻¹ M_⊙, for an average of λ around 0.128. Only 13 clusters are found for λ > 0.2, and 3 for λ > 0.25. This is summarized in Table 1. As expected, we find that δ κ is smaller for Simulation 1 than for Simulations 2 and 3, as on average the rotation of halos is smaller. Nevertheless, as the number of stacked halos in Simulation 1 is bigger, the error bars are much smaller. For the other two, which only have 13 and 3 clusters, the shape noise overwhelms the signal by a factor of ∼10. In all these "realistic" cases, we find null detection of gravitomagnetic effect, given that δ κ is compatible with zero within 1σ.

We perform further checks with Simulations 4 and 5. Simulation 4 is set to understand the relation between the measured δ κ and λ. The values of λ range from 0.3 to 0.8, for fixed number of 400 of stacked clusters. The theoretical prediction, Equation (16), agrees with Simulation 4 data points.

Simulation 5 is designed to study the effect of a misalignment between the true rotation axis of clusters and the ones selected using observational tracers, such as the major axis of central galaxies (Okumura et al. 2009), the distribution of satellite galaxies inside clusters (Baxter et al. 2016), or spin axis of spiral galaxies (Zhang et al. 2015). In general, the tracers are misaligned with their dark counterpart. One famous case is the bullet cluster (Clowe et al. 2006), in which the baryonic distribution significantly differs from the dark matter distribution inferred from weak lensing. To probe such a discrepancy, we choose randomly for each cluster in Simulation 5 the misalignment angle between the tracer axis and the true rotation axis among a normal distribution centered on 0 with scatter: σ_θ = 1°, 5°, 10°, 20°, and 30°. We find that a 5° scatter leads to a damping of ∼10%, which remains consistent with the true input value denoted by the solid black line in the lower panel of Figure 8. For larger scatters, the signal becomes too low to be detected, where even with 10° scatter the signal is already consistent with zero. This highlights the importance of selecting precisely the tracer axis in order to extract the gravitomagnetic distortion signal.

We now proceed with a rough quantification of the rotation axis and speed from the groups of galaxies of SDSS DR7 using redshift selection following Yang et al. (2007). The same method to measure δ κ(δ θ) can be used to measure the mean redshift of member galaxies separated by an arbitrary axis that has an angle of δ θ with respect to the true axis in two-dimensional projected plane. Here, we use ϕ to denote the angle between the true rotation axis and the arbitrary axis from observations in order to distinguish it from δ θ used for simulations. The difference of the mean redshift from the two sides divided by the 2D rotation axis is given by Δz = 〈z₁〉 − 〈z₂〉, where 1 and 2 indicate the two sides of the axis. Following Sofue (2013) in which the rotation curve of our own galaxy is found to be well fitted by a sine function, we fit Δz with the parameterization:

$$\begin{eqnarray}&&{\rm{\Delta }}z(\phi )={z}_{\mathrm{off}}\pm {z}_{\mathrm{amp}}\times \sin (\phi -{\phi }_{0}).\end{eqnarray} \tag{ 26 }$$

The first term on the right-hand side of Equation (26) is the offset of the group along the line of sight, which should be zero since we choose the central galaxies as the reference, one such that the peculiar velocity of the cluster becomes zero (Hwang & Lee 2007). However, we caution that there might be uncertainties in the choice of the brightest central galaxies (BCGs). For our purpose, we consider that a nonzero offset will be much smaller than the error and thus can be neglected. The amplitude z_amp represents the rotation speed along the line of sight, and ϕ₀ is the rotation axis angle with respect to the east to north. The relation between Δz and velocity dispersion along the line of sight is related by (Danese et al. 1980)

$$\begin{eqnarray}&&{v}_{\mathrm{los}}=c\displaystyle \frac{{\rm{\Delta }}z}{1+{z}_{\mathrm{bcg}}},\end{eqnarray} \tag{ 27 }$$

where z_bcg is the redshift of the BCG. With this procedure outlined above, we are thus able to extract all information to evaluate δ κ.

As an illustration, we select the richest group in the Yang et al. (2007) catalog with 623 members with velocity dispersion of about 667.8 km s⁻¹ estimated from the scatter of Δz = z_member − z_bcg as shown in Figure 9, and with a line-of-sight rotation speed of about 195.0 km s⁻¹ based on Equations (26) and (27). The upper right panel of Figure 9 illustrates that the rotation axis alignment can be measured by fitting the rotation curve using a sine function as in Equation (26). We emphasize that, however, one also has to account for the projection effects in order to obtain a reliable measurement of the rotation axis. This will be discussed in a follow-up study based on the combination of observational group catalog and simulations. This leads to λ = 0.292, which is extremely unlikely given the probability distribution P(λ) given in Equation (3) that we used for our simulations: the fraction of clusters with halo mass larger than 10¹⁵ h⁻¹ M_⊙ in SDSS DR7 is about 0.00006 and the probability of a halo with λ >= 0.3 is about 0.0002, yielding a joint probability of such a cluster to appear in this catalog of only 1.2 × 10⁻⁸, according to the probability distribution inferred from simulations (Vitvitska et al. 2002). Yet we note that this is still more likely than the bullet cluster. The probability of forming a bullet cluster-like object with such high colliding speed in the Lambda cold dark matter (ΛCDM) framework is about 3.3 × 10⁻¹¹ (Lee & Komatsu 2010). The reason for such an occurrence may be due to the fact that the velocity dispersion of this cluster is underestimated because SDSS spectroscopic objects need to be brighter than 17.77 mag in the r band, which means that fainter members are not included. Even if we set the velocity dispersion to be 1000 km s⁻¹, λ is still about 0.2. It thus seems that highly rotating clusters are not as rare as predicted by Vitvitska et al. (2002).

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The rotation speed is the most important quantity to estimate gravitomagnetic distortion. If we assume that the rotation speed is 195.0 km s⁻¹, for a cluster halo mass of $\mathrm{log}({M}_{h}/{h}^{-1}{M}_{\odot })\sim 15$, this leads to a value of δ κ = 0.0002. But for such a cluster, the shape noise will be of order 0.003, which is more than 10 times larger than the signal. Manolopoulou & Plionis (2017) have already started this research on the angular dependence of the line-of-sight rotation velocity and rotation axis using Monte Carlo mocks of rotating clusters. They found that using their algorithm up to 28% clusters can be identified as rotating.

The error bars from Figure 9 are estimated by the bootstrap resampling method: we create 600 samples out of 623 member galaxies and take the distribution of mean value of each sample to estimate the velocity and error. The lower panel of Figure 9 shows a 7262 ± 142 offset between the major axis of satellite distribution (red solid line) and rotation axis (blue dashed line), which is simply π/2 + ϕ₀. This means that the major axis from satellites cannot be used as the rotation axis and tends to be antialigned with the major axis in this cluster. Note that we mention the major axis of clusters, as it is of more interest than the minor axis in the sense that it is more likely aligned with large-scale filament structure. The alignment between the two axes is another interesting study, which we will explore in the future.

In this figure we have all the information needed to extract δ κ from observational data, i.e., the velocity dispersion, λ = v_los/σ_v , and the rotation axis. We will further explore if this can be achieved using the group catalog from Yang et al. (2007) and the SDSS DR7 shape catalog of Luo et al. (2017) (C. Tang et al. 2021, in preparation).