Galaxy groups and clusters are structures comprising hundreds to thousands of galaxies bounded together by gravity. According to the concordance cosmology model, cluster sits on the top of the cosmos structural hierarchy. A direct evidence of existence of dark matter is that groups and clusters usually have much larger mass-to-light ratio. The average M/L in solar neighborhood is $\sim 5\Upsilon_\odot$. In large scope, stellar and gas mass-to-light ratio ranges from $0.5 \Upsilon_\odot$ to $30 \Upsilon_\odot$. Larger than $30 \Upsilon_\odot$ usually cannot be explained by stars and gases. Direct probes of group and cluster mass are galaxy velocity dispersion, X-Ray and gravitational lensing.

Galaxy velocities can be empirically fitted by Gaussian, $\exp\left[ -(v-v_0)^2/2\sigma_v^2 \right]$, where $\sigma_v$ is the velocity dispersion. Thus the virial mass of the system can be estimated as, \[ M \sim \frac{5}{3}\frac{R \sigma^2_v}{G}, \\ \Omega \sim \frac{5}{4\pi}\frac{\sigma_v^2}{G R^2 \rho_c}.

\]

90% baryons reside as intracluster media (ICM). These gases are heated to 10 to 100 MK due to gravitational energy and make good X-Ray sources. For ICM, $\left< \epsilon_k \right> \simeq \frac{3}{2} k_B T/\mu M_p $, where $T$ is the temperature and $\mu M_p$ is the gas molecule mass. The virial mass of the system is, \[ M \sim 5 \frac{R}{G \mu M_p} k_B T, \\ \Omega \sim \frac{15}{4 \pi} \frac{k_B T}{G R^2 \mu M_p \rho_c}

\]

Kent and Gunn (1982) measured the velocity dispersion of Coma cluster and found the mass-to-light ratio $M/L \sim 181 h^{-1}_{50} \Upsilon_\odot, \Omega \sim 0.1$, confirming Zwicky's (1933) speculation of dark mass. Carlberg et al. (1997) measured a sample of 14 clusters, containing 1150 galaxies. The average cluster virial mass-to-light ratio is $213 \pm 59 h \Upsilon_\odot$, corresponding to density fraction $\Omega \sim 0.19 \pm 0.1$. In a number of study of our own Virgo supercluster, the velocity dispersion is determined to be around $250 \pm 20$ km/s, implying $\Omega = 0.2 \pm 0.1$. For compilation of investigation of cluster masses and mass-to-light ratio, see Trimble 1987.

These are various issues using virial theorem. As the start, clusters are not totally isolated and there is no clear boundary between them. $R$ has to be some length scale. Furthermore, the virial equilibrium assumption is also questionable, although there is no doubt that the system is equilibrating. Hence virial analysis can only provide an order of magnitude estimation. Virial mass determination can be calibrated and improved by N-body simulation.

The image angle $\theta$ and source angle $\beta$ is related through deflection angle $\alpha$ (See the geometry configuration in Fig. 1), \[ \vec{\beta} = \vec\theta - \frac{D_{ds}}{D_s} \Delta \vec\alpha (\vec\theta) \equiv \vec\theta - \vec\alpha (\vec\theta ). \] The vector hats indicate the 2D nature of these angles. Different $\vec\theta$ may result the same $\vec\beta$, reflecting the possibility of multi-image formation. Under the

The resulted image would a ring, arc, or multiple images depending on the relative position and shapes of the lens and source (See Fig. 3).

As an example, consider Einstein ring (Fig 3a). It occurs when the lens, source and the observer align on a straight line ($\beta = 0 $). The total mass of the lens system is \[ M = \frac{ d_S d_L} {4 G (d_S - d_L)} \theta_E^2 \] where $d$ is angular diameter distance and $\theta_E$ is the angular radius of the ring.

For more general strong lensing, the mass distribution of the lens can be reconstructed from reasonable modeling of the lens and source (see for Koopmans (2005) more technical details). Reconstruction of mass map (See Fig. 4) of CL0024+1654 from strong lensing by Type et al. (1998) shows 98% of the cluster mass is dark matter in $35 h^{-1}$ kpc core.

The image of the background galaxy is slightly distorted, known as cosmic shear. Statistical characteristics of the cosmic shear field provide valuable information of the lens system. Detailed projected mass can be reconstructed from it. Because the cosmic shear is only a few percent effect, it requires careful image processing. It turns out the point spread function (PSF) is one of the major sources of image distortion, which has to be corrected.

Cosmic shear has been observed by Van Waerbeke et al. (2000), Bacon et al. (2000) and Wittman et al.(2000). The observation is consistent with prediction from $\Lambda$-CDM model. Clowe et a. (2006) compared the optical and X-Ray image with weak lensing mass reconstruction of a merging cluster system 1E0657-556, and suggested the > 70% system mass is dark matter (See Fig. 5). The Hubble Space Telescope (HST) Cosmic Evolution Survey (COSMOS) project (2004 - 2005) led by Scoville measured 1.637 square degrees region, shape of half million distant galaxies, and used their distortion image to reconstruct the projected mass and mass evolution with redshift (See Fig. 6 and 7).

### Galaxy Velocity Dispersion and X-Ray Spectrum

Mass of group or cluster can be inferred from their dynamics. Galaxies in groups or clusters bounded together by gravitational wall can be well assumed in*virial equilibrium*. Applying*virial theorem*, \[ \left< \epsilon_k \right> = -\frac{1}{2} \left< \epsilon_p \right> \] where $\left< \epsilon_k \right>$ denotes the average unit kinetic energy and $\left< \epsilon_p \right> \simeq -\frac{3}{5} \frac{G M}{R}$ is the average gravitational potential.Galaxy velocities can be empirically fitted by Gaussian, $\exp\left[ -(v-v_0)^2/2\sigma_v^2 \right]$, where $\sigma_v$ is the velocity dispersion. Thus the virial mass of the system can be estimated as, \[ M \sim \frac{5}{3}\frac{R \sigma^2_v}{G}, \\ \Omega \sim \frac{5}{4\pi}\frac{\sigma_v^2}{G R^2 \rho_c}.

\]

90% baryons reside as intracluster media (ICM). These gases are heated to 10 to 100 MK due to gravitational energy and make good X-Ray sources. For ICM, $\left< \epsilon_k \right> \simeq \frac{3}{2} k_B T/\mu M_p $, where $T$ is the temperature and $\mu M_p$ is the gas molecule mass. The virial mass of the system is, \[ M \sim 5 \frac{R}{G \mu M_p} k_B T, \\ \Omega \sim \frac{15}{4 \pi} \frac{k_B T}{G R^2 \mu M_p \rho_c}

\]

Kent and Gunn (1982) measured the velocity dispersion of Coma cluster and found the mass-to-light ratio $M/L \sim 181 h^{-1}_{50} \Upsilon_\odot, \Omega \sim 0.1$, confirming Zwicky's (1933) speculation of dark mass. Carlberg et al. (1997) measured a sample of 14 clusters, containing 1150 galaxies. The average cluster virial mass-to-light ratio is $213 \pm 59 h \Upsilon_\odot$, corresponding to density fraction $\Omega \sim 0.19 \pm 0.1$. In a number of study of our own Virgo supercluster, the velocity dispersion is determined to be around $250 \pm 20$ km/s, implying $\Omega = 0.2 \pm 0.1$. For compilation of investigation of cluster masses and mass-to-light ratio, see Trimble 1987.

These are various issues using virial theorem. As the start, clusters are not totally isolated and there is no clear boundary between them. $R$ has to be some length scale. Furthermore, the virial equilibrium assumption is also questionable, although there is no doubt that the system is equilibrating. Hence virial analysis can only provide an order of magnitude estimation. Virial mass determination can be calibrated and improved by N-body simulation.

#### Gravitational Lensing

Mass deflects light traveling by. Just light optical lens bends light rays, this effect can also form image, known as gravitational lensing. Although light deflection in gravitational field has been discussed by several authors notably including Newton, Laplace, Cavendish (1978), Soldner (1803) and Einstein (1911, based on*equivalence principle*), it was Albert Einstein (1915) first get the correct deflection angle $\Delta \theta = \frac{4 G M}{r}$, following his triumph of general relativity, and related it to gravitational lensing. Using gravitational lensing to measure nebulae mass was first suggested by Zwicky (1937) and confirmed by Walsh et al. in "Twin Quasar" system in 1979. Einstein first thought gravitational lensing was too rare to be useful. However, today gravitational lensing has superseded dynamical methods, as the most powerful mean of mass measurement.Fig. 1: geometry of gravitational lensing. |

**Strong Lensing**In the strong lensing case, the size of the lens object is much smaller than the lens-source distance and lens-observer distance.The image angle $\theta$ and source angle $\beta$ is related through deflection angle $\alpha$ (See the geometry configuration in Fig. 1), \[ \vec{\beta} = \vec\theta - \frac{D_{ds}}{D_s} \Delta \vec\alpha (\vec\theta) \equiv \vec\theta - \vec\alpha (\vec\theta ). \] The vector hats indicate the 2D nature of these angles. Different $\vec\theta$ may result the same $\vec\beta$, reflecting the possibility of multi-image formation. Under the

*weak field approximation*and*thin lens approximation*, the deflection angle, \[ \vec\alpha = \frac{4G D_{ds}}{D_s} \int \mathrm{d}^3 x' \rho(x') \frac{\vec\zeta - \vec\zeta'}{(\vec\zeta - \vec\zeta')^2} \equiv \frac{4G D_{ds}}{D_s} \int \mathrm{d}^2 \zeta' \Sigma(\vec\zeta') \frac{\vec\zeta - \vec\zeta'}{(\vec\zeta - \vec\zeta')^2}, \] where $\vec\zeta$ and $\vec\zeta'$ are projected position vectors. It is convenient to use angular parameters, \[ \vec\alpha(\vec\theta) = 4 G \int \mathrm{d}^2 \theta' \sigma(\vec\theta')\frac{\vec\theta - \vec\theta'}{(\vec\theta - \vec\theta')^2} = \nabla \psi(\vec\theta), \] where $\sigma(\vec\theta) \equiv \Sigma(D_d \vec\theta') D_{ds}D_d/D_s$ is the projected density, $\psi = 4 G \int \mathrm{d}^2 \theta' \sigma(\vec\theta') \ln \left| \vec\theta - \vec\theta' \right|$ is angular potential. It can be shown it satisfies Poisson equation, \[ \nabla^2 \psi(\vec\theta) = 8 \pi G \sigma(\vec{\theta}). \] The local image distortion (See Fig. 2) is related to the*Jacobian matrix*, \[ \partial \beta_i /\partial \theta_j = \delta_{ij} - \frac{\partial^2 \psi}{\partial \theta_i \partial \theta_j} = \mathbb{1}_{ij} - \left( \begin{array}{c c} \kappa + \gamma_1 & \gamma_2 \\ \gamma_2 & \kappa - \gamma_1 \\ \end{array} \right)_{ij} \]Fig. 2: meaning of distortion parameters, $\gamma \equiv \gamma_1 + i \gamma_2$. Figure from Wikipedia. |

Fig. 3: images of strong lensing. Left: An Einstein ring; Right: An Einstein cross. |

As an example, consider Einstein ring (Fig 3a). It occurs when the lens, source and the observer align on a straight line ($\beta = 0 $). The total mass of the lens system is \[ M = \frac{ d_S d_L} {4 G (d_S - d_L)} \theta_E^2 \] where $d$ is angular diameter distance and $\theta_E$ is the angular radius of the ring.

For more general strong lensing, the mass distribution of the lens can be reconstructed from reasonable modeling of the lens and source (see for Koopmans (2005) more technical details). Reconstruction of mass map (See Fig. 4) of CL0024+1654 from strong lensing by Type et al. (1998) shows 98% of the cluster mass is dark matter in $35 h^{-1}$ kpc core.

Fig. 4: mass reconstruction of CL0024+1654 from strong lensing. Galaxies are shown in blue, dark matter distribution is shown in orange. |

**weak lensing**In weak lensing, light is deflected by large number of non-uniformly distributed mass a long its path. Thus the thin lens approximation fails. The Jacobi matrix has to be generalized as, \[ \partial \beta_i / \partial \theta_j = \delta_{ij} - \int_0^{\chi_h} \mathrm{d}\chi g(\chi) \frac{\partial^2 \Psi}{\partial \zeta_i \partial \zeta_j}, \] where $\zeta$ is transverse distance, $\Psi$ is Newtonian potential, $\chi$ is the comoving distance. The weight function is $g(\chi) = 2 r \int_{\chi}^{\chi_h} \mathrm{d} \chi' \; n(\chi') r(\chi - \chi')/r(\chi') $, where $n(\chi)$ is normalized lensing object distribution, $r = d_A/(1+z), d_A$ is the angular diameter distance, $\chi_h$ is the comoving distance to the horizon.The image of the background galaxy is slightly distorted, known as cosmic shear. Statistical characteristics of the cosmic shear field provide valuable information of the lens system. Detailed projected mass can be reconstructed from it. Because the cosmic shear is only a few percent effect, it requires careful image processing. It turns out the point spread function (PSF) is one of the major sources of image distortion, which has to be corrected.

Cosmic shear has been observed by Van Waerbeke et al. (2000), Bacon et al. (2000) and Wittman et al.(2000). The observation is consistent with prediction from $\Lambda$-CDM model. Clowe et a. (2006) compared the optical and X-Ray image with weak lensing mass reconstruction of a merging cluster system 1E0657-556, and suggested the > 70% system mass is dark matter (See Fig. 5). The Hubble Space Telescope (HST) Cosmic Evolution Survey (COSMOS) project (2004 - 2005) led by Scoville measured 1.637 square degrees region, shape of half million distant galaxies, and used their distortion image to reconstruct the projected mass and mass evolution with redshift (See Fig. 6 and 7).

Fig. 5: observation of 1E0657-556. Left: optical band; Right: X-Ray band. The mass contours are reconstructed from weak lensing. |

Fig. 7: Three-dimensional reconstruction of the dark matter distribution from COSMOS project. |

**microlensing**microlensing is a gravitational lensing by pass-by massive compact astronomical objects that causes background light source apparent brightening in a certain a mount of time (several months to years). Microlensing involves small and usually faint lenses such as dwarfs and blackholes. It provides a major mean to detect massive compact astronomical halo objects (MACHOs) including planets, red, brown and black dwarfs, neutron stars and black holes etc. Microlensing is very useful for baryonic dark matter detection. However, two surveys of MACHOs shows that they are insignificant of the overall halo mass.
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