Dec 31, 2012

text censor/keyboard focus lost issues in Ubuntu/Linux

As the title states, in some older versions of Ubuntu/Linux (confirmed on 8 - 10, not know if still exists in 11, 12), the text censor focus sometimes just loses when auto-complete is up. The keyboard seems frozen and one has to click other places first and then click again to gain focus back. This bug is particularly annoying when working with IDEs with auto-complete features on, such as Eclipse. This issue is caused by SCIM (SCIM daemon is on, although one doesn't explicitly summon it.) As the SCIM development has been off for many years, this bug has not been fixed so far.

Nevertheless, there are several temporary solutions,

1. use scim-bridge:
im-switch -s scim-bridge
This method does not work on my system (karmic koala, 9.10). It is also reported that this may cause SCIM crashes.

in System/Administration/Language Support, uncheck the "Enable support to enter complex characters".
There is no such an option at all in karmic koala.

3. use ibus/shut down SCIM when using Eclipse

4. delete ~/.xinput.d/en_US
This method works for me, but SCIM are not stable when working with Google-Chrome browser. Note that you may need to logout and login several times.

Dec 8, 2012

Dark Matter VI: Beyond Standard Model

To explain the microscopic nature of dark matter requires physics beyond standard model. In fact, virtually all particle physicists believe that standard model is only an effective theory of some more fundamental theory. The following problems of standard model are the main motivation.
  • Gauge hierarchy problem. Gauge hierarchy problem asked why Higgs mass ($\sim$100 GeV) is much smaller than Planck scale energy, the believed grand unification scale. A possible explanation is there is a TeV scale physics, which affects Higgs mass through quantum fluctuation.

    Moreover, calculation shows if Higgs mass is smaller than 134 GeV, the quantum fluctuation will cause the collapse of Higgs field vacuum. The current preliminary Higgs mass from LHC (125.7 GeV) demands a mechanism to stabilize the Higgs field vacuum. All these problem can be solved by TeV physics including super-symmetric models and universal extra dimension models.

  • Neutrino mass, left-handed neutrino problem. In standard model, fermions acquire mass via Higgs mechanism. But fermion masses extend 14 orders of magnitude. This fact can hardly be viewed as a coincidence. Moreover, if neutrinos are massive, right-handed neutrinos must exist. One possibility is that there exist massive, non-interacting right-handed neutrinos, called sterile neutrinos. Sterile neutrino can solve the neutrino mass problem through see-saw mechanism.

    Fig. 1: fermion masses.

  • Strong CP problem. In QCD, a term proportional to $\theta \epsilon_{\mu\nu\rho\lambda} F^{\mu\nu} F^{\rho\lambda}$ is allowed, which, if exists, will break CP-symmetry and lead to phenomena like nonzero neutron electric dipole moment. The current experimental observation constrains this term to be close to zero. Physicists do not believe this is a coincidence. Introducing axions can solve this problem

There have been many well motivated extension of standard model. Each of them is designed to solve one or more problems listed above.


Supersymmetric standard models introduce an additional symmetry between fermions and bosons called supersymmetry. Supersymmetry assigns each boson a fermionic supersymmetric partner. The lightest supersymmetric neutral particles (LSPs) are natural dark matter candidates, as protected by $R$-parity from decaying into lighter SM particles. If supersymmetry is broken at TeV scale, gauge hierarchy and Higgs vacuum stability problem can be solved elegantly.

Another strong support of supersymmetry comes from attempt towards the grand unification. Physicists postulate all interactions have the same origin at Plank scale $10^{18} \sim 10^{19}$ GeV. If this is true, strong weak and electromagnetism forces should be unified at around $10^{16}$ GeV. However, study of the evolution of three standard model couplings shows a discrepancy from unification at Planck scale. The discrepancy can be greatly reduced by introducing supersummetry (See Fig. 2).

Fig. 2: Left, evolution of couplings within standard model; Right, evolution of couplings by introducing a pair of vector like fermions carrying SM charges and masses of order 300 GeV-1 TeV.
For all these reasons, supersymmetric models are among the most favored standard model extensions. But one common issue of these models is that they have a vast set of parameters. The Minimal Supersymmetric Standard Model (MSSM) for example already has 63 parameters. It is impossible to explore the whole parameter space. By using theoretical arguments and/or current experimental data, majority of the parameter can be fixed to certain reasonable values. The left parameter set, containing 5 - 10 parameters, can be constrained by experiment from the upcoming particle physics experiments and astrophysics observation if the model is designed for dark matter (See Fig 3).
Fig. 3: The parameter space of various MSSM models. The shaded area is excluded by measurement of $B_s \to \mu^+ + \mu^-$ and $B_d → \mu^+ \mu^-$ branching ratios.

Extra Dimensions

Extra Dimension models (ED) assume there exist extra dimensions besides the usual 3+1 spacetime. The shape of the extra dimensions are usually curled, such as a small circle, to explain their invisibility. In ED, each normal particle corresponds to a set of excitations in the extra dimension, known as Kaluza-Klein (KK) tower. The mass of KK excitations in the tower, \[ m^2_\mathrm{KK} = m^2_\mathrm{SM} + n^2 / R^2, \quad n = 0, 1, 2, \dots \] where $R$ is the radius of the extra dimension, $R^{-1} \gtrsim$ 300 GeV to explain why they have not been observed in current colliders. Lightest KK particles (LKPs) are also natural dark matter candidates, as protected by KK-parity. Meanwhile, if the radius of the extra dimension is about $\mathrm{TeV}^{-1}$ , ED extension of standard model known as Universal Extra Dimension (UED) can replace Higgs mechanism to break the electroweak gauge symmetry, hence solves the gauge hierarchy problem and Higgs vacuum stability problem.

Just like supersymmetry, ED may also modify gravity. In the large extra dimension model of Arkani-Hamed, Dimopoulos and Dvali (ADD), standard model is lived on TeV scale 4D surface (called brane), whereas gravity and only gravity can penetrate from Planck scale brane and the TeV scale brane by propa- gating in the extra dimensions. EDs are constrained by various collider experiments, gravity tests and dark matter relic density. In theminimal case (mUED), UED only has one free parameter, the radius R of the curled extra dimension (See Fig. 4).

Fig. 4: Combined collider constraints on mUED. MH is standard model Higgs mass. By Belanger et al.(2012)

Peccei-Quinn Theory

Peccei-Quinn theory is proposed to solve the strong CP problem in QCD. Peccei- Quinn theory postulates the coupling constant θ of strong CP term is a dynamical field with a new global symmetry Peiccei-Quinn symmetry. The new field is called axion. In some models, axion is also a dark matter candidate.

See-Saw Mechanism and Neutrino Masses

To explain neutrino mass, one may extend standard model to include neutrino mass via Higgs mechanism as other fermions. So one need a right-handed neutrino. Right-handed neutrino is a isospin singlet. It does not interaction with gauge bosons. Meanwhile, it can also acquire a Majorana mass term. \[ \mathcal{L} = m_D \bar{\nu}_R \nu_L + m_M \bar{\nu}^c_L \nu_R + \mathrm{h.c.} \] The mass matrix is, \begin{equation} \left( \begin{array}{c c} 0 & m_D \\ m_D & m_M \\ \end{array} \right) \end{equation} $m_M \sim M_\text{Pl} \gg m_D$. there are two mass eigenstates, $m_1 \simeq m_M, m_2 \simeq -\frac{m_D^2}{m_M}$. $m_2 \ll m_D \ll m_M$. This is the so-called see-saw mechanism. It explains why neutrino mass $m_2$ is much smaller than fermion mass $m_D$. The first mass eigenstate is a massive nearly non-interacting neutrino, called sterile neutrino. Sterile neutrino is a very good dark matter candidate.

It should be noted that these models may appear together, even producing new dark matter candidates. For example, the supersymmetric partner of axion, called axino is also a dark matter candidate.

Weakly Interacting Massive Particles

A large class of dark matter candidates is weakly interaction massive particles (WIMPs). WIMPs are neutral stable particles with mass of weak scale $\sim$ 100 GeV, and coupling with standard model particles on weak coupling scale $\alpha \sim 10^{-2}$. Candidates in WIMP class include lightest supersymmetric particles (LSPs) such as neutralinos (the linear superposition of photino, Zino and Higgsino), lightest Kaluza-Klein excitations (LKPs) from universal extra dimension models.

WIMPs may be produced in early universe as the thermal relic of Big Bang. At very early universe, WIMPs are in equilibrium. As the temperature falls below its $m_{\chi}$ , there are more WIMPs annihilating into lighter particles than the converse reaction. The WIMP density drops exponentially, until the reaction rate below the Universe expansion. Then WIMPs freeze-out from equilibrium as the thermal relic and their co-moving number density remains fixed as the relic density. The above description can be formally modeled by Boltzmann equation: \[ \frac{\mathrm{d}n_\chi}{\mathrm{d} t} + 3 H(t) n_\chi = -\left< \sigma_A v \right> (n_\chi^2 - \bar{n}^2_\chi ) \] where $H$ is the Hubble parameter, $\bar n$ is the equilibrium number density.  The equation can be solved numerically. In the simplest case, the frozen-out temperature is about $T_f \sim m_\chi^2 /20$ and the relic density is, \[
\Omega_\chi h^2 \simeq \frac{ 3\times 10^{-27} \mathrm{cm^3 s^{-1}}}{\left< \sigma_A v \right>}
\]. The cross section of WIMPs is about $\left< \sigma_A v \right> \sim \frac{\alpha^2}{m_\chi^2} (a + b v^2 + \mathcal{O}(v^4)) $. The first two terms represent $s$-wave and $p$-wave cross section respectively. $v \ll 1$, so higher order terms can be neglected. For the case of s-wave dominated annihilation, $a \sim \mathcal{O}(1)$, then $\Omega_\chi \sim 10^{-3} - 10^{-1}$, where we have allowed one order of magnitude fluctuation. The cursory estimation shows,  particles with weak scale mass and coupling have naturally the correct relic density as dark matter. This is called the "WIMP miracle". These properties make WIMPs the leading dark matter candidates.

Fig. 4: The WIMP relic density. The dashed line is relic density without thermal frozen-out. The solid lines are the actual relic density. The strips represents a variation of the cross section of orders of magnitude from the one with the "correct relic density".

Dec 4, 2012

Newton's Law of Gravity for Solar System Planets (visualization)

According to Newtonian gravity theory \[ v(r) =\sqrt{ \frac{G M}{r}} \] If the orbit is circular, the speed is simply proportional to $1/\sqrt{r}$. For general cases, however, after some derivation, the average speed, \[ \bar{v} = \frac{1}{2\pi}\int_0^{2\pi} \mathrm{d}\theta v(\theta) = \sqrt{\frac{GM}{a} } \frac{2 \mathrm{E}(\frac{2\epsilon}{1+\epsilon})}{\pi \sqrt{1-\epsilon}}, \] where $\mathrm{E}(z) = \int_0^{\frac{\pi}{2}} (1-z \sin^2\theta)^{1/2} \mathrm{d}\theta$ is the elliptic function. Note that $\mathrm{E}(0) = \frac{\pi}{2}$, restoring the circular motion result. So, the average speed is proportional to $\frac{1}{\sqrt{a}}$ where $a$ is the semi-major axis.

Fig. 1: the semi-major axis vs. average orbital speed for solar system planets in linear coordinates

Fig. 2: the semi-major axis vs. average orbital speed for solar system planets in logarithmic coordinates
The best fit of the slope gives 29.779763 km/s/AU, which is about the earth average orbital speed. Using the data of solar mass and gravitational constant, the average eccentricity is about 0.0195386. This is the absolute value. 

Dec 1, 2012

Dark Matter V: Galaxy Rotation Curves

According to Newtonian gravity, the rotational velocity $v(r)$ of an object is related to the mass enclosed by its orbit (integral mass): \[ v^2(r) = \frac{G M(r)}{r}. \] Rotation curves plot orbital velocity of a star vs. its distance from the center of the galaxy. Rubin et al. (1978) measured the rotation curves of various galaxies and found that the rotation curve at the edge of the luminous galaxy does not decline as expected $v(r) \propto 1/\sqrt{r}$. Instead, the curve is fairly flat or even increase (See Fig. 1 and Fig. 2). If Newtonian gravity is correct in these systems, there must be some invisible mass extended over the luminous part to provide gravity. These are the classical evidences of existence of dark matter.
Fig. 1: rotational curves of several spiral galaxies, with the contribution from luminous components (dashed), gas (dotted) and dark matter halo (dash-dot). The square blocks are data. The solid line is dark matter model fitting. By Begeman et al. (1991)
Fig. 2: rotation curves of several spiral galaxies. By Rubin et al. (1978)
The dark matter distribution can be inferred from the rotation curve. The flatness of rotation curves in spiral galaxies indicate an integral mass $M(r) \propto r^{1+\delta}$. It has been shown by N-body simulation that there exists universal dark matter halo profile, \[
\rho(r) = \rho_0 \left( \frac{r_0}{r} \right)^\gamma \left[ \frac{1 + \left( \frac{r_0}{a} \right)^\alpha}{1 + \left( \frac{r}{a} \right)^\alpha} \right]^{\frac{\beta - \gamma}{\alpha}} \] Some widely used dark matter halo profiles are, Kravtsov et al. (1998)$ (\alpha, \beta, \gamma) = (2, 3, 0.4), a = r_0 = 10$ kpc, Navarro et al. (1995) $(\alpha, \beta, \gamma) = (1, 3, 1), a = r_0 = 20$ kpc, Moore et al. (1999) $(\alpha, \beta, \gamma) = (1.5, 3, 1.5), a = r_0 = 28$ kpc and Bergstrom et al. (1998) $(\alpha, \beta, \gamma) = (2, 2, 0), a = r_0 = 3.5$ kpc

Dark Matter Distribution

In summary, evidences of existence of dark matter can be found in galaxies, clusters and from cosmological probes. Cosmological observations suggest overall density fraction of invisible mass $\Omega_{DM} \simeq 0.23 $. Bahcall et al. (1995) collected the mass-to-light ratio on different scales (See Fig. 3). Their study shows mass-to-light ratio of dark matter on large scale structures is consistent with the cosmological constraints. This conclusion also suggests dark matter from cosmology is the same thing with the dark matter in galaxy and clusters.
Fig. 3


Dark Matter IV: Galaxy Groups and Clusters

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 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.
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).

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. 6: Large scale structure mass distribution from COSMOS project. The total projected mass from weak lensing, is shown as contours in a and gray level in b c d; Stellar mass in blue; galaxy number density in yellow; hot gass in red.
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.

Dark Matter III: Cosmology

Evidences of dark matter can be tracked back to Oort (1932) and Zwicky (1933). In 1970s, study of galaxy rotation curves shows clear evidence of missing mass. Since then, overwhelming evidences, direct or indirect, emerge from galaxy velocity dispersion, cluster X-Ray spectroscopy, strong and weak gravitational lensing, N-body simulation, large scale structure, comic microwave background (CMB), baryon acoustic oscillation (BAO), Type-Ia supernovae (SNe) and Ly-$\alpha$ forest.

The concordant cosmology model $\Lambda$-CDM postulates existence of significant amount of dark energy and cold dark matter. $\Lambda$ stands for Einstein's cosmology constant, i.e. dark energy; CDM stands for cold dark matter. Many other competing cosmology models, such as CHDM, OCDM, SCDM and $\tau$CDM, also require some amount of dark matter. The success of CDM cosmology models forces people to take their postulates seriously.

Modern cosmology is based on three observation facts of the universe: in large scales ($\gtrsim$ 10 Mpc ), the universe is homogeneous (See Fig. 0), isotropic and the Doppler-shift velocity of an observable object is proportional to their distance to the earth. The third observation is well known as Hubble's law. The proportionality of the recession velocity to the distance $v/d = H(t)$ is called Hubble's parameter. The current value is called Hubble constant, $H_0 = 71 \pm 4$ km/(s$\cdot$Mpc). Two dimensionless constants are frequently used, $h = H_0/100$ km/(s$\cdot$Mpc) and $h_{50} = H_0/50$ km/(s$\cdot$Mpc). Hubble's law implies the universe is undergoing a homogeneous expansion.

Fig. 0: Density fluctuations vs scale. Image from Sloan Digital Sky Survey project.
To generalize our limited knowledge of the universe, one also need to adopt a modern version of Copernican principle, formally termed Cosmological Principle. Cosmological principle extends the three observation facts to the whole observable universe. Mathematically it assigns the universe a symmetry and leads to Robertson-Walker (RW) metric: \[ \mathrm{d}s^2 = - \mathrm{d}t^2 + a^2(t)\left( \frac{\mathrm{d} r^2 }{1 - k r^2} + r^2 \mathrm{d}\Omega^2_{\theta,\phi} \right), \] where $r$ is called comoving distance, $a(t)$ is called scalar factor and related to Hubble parameter $H(t) = \dot{a}/a $. RW metric describes three classes of universe, open $(k < 0)$, flat $(k = 0)$ and close $(k > 0)$.

Fig. 1: three types of universe, closed, open and flat. According to General Relativity,  the geometry of the universe is determined by the ratio $\Omega_0$ of the total energy density and the critical density. Credit: NASA/WMAP Science Team
According to Einstein's Equation of Field (EEF), the geometry of the universe if ultimately determined by the mass distribution $T^{\mu\nu}$ and probably also cosmological constant $\Lambda$. EEF implies: \[ \begin{split} & \left( \frac{\dot{a}}{a} \right)^2 + \frac{k}{a^2} = \frac{8\pi G }{3 c^2} (\rho + \rho_\Lambda)\\ & 2 \frac{\ddot{a}}{a} + \left( \frac{\dot{a}}{a} \right)^2 + \frac{k}{a^2} = \frac{8 \pi G}{c^2} \left( \rho_\Lambda - p/c^2 \right) \end{split}, \] where $\rho$ is the total energy density, $p$ is the pressure owing to $\rho$ and the motion, $\rho_\Lambda = \Lambda /8\pi G$ is known as dark energy.

Similarly, introduce Hubble density $\rho_H \equiv \frac{3}{8\pi G} H^2$ and curvature density $\rho_k = -\frac{3 k}{8 \pi G}$, total energy density $\rho_0 = \rho + \rho_\Lambda$. The current value of Hubble density is called critical density $\rho_c \equiv \frac{3}{8\pi G} H_0^2$. It is convenient to work with dimensionless density ratios $\Omega_i = \rho_i/\rho_c$, also known as density fractions or relic fractions. EEFs become, \[ \begin{split} & 1 = \Omega_k + \Omega_0 \\ & \frac{\ddot{a}}{a} = - \frac{H_0^2}{2} \sum_i \Omega_i (1+3w_i) \end{split}.  \] The main contributions for $\Omega_0$ include dark energy $\Omega_\Lambda$, dark matter $\Omega_{DM}$ (cold $\Omega_{CDM}$, warm $\Omega_{WDM}$ and/or hot $\Omega_{HDM}$), neutrinos $\Omega_\nu$, baryonic matter $\Omega_b$ and mosmic microwave background radiation (photons) $\Omega_R$ etc. $w_i = p_i/ \rho_i$ is called equation of state. For idea gas, $w_i$ are constants, which is indeed true in cosmology since the typical density is only $\rho_c \sim 10^{-26} \; \mathrm{ kg / m^3}$. For non-relativistic matters such as baryons and cold dark matter, $w = v_s^2 / c^2 \simeq 0 $, where $v_s$ is the speed of sound in such a matter medium. For relativistic particles such as photons, neutrinos and hot dark matter, $w = 1/3$. Energy conservation $ \nabla_\nu T^{\mu\nu} = 0$ (also implied from EEFs) implies $\rho(t) \propto a^{-3(1+w)}$. Dark matter density $\rho_\Lambda$ stays constant while the universe expands, implying $w_\Lambda = -1$. In some other models, there exists another hypothetical energy similar to dark energy, called quintessence. Quintessence may have $w \leq -1$. Measurement of the deviation of $w_\Lambda$ from -1 provides information of existence of quintessence.

In summary, the geometry of the universe is determined by the energy density ratio $\Omega_0$; the acceleration of the universe expansion is determined by the relative abundance of energy and matter. Conversely, we can determine the values of these density ratios by measuring the observables associated with the cosmos geometry and/or expansion acceleration. The existence of dark matter as well as dark energy thus is measurable.

The density of a galaxy and other astronomical structures can be obtained from their mass-to-light ratio. As we will see in following sections, it may provide direct evidence of existence of dark matter. Given the mass-to-light ratio $\Upsilon$ (V-band), the density fraction $\Omega = (6.12 \pm 2.16) \times 10^{-4} h^{-1} \Upsilon/\Upsilon_\odot$. Typical cluster mass-to-light ratio is 200 - 300 $\Upsilon_\odot$, indicating a density fraction 0.17 - 0.26.

The geometry of the universe affects the anisotropy of the CMB power spectrum. Cosmic Background Explorer(COBE) (1989 - 1996) and Wilkinson Microwave Anisotropy Probe (WMAP) (2001 - ) and others measured the full sky CMB angular power spectrum (See Fig. 2 & 3).

Fig. 2: 7 years WMAP full sky image of CMB. Credit: NASA/WMAP Science Team
Fig. 3: the angular power spectrum of CMB from WMAP. Credit: NASA/WMAP Science Team

The deviation of high red-shift Type-Ia supernovae from standard candle reflects the acceleration of the universe expansion. In 1998, High-Z supernova Search Team and Supernova Cosmology Project report evidence that the universe expansion is accelerating (See Fig 4).
Fig. 4: measurement of high-z supernovae magnitudes vs. red-shift z.
Similar to CMB fluctuation, BAO is the fluctuation of baryon density in the universe. Sloan Digital Sky Survey (SDSS) project provides an image of  the distribution of matter (See Fig. 5). SDSS Team report discovery of baryon acoustic peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies, which provides information of cosmology model parameters (See Fig. 6).

Fig. 5: a SDSS map of local galaxies. See a 3D map of SDSS-III galaxies below.

Fig. 6: SDSS galaxies two point correlation function vs. distance. The magenta curve is pure CDM model $\Omega_m h^2 = 0.105$, which lacks acoustic peak at around 100 Mpc/h. Other models are green, red, blue $\Omega_m h^2 = 0.12, 0.12, 0.14, 0.15$. All models take $\Omega_b h^2 = 0.024, n = 0.98$.
Combine all these cosmological probes, the current data suggest (See Fig. 7), \[
\Omega_\Lambda \simeq 0.73, \\
\Omega_M \simeq 0.27, \\
\Omega_{R} \simeq 6 \times 10^{-5}, \\
\Omega_\nu < 0.0062, \\
\Omega_0 \simeq 1.00, \\
w_\Lambda \simeq -1 \pm 0.053.
\] This result is also consistent with the typical mass-to-light ratios measured in galaxies and clusters. We conclude from these data: (1): our universe is ( at least close to ) a flat universe ($k\simeq 0$); (2): there is significant amount of dark energy ($\Omega_\Lambda > \Omega_M \gg \Omega_\text{others}$); (3): The universe expansion is accelerating ($ \ddot{a}/a > 0$).

Fig. 7: $\Omega_\Lambda$ vs. $\Omega_M$ using compilation of various cosmological probes.

Modern cosmology postulates the universe is born in a big bang in around 13.7 billion years ago. The early universe is in equilibrium at very high temperature. As it expands, temperature drops, the number of heavier particles drops, until their annihilation rates below the universe expansion rate. Thus they freeze out from reaction and become the relic. The big bang nucleosynthesis predicts abundance of elements based on the baryon-to-photon ratio $\eta$. Measurement of element relative ratios determines $\eta$, hence the baryon relic density, since photon relic density is known from CMB. The current data suggest, \[ \Omega_b h^2 = 0.0214 \pm 0.0020 \quad (9.3\%). \]

Now the evidence of dark matter arises: $\Omega_M \simeq 0.3$ but $\Omega_b \simeq 0.04$, suggesting there are significant amount of dark matter $\Omega_{CDM} \simeq 0.23$ (See Fig. 8).

Fig. 8: the contents of the universe. Credit: NASA/WMAP Science Team.