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.


Nov 28, 2012

$f(R)$ gravity and dark matter

Consider Einstein-Hilbert action for gravitational field, \[ S = \int \mathrm{d}^4 x \sqrt{-g} \left\{ \frac{1}{2\kappa}R - \frac{1}{\kappa}\Lambda + \mathcal{L}_M \right\}\] where $R = R^{\mu\nu} g_{\mu\nu}$ is the curvature scalar, $R^{\mu\nu} = R^\lambda_{\mu\lambda\nu}$ is the Racci tensor, $\kappa = 8 \pi G/c^2$.

Einstein-Hilbert action is the most simple Lorentzian invariant action that encodes space-time curvature and gives correct gravity (Newtonian gravity) in weak field limit. A straightforward generalization of Einstein-Hilbert action is the $f(R)$ gravity: $R \to f(R) = a_0 + a_1 R+ a_2 R^2 + \cdots$. From the aesthetic point of view, one may argue the higher order terms shouldn't be there. But there terms may arise for a good reason. We know that at high energy $\sim T_\text{Pl}$, general relativity (GR) will be superseded by quantum gravity. This in fact may even happen before Planck scale. General relativity is merely a low energy effective theory. Effective theories are not always neat. In fact, it is well known that quantized general relativity is non-renormalizable. Therefore, quantum fluctuation would bring infinite terms into the effective Lagrangian.

A natural question is, if these term has any observable effect. Let's restrict ourselves to classical theory only. The question is, how well has general relativity been tested [1]? $f(R)$ gravity and many other competing theories of gravity shares the same foundations with general relativity, except for the amount of gravity produced by the same energy. The relevant test includes deflection of light ray and time delay (Fig. 1), Mercury perihelion shift and spin precession (Table 4), change of Newton's constant (Table 5),  mass of graviton (or propagation distance of gravity) etc.

Fig. 1: test of deflection of light, $ \delta \theta = \frac{1+\gamma}{2}\frac{4 G M}{d} $ and test of time delay experiment $ \delta t = 2(1+\gamma ) GM \ln \left( \frac{(r_\oplus + \mathbf{x}_\oplus\cdot \mathbf{n})(r_e - \mathbf{x}_e\cdot \mathbf{n})}{d^2}\right)$. General relativity predicts $\gamma = 1$.

Table 5: test of change of Newton constant.
Note that most of these tests are conducted with in solar system. They cannot exclude gravity theory with only large scale effects. It has been known [2-3] $R^2$ and/or $R^{\mu\nu}R_{\mu\nu}$ term can produce a massive scalar field in addition to graviton. This field couples with matter through Yukawa potential. Its mass can be chosen such than too heavy to propagate a observable distance, but too light $\ll M_\text{Pl}$ to modified gravity. In other words, it is a massive, very weakly interaction particle. Naturally, it is a dark matter candidate, since it only manifests itself in large scales.

Now let's see how it happens. Consider $R^n$ term: \[ S_n = \frac{1}{\kappa}\int \mathrm{d}^4 x \sqrt{-g} R^n \] Now let's take functional derivative with respect to $g^{\mu\nu}$. \[ \frac{\delta \sqrt{-g} R^n}{\delta g^{\mu\nu}} = \sqrt{-g} R^{n-1} \left( n \frac{\delta R}{\delta g^{\mu\nu}} + \frac{R}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}} \right)
\] Now, according to Jacobi's formular, $\delta g = g g_{\mu\nu} \delta g^{\nu\mu}$. So \[ \frac{1}{\sqrt{-g}}\frac{\delta\sqrt{-g}}{\delta g^{\mu\nu}} = -\frac{1}{2}g_{\mu\nu}.
\] \[ \frac{\delta R}{\delta g^{\mu\nu} } = R_{\mu\nu} + g^{\alpha\beta} \left( \frac{\delta \Gamma_{\alpha\beta;\lambda}^\lambda}{\delta g^{\mu\nu}} - \frac{\delta\Gamma_{\alpha\lambda;\beta}^\lambda}{\delta g^{\mu\nu}} \right) \] Note that $ \delta R^\lambda_{\;\;\alpha\lambda\beta} = \delta\Gamma^\lambda_{\alpha\beta;\lambda} - \delta\Gamma^\lambda_{\alpha\lambda;\beta}$. $\delta \Gamma$'s are in fact tensors: \[ \delta\Gamma^\lambda_{\alpha\beta} = \frac{1}{2} g^{\lambda\rho}\left\{ \delta g_{\rho\alpha;\beta} + \delta g_{\rho\beta;\alpha} - \delta g_{\alpha\beta;\rho} \right\}. \]

Hence \[ \frac{\delta R}{\delta g^{\mu\nu} } = R_{\mu\nu} + \frac{\delta g^{\alpha\beta}_{ \;\;\;\;;\alpha;\beta}}{\delta g^{\mu\nu}} - \frac{\delta g^{\alpha \;\;;\beta}_{\;\; \alpha \;\;;\beta}}{\delta g^{\mu\nu}} \]

Therefore, the action \[ \delta S_n = \frac{1}{\kappa} \int \mathrm{d}^4 x \sqrt{-g} \delta g^{\mu\nu} R^{n-1} \left\{ n R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \frac{n-1}{R} \left(R_{;\mu;\nu} - g_{\mu\nu} R^{;\alpha}_{\;\;;\alpha} \right) + \frac{(n-1)(n-2)}{R^2} \left( R_{;\mu}R_{;\nu} - g_{\mu\nu} R^{;\alpha} R_{;\alpha} \right) \right\} \] yields the equation of motion: \[ \sum_n a_n R^{n-1} \left\{ n R_{\mu\nu} - \frac{1}{2} g_{\mu\nu} R + \frac{n-1}{R} \left(R_{;\mu;\nu} - g_{\mu\nu} R^{;\alpha}_{\;\;;\alpha} \right) + \frac{(n-1)(n-2)}{R^2} \left( R_{;\mu}R_{;\nu} - g_{\mu\nu} R^{;\alpha} R_{;\alpha} \right) \right\} = \frac{\kappa}{2} T_{\mu\nu} \]

For $n = 1, a_1 = \frac{1}{2} $ it reduces to GR. One of the most distinguish feature of this formulation is for $n=1$ (GR), two extra terms vanish. If we keep terms up to $n=2$, the equation of motion simplifies as:\[
(1 + 2 a_2 R) G_{\mu\nu} + a_2 g_{\mu\nu} R^2 + 2 a_2 \left( R_{;\mu;\nu} - g_{\mu\nu} R^{;\alpha}_{\;\;;\alpha} \right) = \kappa T_{\mu\nu}

Now, let's identify the particle contents. Define $h_{\mu\nu} = g_{\mu\nu}-\eta_{\mu\nu}$, where $\eta_{\mu\nu}$ is the Minkowski  metric. In weak field limit, we can simply take $h_{\mu\nu} \to \delta g_{\mu\nu}$ and $g_{\mu\nu} \to \eta_{\mu\nu}$. But graviton is usually defined as the inverse trace $\bar{h}_{\mu\nu} = h_{\mu\nu} - \frac{1}{2}\eta_{\mu\nu} h$, where $ h = h^\mu_\mu$. Note that, $G_{\mu\nu} = -\frac{1}{2} \Box \bar{h}_{\mu\nu} $ and $ R = \bar{h}^{\mu\nu}_{\;\;\;\; ,\mu\nu} - \frac{1}{2} \Box h $. The lowest order terms are: \[ S = \frac{1}{\kappa}\int \mathrm{d}^4 x \left\{ -\frac{1}{4} \bar{h}^{\mu\nu} \Box \bar{h}_{\mu\nu} + \frac{1}{8} h ( 2 a_2 \Box^2 + \Box ) h - \frac{a_2}{2} \bar{h}^{\mu\nu} \partial_\mu\partial_\nu \Box h \right\} \]

As we can see, there is an additional field besides the conventional graviton field, with mass $m^2_h = -\frac{1}{2 a_2}$. This field interacts with matter via Yukawa coupling: $\sqrt{-g} \simeq 1 + \delta \sqrt{-g} = 1 + \frac{1}{2} h$. Therefore, it propagates $e^{- r /m_h}$. Taking $R \ll m^2_h \ll M_\text{Pl}$, $R^2$ gravity produces a massive weakly interacting particle.

[1]: Clifford M. Will, The Confrontation between General Relativity and Experiment , Living Rev. Relativity, 9, (2006), 3,
[2]: Jose A. R. Cembranos, Dark Matter from R2 Gravity, Phys. Rev. Lett. 102, 141301 (2009)
[3]: S. Stelle, Gen. Relativ. Gravit. 9, 353 (1978)

Nov 26, 2012

Jungman's Supersymmetric Dark Matter Review Compilation

Gerard Jungmana , Marc Kamionkowskic, and Kim Griestb,
"Supersymmetric Dark Matter",
Phys.Rept. 267 (1996) 195-373, arXiv:hep-ph/9506380
[src] [pdf] [ps]

My pdf readers cannot display some of the fonts. I used his src code and compiled a new version of pdf. [pdf] or url: []

Nov 21, 2012

Dark Matter II: Standard Model

The microscopic nature of dark matter is crucial yet still unanswered. The most successful description of the microscopic world is particle physics standard model (abbr. SM). However, standard model could not offer satisfactory answer for dark matter.

Fig. 1: the standard model stew
In order to be "invisible", a dark matter candidate has to be neutral, stable and massive. Most of standard model particles and hadrons are either unstable or charged, or both. Two possibilities remains in standard model, neutrinos and neutrons. But neutrons themselves are not stable. They has to form nuclei and in addition elements together with protons and/or electrons. More broadly, we consider a class of massive astronomical compact objects (abbr. MACHOs) composed of baryonic matter.


In the simplest version of standard model, all neutrinos are left-handed massless particles. Neutrino  oscillation reveals that neutrinos have non-zero mass. The current experimental values $\left|\Delta m _{23}^2 \right| \simeq \left| \Delta m_{13}^2 \right|= 2.43^{+0.13}_{-0.13} \times 10^{-3} \; \mathrm{eV}^2$, $\Delta m_{12}^2 = 7.59^{+0.20}_{−0.21} \times 10^{−5} \; \mathrm{eV}^2$ gives a lower bound $\geq 0.05 $ eV for the heaviest neutrino. In the simplest case, neutrino masses are just $m_1 \simeq 0, m_2 \simeq 0.01 \;\mathrm{eV}, m_3 \simeq 0.05 \;\mathrm{eV} $. Of course it is possible that neutrino masses are nearly degenerate: $m_1 \sim m_2 \sim m_3 \gg \Delta m^2$. All kinematic measurement so far, however, has failed. Therefore a direct constraint of neutrino mass is particularly important.

Such upper bounds comes from cosmological probes. According to the big bang theory, neutrinos in cosmos were created in the early universe and decoupled from synthesis in the lepton epoch. The relic neutrino density is related to their masses [2]:\[\Omega_\nu = \frac{\sum_f m_{\nu_f}}{93.2 \;\mathrm{eV} H_0^2} \times (100 \;\mathrm{km s^{-1} Mpc^{-1}})^2. \] Presence of massive neutrinos affects structures in the Universe. Analysis of observational data from various cosmological probes based on $\Lambda$-CDM model, with moderate statistical error roughly gives 0.5 - 1 eV neutrino mass upper bound.

On the other hand, simple argument from Pauli exclusive principle (the so-called phase space argument) could offer a lower bound of fermionic dark matter mass. In order to be bounded by gravity, the Fermion velocity should be smaller than the escape velocity,\[ m_f \geq \left( \frac{9\pi\hbar^3}{4\sqrt{2} g M^\frac{1}{2} R^\frac{3}{2} G^{\frac{3}{2}}} \right)^{\frac{1}{4}} \] where $g \geq 1$ is the internal degrees of freedom. Carrying out the analysis to the actually dark matter dominating system, we can conclude the mass of fermionic dark matter $ \gtrsim 1 \;\mathrm{keV}$.

Moreover, if dark matter are neutrinos, they are hot matter. So far, hot dark matter models have various issues. First of all, hot dark matter tends to smooth out the matter fluctuation observed by Sloan Digital Sky Survey and other observations. It is also incompatible with the angular power spectrum of cosmic microwave background (See section 2.1). Secondly, in hot dark matter model (HDM), the structural formation is in the opposite order (i.e. large structures form early) of what we observed. Numerical simulation of galaxy formation in hot dark matter halo so far could not agree with observation.

Therefore, neutrinos are unlikely dark matter candidate.

Fig. 2: neutrino oscillation is caused by the difference between
flavor eigenstates and mass eigenstates.


MACHOs such as red dwarfs and brown dwarfs may also show high mass-to-light ratio. However, large amount of baryonic matter will breaks the baryonic density constraint put by various astronomical probes based on cosmological models. Furthermore, observation using Hubble Space Telescope shows halo red dwarfs and brown dwarfs density is about 0.25% ~ 0.67% of the halo density, hence insignificant. The EROS project searching Magellanic clouds for microlensing events caused by MACHOs also reported that MACHOs make up to less than 8% of the halo mass.

Fig. 3: Left: An artist vision of a Y-dwarf; Right: An artist vision of a red dwarf.
In summary, standard model particles are almost ruled out as dark matter candidates.

Nov 20, 2012

Dark Matter I: Introduction

I was asked to write a review on dark matter, as the final report of General Relativity (GR) class. I'd like to share my reading here.

Fig. 1: Dark matter forms a halo around a galaxy.
In astronomy and physics, dark matter (abbr. DM) is a term for non-luminous matters in the Universe. It is proposed to explain the anomalous mass-to-light ratio (abbr. M/L). Dark matter can not be observed but leaves their trace in galaxy motion among others. It is generally postulated dark matter forms a halo around a galaxy. They initiate the formation of galaxies and galaxy clusters, dominates the mass of large scale structures. At present, dark matter is part of the concordant cosmology model, the $\Lambda$-CDM model, where CDM is abbreviation for cold dark matter. It is generally believed dark matter comprises about 23% of the total mass of the Universe. Together with dark energy 72%, they dominate the present Universe.

Fig. 2: Left: The contents of the Universe according to $\Lambda$-CDM model. Only 4.6% of the Universe is ordinary matter. Dark matter comprises 23% of the Universe. 72% of the Universe, is composed of dark energy. This energy, distinct from dark matter, is responsible for the present-day acceleration of the universal expansion. Right: Timeline of the Universe according to $\Lambda$-CDM model.

However, the nature of dark matter is still unclear. It has been shown, none of the elementary particles in the standard model (abbr. SM) could be dark matter. Perhaps some suspicion remains for neutrinos, given their masses indeterminate. Therefore, the existence of dark matter naturally postulates physics beyond the particle physics standard model. In fact, for various theoretical speculations, particle physicists tend to believe standard model is merely an effective theory of an more advanced theory. The demands of dark matter, whether truly relevant or not, has been as a strong motivation for extension of standard model in particle physics. Of course not all new particles are dark matter candidates. It is generally plausible to assume dark matter are some weakly interacting massive particles (abbr. WIMPs), raising from some TeV scale new physics. The favored WIMP candidates include neutralinos from supersymmetric (abbr. SUSY) models and the first Kaluza-Klein excitations from universal extra dimension (abbr. UED) models among others.
Fig. 3: The standard model elementary particle zoo
On the other hand, dark matter may also merely be a misleading paradigm. The existence of dark matter, inferred from large gravitational mass to luminosity ratio, is based on the assumption that general relativity with its flat space-time approximation Newtonian gravity holds up to cosmological scale. Milgrom and others has shown that it is possible to modified Newtonian gravity to explain the large mass-to-light ratio in galaxies. If confirmed, however, mankind's understanding of the cosmos will be completely overthrown.

Just as Richard P. Feynman had said, "Experiment is the sole judge of scientific truth". Some of the proposals lies within the current experimental and observational scope. People have conducted various experiments and observations to detect possible dark matter directly or indirectly, from space telescope to ground based detector and colliders. Current results, mainly null results with some suspicious signals, have excluded a large class of theories.

In this paper, I review the physics of dark matter. The aim is to give a pedagogical introduction to general interested readers, like myself. In the next few posts, I will introduce the evidences and motivations for dark matter; then review the candidates and their properties. Focus will be put on popular models such as WIMPs, gravitino, axions and sterile neutrinos. After that, I will talk about the undergoing experiments and observations in astronomy and physics. Their results and the constraints on models will be discussed. In the end, I will visit alternative theories and other speculations.

Nov 17, 2012

The Top 10 Supercomputers

parameters of top 10 supercomputers (Nov. 2012) data source:
comparison of top 10 supercomputers (Nov. 2012) data source:
comparison of efficiency of top 10 supercomputers
the power law of performance

Nov 3, 2012

On the Pronunciation of the Name of Greek Letters

There are, generally speaking, two main uses of Greek letters in English, the name of honor societies and academia. As a graduate student, I interact with both people from academia and collage students. I found there are roughly three ways for English speakers to pronounce the name of Greek letters
1. as the Greek pronunciation, very common in academia, though most people may not follow it exactly; 2. as the English name, very common in English speakers; 3. as the name of the corresponding English letter, common for collage students who is not STEM majors.

Either way is perfectly okay. But sometimes, confusion arises when one mix these three ways. The most notorious example is the letter xi and psi. Some people call both /sai/. Moreover, fancy fonts of English letters may be confused with Greek letters.

Oct 18, 2012

Coulomb's Law in $d$-Dimension

In 3+1 dimension, Coulomb's law and Newton's law of gravity takes the form of distance inverse-squared,
\[ f = \frac{1}{r^2} \] with proper definition of the source and distance. What is Coulomb's law and Newton's law look like in high dimensions?

To answer this question, we have to make assumptions. We assume the Lagrangian stays the same form in $d+1$ dimension. It means, the Maxwell equations hold; or equivalently, Poisson equation holds \[ \nabla^2 \varphi(\mathbf{x}) = 0. \]
Solving $\varphi$ in free space will produce the potential hence the force. By doing Fourier transform, \[ \varphi(\mathbf{x}) = \int \frac{\mathrm{d}^d k}{(2\pi)^d} \frac{ e^{i \mathbf{k}\cdot \mathbf{x}}}{k^2}. \]

Solving $\varphi(r)$

\[\varphi = \frac{1}{(2\pi)^d}\int \mathrm{d} k \; k^{d-3} \mathrm{d}^{d-1}\Omega {e^{i k r \cos \theta_1}} \], where $\mathrm{d}^{d-1} \Omega $ is the $(d-1)$-D angular element. Now this integral involves the integral of one azimuthal angle $\theta$. We can parametrize the coordinate in $d$-D spherical coordinate as: \[ x_1 = r \cos\theta_1; x_2 = r \sin\theta_1 \cos\theta_2; \cdots; x_d = r \sin\theta_1 \sin\theta_2\cdots \sin\theta_{d-2}\cos\phi; \] Then the surface element becomes: $\mathrm{d}^{d-1} \Omega = \sin^{d-2}\theta_1 \sin^{d-3}\theta_2 \cdots \sin \theta_{d-2} \mathrm{d}\theta_1 \mathrm{d}\theta_2 \cdots \mathrm{d}\theta_{d-2} \mathrm{d} \phi $. The integral over angles except $\theta_1$ is just the surface area of a $(d-2)$-D hypersphere (See appendix for a derivation) \[ S_{d-2} = \frac{2 \pi^{\frac{d-1}{2}}}{\Gamma\left( \frac{d-1}{2} \right) } \]

So $\varphi = \frac{ S_{d-2}}{ (2\pi)^d r^{d-2}} I_{d} $, where \[ I_d = \int_0^\infty \mathrm{d}\xi \; \int_0 ^\pi \mathrm{d}\theta \; \xi^{d-3} \sin^{d-2}\theta \exp\left[ i \xi \cos\theta \right]. \]
It's tempted to do the $\xi$ integral first, because it gives gamma function and leaves the rest a integral over $\tan\theta$: $ \int_0^{\pi/2} \mathrm{d}\theta \tan^{d-2}\theta + (-1)^{d-2}\int^{\pi/2}_\pi \mathrm{d}\theta \tan^{d-2}\theta$. The problem is that integral of $\tan$ function at $\pi/2$ is singular . We can do the $\theta$ integral first, which yields (using mathematica):\[ I_d = \int_0^\infty \mathrm{d}\xi \; \sqrt{\pi} \Gamma\left( \frac{d-1}{2} \right) \frac{{ }_0F_1\left( \frac{d}{2}, -\frac{\xi^2}{4} \right)}{\Gamma\left( \frac{d}{2} \right)} \xi^{d-3} = 2^{d-3} \sqrt{\pi} \Gamma\left( \frac{d-2}{2} \right) \Gamma\left( \frac{d-1}{2} \right). \]
Therefore, \[ \varphi = \frac{ S_{d-2}}{ (2\pi)^d r^{d-2}} 2^{d-3} \sqrt{\pi} \Gamma\left( \frac{d-2}{2} \right) \Gamma\left( \frac{d-1}{2} \right) = \frac{\Gamma\left( \frac{d-2}{2}\right)}{4 \pi^\frac{d}{2} }\frac{1}{r^{d-2}} \]
Coulomb potential in higher dimensions

Gauss Law

There is a much easier method to solve this problem. We note that Gauss theorem (in mathematics) hence Gauss law (in physics) still holds. \[ E(r) \cdot S_{d-1} = 1 \] where $S_{d-1}$ is the area of a $d-1$ D hypersphere. So we get Coulomb's law in $d+1$ dimension as:\[ f = \frac{\Gamma\left( \frac{d}{2} \right) }{2 \pi^\frac{d}{2}} \frac{1}{r^{d-1}}. \]

It can be checked, $-\frac{\partial}{\partial r} \varphi(r) = E(r)$, Just as we expected. Of course, the direct integration can be used in where Gauss law does not hold.

Coulomb's law for massive boson exchange

Another interesting result is the Coulomb's law for classical theories with massive intermediate boson in higher dimentions. The Poisson equation becomes \[ (\nabla^2 - m^2) \varphi(\mathbf{x}) = 0. \]
By doing Fourier transform, \[ \varphi(\mathbf{x}) = \int \frac{\mathrm{d}^d k}{(2\pi)^d} \frac{ e^{i \mathbf{k}\cdot \mathbf{x}}}{k^2+m^2}. \] Apply the same technique again (except the Gauss Law), \[
\varphi(r) = \frac{ (m r)^{\frac{d}{2}-1} K_{\frac{d}{2}-1} (mr)}{(2 \pi)^\frac{d}{2}}\frac{1}{ r^{d-2}} \] where $K_n(x)$ is the Bessel function of the second kind.
Comparison of field potential of massless and massive boson exchange in higher dimensions
Comparison of field potential of massless and massive boson exchange in higher dimensions at large $r$

Appendix: the surface area of a $(d-1)$-D hypersphere

Consider the following Gaussian integeral: \[ \int \mathrm{d}^n x \exp\left( - \mathbf{x}^2 \right) = \left( \int \mathrm{d}x \exp\left[ - x^2 \right] \right)^n = \pi ^{\frac{n}{2}} \]
The left hand side can be written as $ \int \mathrm{d}r \; r^{n-1} \exp\left[ -r^2 \right] S_{n-1}$. So $ \frac{1}{2} \Gamma\left(\frac{n}{2}\right) S_{n-1} = \pi^\frac{n}{2} $. \[ S_{n-1} = \frac{2 \pi^\frac{n}{2} }{\Gamma\left(\frac{n}{2}\right)}. \]

See also:

update, March 21, 2014:

  • I was solving the Green's function with free space boundary condition. The Poisson equation should have been    

\[ \nabla^2 \varphi(\mathbf{x}) = \delta(\mathbf{x}) \]

  • This is not a the only generalization. It may not even be the natural generalization, from the point view the Newtonian approximation in general relativity. In GR, one should write down the Einstein equation in $d+1$ D and do the linearization there, as our commentator explained. (That means one fix G, which is not always appreciated.)

Oct 4, 2012

ubuntu/linux 无法启动问题

今天开机启动linux, 又一次遇到了无法启动进入busybox的问题. 上次是在中国, 搜寻多次未果, 最后用liveCD导出了数据重新安装的linux. 这次在Sasha的帮助下解决了问题.
问题如下: 开机启动, grub 选择系统(ubuntu 9.10+ vista)登录. 一切顺利. 然后ubuntu的logo还能闪亮. 到这里为止一切正常. 接下来就不能继续了. 大约过了20s, 进入了busybox 如下:
Gave up waiting for boot device. Common problems:
-Boot args (cat /proc/cmdline)
-check rootdelay
-check root
-Missing modules(cat /proc/modules; ls /dev)
ALERT! /dev/disk/by-uuid/d286688a-8587-4977-99b9-1414cf09c374 does not exist. Dropping to a shell!

BusyBox v1.1.3 (Debian 1:1.1.3-5ubuntu12) Built-in shell (ash)
Enter 'help' for build in commands.

这个界面里可以键入命令, 但是很有限. 不能成为管理员, 甚至无法重启(用ctrl+alt+delete)可以. 也无法访问自己的/home/

解决的办法: Sasha使用USB制作的liveCD (在BIOS里选择USB启动)进入系统, 挂载原磁盘, 然后找到grub.d/ 文件夹里的 grub.cfg 文件. 把
linux /boot/vmlinuz-2.6.31-17-generic root=UUID=a026ae5a-4c0b-42cd-8b46-b57bfb433ac7 ro quiet splash
linux /boot/vmlinuz-2.6.31-17-generic root=/dev/sda1 ro quiet splash
注意是要把root=UUID=(UUID) 替换成 root=/dev/sda(x). 其中sda(x)是你的根目录所在的磁盘.

这个问题是, linux 内核似乎有时候无法按照 UUID 识别磁盘. 所以改成用绝对文件路径来查找.

参考: 1.

c++ notes: I/O

  1. 记得包含 <iomanip> <fstream>
  2. 少用 std::cin >> ...; 处理字符和字符串. 如果使用记得加一个 std::cin.ignore(); 来吃掉回车. std::cin.sync(); 来清空buffer
  3. 使用std::cin.getline( const char* str, size_t n, char ch );   /*对于文件流也一样适用*/ ch设置为 '\n'
  4. 对于使用string 类型的操作, 使用相应的版本: getline( iostream std::cin, string str);
  5. 使用GNU C的系统库函数来实现系统相关的操作, 比如文件操作. 比较有用的一个函数是 int scandir(const char* directory, struct ***dirent namelist, int (*selector)(void * ), alphasort) ; 来扫描文件. 其中 返回文件个数, directory 是目标文件夹(路径), namelist 存储扫描得到的文件(夹) 一般定义为: struct ** storagelist 并使用 &storagelist 作为参量. 不用担心其大小. 因为它只是一个指针, 指向的部分是由malloc 分配的. 使用完之后最好加以free.; selector 是一个自定义的比较函数. 返回值为非零的文件将会被选择; alphasort 也本应该是一个自定义的文件, 表示返回文件的排列方法. 不过系统提供了几个函数. alphasort就是一个常用的.
  6. 使用 定义在 <cstdlib.h> 中的库函数来读取环境变量: system(const char* str);来执行shell命令. getenv(const char* str) 来获得环境变量.
  7. 使用系统预编译宏: __TIME__ __DATE__ ; __LINE__ 等等

Sep 7, 2012

Faster Mathematica II: List Manipulation

The most important datatype in math is list. Lists in Mathematica are like set in mathematics. List is also used to hold data. In the case of large amount of data in input/output or intermediate stage, fast list operations certainly would speedup the code. One surprising fact is, seemly the same list operations may have different performance.

1. List Construction

1-0 Table is fast when creating block list

The most simple way to construct a list is use Table. Table[] is the suitable choice in generic cases. But making use of knowledge of the list, you can get better performance.
Table[], Array[], and function apply

1-1 use build-in functions

Some build-in functions support generation of a list. They are usually faster that explicit call them in a list constructor like Table[].  
Build-in random number generator is way faster.

1-2 use Reap-Sow, instead of Insert, Append, AppendTo, Prepend or PrependTo

1-3 Use NestList for recursable list  

Recursable list means you can get the n+1-th from the n-th. 

2. List Traversal

2-1. Listable attribute

Many function as Listable attribute. This means, when hitting a list, the function automatically hit on each element of the list. And the result will be the list of all hits. Using Listable functions is the fastest  list traversal. 

One can also Map (/@), Apply (@@, @@@), Thread, MapThread the function (head) to a list if Listable attribute is not available. 

Note that some binary operations have been defined between lists and between list and scalar. This build-in operations between lists are fast and can be in parallel. 

2-2. iterate a list

If one has to explicitly traverse a list,  Mathematica allows iterating a list directly. It is faster than using iterators. The iteration concept is not limited to list. In Do and other iterating cases, list can also be iterated directly. 

 2-3. use build-in list manipulation

Mathematica has defined many build-in list operations. Using these build-in operations are generally faster.