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 We know that at high energy $\sim T_\text{Pl}$, general relativity (GR) will be superseded by quantum gravity.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.