Dec 1, 2012

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.


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